Удовлетворение ограничений и программирование в

ÓÄÊ 519.68 Óäîâëåòâîðåíèå îãðàíè÷åíèé è ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ1 Î.À. Ùåðáèíà University of Vienna, Vienna 1090, Austria. E-mail: oleg.shcherbina@univie.ac.at Àííîòàöèÿ. Â ñòàòüå ïðåäñòàâëåí îáçîð îñíîâíûõ íàïðàâëåíèé óäîâëåòâîðåíèÿ îãðàíè÷åíèé (ÓÎ) è ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ. Öåëüþ çàäà÷è ÓÎ ÿâëÿåòñÿ íàõîæäåíèå çíà÷åíèé ïåðåìåííûõ, óäîâëåòâîðÿþùèõ îïðåäåëåííûì îãðàíè÷åíèÿì. Â èñêóññòâåííîì èíòåëëåêòå ïàðàäèãìà ÓÎ ïðèçíàíà â êà÷åñòâå óäîáíîãî è ýôôåêòèâíîãî ñïîñîáà ìîäåëèðîâàíèÿ è ðåøåíèÿ ìíîãèõ ïðèêëàäíûõ êîìáèíàòîðíûõ çàäà÷. Âàæíàÿ çàäà÷à îöåíêè êîíúþíêòèâíûõ çàïðîñîâ â òåîðèè áàç äàííûõ ìîæåò ðàññìàòðèâàòüñÿ êàê çàäà÷à ÓÎ. Êðîìå òîãî, çàäà÷è ÓÎ ïðèâëåêàþò áîëüøîå âíèìàíèå â òåîðèè ñëîæíîñòè, òàê êàê ðàçëè÷íûå âåðñèè çàäà÷ ÓÎ íàõîäÿòñÿ â ñåðåäèíå ìíîãèõ ñòàíäàðòíûõ êëàññîâ ñëîæíîñòè, è ïîñêîëüêó ó íèõ èìååòñÿ òåíäåíöèÿ íå èìåòü ïðîìåæóòî÷íîé ñëîæíîñòè, òî åñòü îíè áûâàþò ëèáî ëåãêî ðàçðåøèìûìè, ëèáî ïîëíûìè äëÿ ñòàíäàðòíûõ êëàññîâ ñëîæíîñòè. Ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ ÿâëÿåòñÿ ïàðàäèãìîé ïðîãðàììèðîâàíèÿ äëÿ äåêëàðàòèâíîãî îïèñàíèÿ è ýôôåêòèâíîãî ðåøåíèÿ êîìáèíàòîðíûõ çàäà÷, òåñíî ñâÿçàííîé ñ òåîðèåé ÓÎ. Ðàññìîòðåíû ïðèìåðû èñïîëüçîâàíèÿ ìîäåëåé ÓÎ è êîíêðåòíûå ïðèëîæåíèÿ. 1. Ââåäåíèå 1.1. Óäîâëåòâîðåíèå îãðàíè÷åíèé Èñïîëüçîâàíèå ïîäõîäîâ è àëãîðèòìîâ èñêóññòâåííîãî èíòåëëåêòà (ÈÈ) ïîçâîëÿåò ðåøàòü ìíîãèå ïðèêëàäíûå çàäà÷è, òàêèå, êàê çàäà÷è òåîðèè ðàñïèñàíèé [45], çàäà÷è ïðîåêòèðîâàíèÿ ýêñïåðòíûõ ñèñòåì è ñèñòåì ïîääåðæêè ïðèíÿòèÿ ðåøåíèé [13], äîêàçàòåëüñòâî òåîðåì, çàäà÷è òåñòèðîâàíèÿ ýëåêòðîííûõ ñõåì, îáðàáîòêà èçîáðàæåíèé. Îäíîé èç âàæíûõ çàäà÷ ÈÈ ÿâëÿåòñÿ çàäà÷à óäîâëåòâîðåíèÿ îãðàíè÷åíèé (ÓÎ) (constraint satisfaction problem) [142], [171], [364]. Òåîðèÿ ÓÎ ïðåäëàãàåò óäîáíûé àïïàðàò è ïðîñòóþ ôîðìàëüíóþ ñõåìó äëÿ ïðåäñòàâëåíèÿ è ðåøåíèÿ êîìáèíàòîðíûõ çàäà÷ èñêóññòâåííîãî èíòåëëåêòà. Öåëüþ ðåøåíèÿ çàäà÷è ÓÎ ÿâëÿåòñÿ íàõîæäåíèå çíà÷åíèé ïåðåìåííûõ, óäîâëåòâîðÿþùèõ îïðåäåëåííûì îãðàíè÷åíèÿì. Ïàðàäèãìà ÓÎ ÿâëÿåòñÿ îáîáùåíèåì ïðîïîçèöèîíàëüíîé ëîãèêè, â êîòîðîé ïåðåìåííûì ìîãóò áûòü ïðèñâîåíû çíà÷åíèÿ 1 Ðàáîòà N13 âûïîëíåíà ïðè ôèíàíñîâîé ïîääåðæêå àâñòðèéñêîãî ôîíäà FWF, ãðàíò No. P20900- 1 èç ìíîæåñòâà ëþáîãî ÷èñëà ïåðåìåííûõ, à íå òîëüêî ¾èñòèíà¿ è ¾ëîæü¿ [138]. Ïðîáëåìà ñóùåñòâîâàíèÿ ðåøåíèé çàäà÷è ÓÎ ÿâëÿåòñÿ NP-ïîëíîé. Â èñêóññòâåííîì èíòåëëåêòå ïàðàäèãìà ÓÎ ïðèçíàíà â êà÷åñòâå óäîáíîãî è ýôôåêòèâíîãî ñïîñîáà ìîäåëèðîâàíèÿ è ðåøåíèÿ ìíîãèõ ïðèêëàäíûõ çàäà÷, òàêèõ êàê ïëàíèðîâàíèå [240] è êàëåíäàðíîå ïëàíèðîâàíèå [343], [166], [74], [49], [308], [321], [357], çàäà÷è íàçíà÷åíèÿ ÷àñòîòû [157], îáðàáîòêà èçîáðàæåíèé [289], ïðè òåñòèðîâàíèè ñâåðõáîëüøèõ èíòåãðàëüíûõ ñõåì ÑÁÈÑ (VLSI) [211], àíàëèç ÿçûêîâ ïðîãðàììèðîâàíèÿ [294] è ïîíèìàíèÿ åñòåñòâåííîãî ÿçûêà [31]. Â òåîðèè áàç äàííûõ ïîêàçàíî, ÷òî êëþ÷åâàÿ çàäà÷à îöåíêè êîíúþíêòèâíûõ çàïðîñîâ ìîæåò ðàññìàòðèâàòüñÿ êàê çàäà÷à ÓÎ [196], [247]. Êðîìå òîãî, íåêîòîðûå öåíòðàëüíûå çàäà÷è êîìáèíàòîðíîé îïòèìèçàöèè ìîãóò áûòü ïðåäñòàâëåíû â âèäå çàäà÷ ÓÎ [125], [161], [225], [241]. Îïûò ïðèìåíåíèÿ íåäîîïðåäåëåííûõ ìîäåëåé ê èññëåäîâàíèÿì ýêîíîìèêè Ðîññèè ÿâëÿåòñÿ âïîëíå îáíàäåæèâàþùèì [9]. Ïîìèìî ýòîãî, çàäà÷è ÓÎ ïðèâëåêàþò áîëüøîå âíèìàíèå â òåîðèè ñëîæíîñòè, òàê êàê ðàçëè÷íûå âåðñèè çàäà÷ ÓÎ íàõîäÿòñÿ â ñåðåäèíå ìíîãèõ ñòàíäàðòíûõ êëàññîâ ñëîæíîñòè, è òàê êàê ó íèõ èìååòñÿ òåíäåíöèÿ íå èìåòü ïðîìåæóòî÷íîé ñëîæíîñòè, òî åñòü îíè áûâàþò ëèáî ëåãêî ðàçðåøèìûìè, ëèáî ïîëíûìè äëÿ ñòàíäàðòíûõ êëàññîâ ñëîæíîñòè [76], [77], [89], [94], [95], [125], [161], [234], [244], [314]. Ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ ÿâëÿåòñÿ ïàðàäèãìîé ïðîãðàììèðîâàíèÿ äëÿ äåêëàðàòèâíîãî îïèñàíèÿ è ýôôåêòèâíîãî ðåøåíèÿ êîìáèíàòîðíûõ çàäà÷, òåñíî ñâÿçàííîé ñ òåîðèåé ÓÎ (ñì. [19], [266], [322], [377]). ÓÎ ÿâëÿåòñÿ ìîùíîé ïàðàäèãìîé, ïîçâîëÿþùåé àäåêâàòíî ìîäåëèðîâàòü ìíîãèå êîìáèíàòîðíûå çàäà÷è [125], [263], [266], [364]. Ìíîãèå êëàññè÷åñêèå êîìáèíàòîðíûå çàäà÷è, òàêèå êàê èçâåñòíàÿ òåîðåìà Ôåðìà, çàäà÷à âûïîëíèìîñòè (SAT) èç ïðîïîçèöèîíàëüíîé ëîãèêè, çàäà÷à ðàñêðàñêè ãðàôà è çàäà÷à èçîìîðôèçìà ãðàôîâ èç òåîðèè ãðàôîâ, çàäà÷à BANDWIDTH èç èññëåäîâàíèÿ îïåðàöèé ìîãóò ôîðìóëèðîâàòüñÿ â âèäå çàäà÷ ÓÎ (ÇÓÎ). Îñòàíîâèìñÿ ïîäðîáíåå íà îäíîé èç äàâíî ïîñòàâëåííûõ çàäà÷ â ìàòåìàòèêå çàäà÷å î ðàñêðàñêå ãðàôà (ðàñêðàñêà êàðòû ÷àñòíûé ñëó÷àé ýòîé çàäà÷è). Ñîãëàñíî äàííûì, ïðèâåäåííûì â (Biggs et al., 1986) [67], ãèïîòåçó î ÷åòûðåõ öâåòàõ (â ñîîòâåòñòâèè ñ êîòîðîé ëþáîé ïëàíàðíûé ãðàô ìîæíî ðàñêðàñèòü ñ ïîìîùüþ ÷åòûðåõ èëè ìåíüøåãî êîëè÷åñòâà öâåòîâ) âïåðâûå âûäâèíóë Ôðåíñèñ Ãóòðè â 1852 ãîäó. Ýòà çàäà÷à íå ïîääàâàëàñü ðåøåíèþ äî òåõ ïîð, ïîêà äîêàçàòåëüñòâî íå áûëî ïîëó÷åíî ñ ïîìîùüþ êîìïüþòåðà â (Appel & Haken, 1977) [33]. Ôîðìóëèðîâêà çàäà÷è î ðàñêðàñêå â âèäå çàäà÷è ÓÎ ñòàâèò â ñîîòâåòñòâèå âåðøèíàì ðàñêðàøèâàåìîãî ãðàôà ïåðåìåííûå, âîçìîæíûå öâåòà ïðåäñòàâëÿþò ñîáîé äîìåíû (îáëàñòè îïðåäåëåíèÿ) ïåðåìåííûõ, à îãðàíè÷åíèÿ-íåðàâåíñòâà ìåæäó ñìåæíûìè âåðøèíàìè ÿâëÿþòñÿ îãðàíè÷åíèÿìè çàäà÷è. Áîëåå ñëîæíûì, íî è áîëåå ðåàëèñòè÷åñêèì ïðèìåðîì ðàñïðåäåëåíèÿ ðåñóðñîâ ÿâëÿåòñÿ çàäà÷à ðàçìåùåíèÿ âûõîäîâ â àýðîïîðòó. Îáû÷íî íåîáõîäèìî ðàññìàòðèâàòü êàê ôèçè÷åñêèå îãðàíè÷åíèÿ (íàïðèìåð, òåëåòðàïû îïðåäåëåííîãî âèäà ìîãóò ïîäõîäèòü òîëüêî äëÿ îïðåäåëåííûõ òèïîâ ñàìîëåòîâ), òàê è ïîëüçîâàòåëüñêèå ïðåäïî÷òåíèÿ (íàïðèìåð, ðàçëè÷íûå àâèàëèíèè ïðåäïî÷èòàþò ïàðêîâêó â 2 îïðåäåëåííûõ ÷àñòÿõ àýðîïîðòà). Îïèñàííàÿ çàäà÷à ìîæåò áûòü ïðåäñòàâëåíà â âèäå çàäà÷è ÓÎ. Ñïåöèàëüíûå êëàññû çàäà÷ ÓÎ ðàññìàòðèâàëèñü íà ïðîòÿæåíèè âñåé èñòîðèè ðàçâèòèÿ èíôîðìàòèêè. Îäíèì èç ñàìûõ ïåðâûõ ïðèìåðîâ ðåøåíèÿ òàêèõ çàäà÷ áûëà ñèñòåìà Sketchpad (Sutherland, 1963) [356], êîòîðàÿ ðåøàëà ãåîìåòðè÷åñêèå îãðàíè÷åíèÿ íà ÷åðòåæàõ è áûëà ïðåäøåñòâåííèöåé ñîâðåìåííûõ ïðîãðàìì ðèñîâàíèÿ è èíñòðóìåíòàëüíûõ ñðåäñòâ ÑÀÏÐ. Ñïèñîê ïðèëîæåíèé ÓÎ è ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ äîñòàòî÷íî îáøèðåí. Çäåñü è âåðîÿòíîñòíûå ñåòè (belief maintenance) ((de Kleer, 1989 ) [130]; (Dechter, 1987) [131]; (Dechter & Dechter, 1988) [132]; (Doyle, 1979) [152]; (McDermott, 1991) [272]), êðîññâîðäû ((Nadel, 1989) [293]), òåìïîðàëüíûé âûâîä ((Allen, 1984) [30]; (Rit, 1986) [319]; (Tsang, 1987) [363]), ãðàôîâûå çàäà÷è ((Bruynooghe, 1985) [87]; (Fowler & Haralick, 1983) [165]; (McGregor, 1979) [273]; (Ullman, 1976) [366]), çàäà÷è ïðîïîçèöèîíàëüíîé âûïîëíèìîñòè ((Ginsberg & McAllester, 1994) [190]; (Zabih & McAllester, 1988) [387]), ëîãè÷åñêîå ïðîãðàììèðîâàíèå ((Borning et al., 1989) [79]; (van Hentenryck, 1989) [368]), èìèòàöèîííîå ìîäåëèðîâàíèå ñèñòåì ((Kuipers, 1994) [254]; (Miguel & Shen, 1998) [282]; (Shen & Leitch, 1993) [341]), àíàëèç ýëåêòðîííûõ ñõåì ((Stallman & Sussman, 1977) [355]), îáðàáîòêà åñòåñòâåííîãî ÿçûêà [277], ïðîåêòèðîâàíèå ýêñïåðòíûõ ñèñòåì (Bartak, 1997) [51]. Â ïîñëåäíåå âðåìÿ ìåòîäû ãðàôîâîé äåêîìïîçèöèè è àëãîðèòìû ðàñïðîñòðàíåíèÿ îãðàíè÷åíèé áûëè òàêæå èñïîëüçîâàíû è äëÿ çàäà÷ ÓÎ ñ íåïðåðûâíûìè ïåðåìåííûìè [73], [117], [331] è äëÿ çàäà÷ ñ îãðàíè÷åíèÿìè â âèäå äèôôåðåíöèàëüíûõ óðàâíåíèé [148] . Â ëèòåðàòóðå ìîæíî íàéòè íåñêîëüêî õîðîøèõ îáçîðîâ ñ îïèñàíèåì ìåòîäîâ ðåøåíèÿ çàäà÷ ÓÎ, âêëþ÷àÿ (Kumar, 1992) [255], (Dechter & Frost, 1999) [138], (Bartak, 2001) [46], [278], [284], à òàêæå ñòàòüè â ýíöèêëîïåäè÷åñêèõ ñáîðíèêàõ (Dechter, 1992) [135], [214] è (Mackworth, 1992) [263]. Ê ñîæàëåíèþ îáçîðîâ íà ðóññêîì ÿçûêå, ïîñâÿùåííûõ ïðîáëåìàì òåîðèè ÓÎ, â íàñòîÿùåå âðåìÿ íåò, õîòÿ èìåþòñÿ ïóáëèêàöèè, ïîñâÿùåííûå îòäåëüíûì àñïåêòàì ÓÎ [1], [5], [6], [8], [7], [10], [11], [14], [15], [16], [17], [18], [20], [22], [24]. Ðàçóìååòñÿ, â ðàìêàõ äàííîé îáçîðíîé ñòàòüè íåâîçìîæíî ïîäðîáíî îïèñàòü âñå àñïåêòû è íàïðàâëåíèÿ òåîðèè óäîâëåòâîðåíèÿ îãðàíè÷åíèé è ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ, ïîýòîìó áîëåå ïîëíóþ èíôîðìàöèþ ìîæíî íàéòè â ìîíîãðàôèÿõ [34], [142], [176], [266], [324], [368], [371], [374], [376]. Ê ÷èñëó ïåðåâîäíûõ ìîíîãðàôèé, â êîòîðûõ îñâåùàþòñÿ âîïðîñû ÓÎ ìîæíî îòíåñòè êíèãó Ðàññåë Ñ., Íîðâèã Ï. Èñêóññòâåííûé èíòåëëåêò: ñîâðåìåííûé ïîäõîä. 2å èçä.: Ïåð. ñ àíãë. Ì.: Âèëüÿìñ, 2006. 1408 ñ. [12]. Îáçîð îñíîâíûõ íàïðàâëåíèé ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ äî 2000 ã. ñäåëàí â ðàáîòå [19]. Íàñòîÿùàÿ ñòàòüÿ ïðèçâàíà çàïîëíèòü óêàçàííûé ïðîáåë è äàòü íàøåìó ÷èòàòåëþ ïðåäñòàâëåíèå îá îñíîâíûõ íàïðàâëåíèÿõ òåîðèè ÓÎ è ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ. Êîñíåìñÿ âíà÷àëå òåðìèíîëîãèè è èñòîðèè âîçíèêíîâåíèÿ ìåòîäîâ ÓÎ. Montanari [289] ïðåäëîæèë èñïîëüçîâàòü ìîäåëè ÓÎ äëÿ îïèñàíèÿ ðÿäà êîìáèíàòîð3 íûõ çàäà÷, âîçíèêàþùèõ ïðè êîìïüþòåðíîé îáðàáîòêå èçîáðàæåíèé, è íàçâàë ýòè çàäà÷è ÓÎ ¾ñåòÿìè îãðàíè÷åíèé ¿ (networks of constraints). Ýòî ñâÿçàíî ñ òåì, ÷òî ñèñòåìó îãðàíè÷åíèé ìîæíî ïðåäñòàâèòü â âèäå íåîðèåíòèðîâàííîãî ãðàôà ñ âåðøèíàìè-ïåðåìåííûìè è ðåáðàìè, ñîîòâåòñòâóþùèìè îãðàíè÷åíèÿì ìåæäó äâóìÿ ïåðåìåííûìè. Ïî ìíåíèþ Dechter [142], ñåòè îãðàíè÷åíèé ÿâëÿþòñÿ ãðàôîâûì ïðåäñòàâëåíèåì, èñïîëüçóåìûì äëÿ ïîèñêà ñòðàòåãèé ðåøåíèÿ çàäà÷ ÓÎ. Äîñòàòî÷íî áûñòðî ýòîò ïîäõîä áûë èñïîëüçîâàí äëÿ ðåøåíèÿ ãîðàçäî áîëåå øèðîêîãî êëàññà çàäà÷. Â ëèòåðàòóðå èñïîëüçóþòñÿ îáà ýòèõ òåðìèíà ñåòè îãðàíè÷åíèé è çàäà÷è óäîâëåòâîðåíèÿ îãðàíè÷åíèé (Mackworth [263]). 1.2. Î ñâÿçè çàäà÷ ÓÎ ñ äðóãèìè ìàòåìàòè÷åñêèìè ìåòîäàìè Íåêîòîðûå èç ïðèêëàäíûõ çàäà÷, ðåøàåìûõ ñ ïîìîùüþ ìåòîäîâ ÓÎ, ìîãóò áûòü òàêæå ðåøåíû ñ ïîìîùüþ ìåòîäîâ èññëåäîâàíèÿ îïåðàöèé: ÓÎ ïðåäëàãàåò îáùíîñòü, ãèáêîñòü, ñðåäó ìîäåëèðîâàíèÿ âûñîêîãî óðîâíÿ, óïðàâëåíèå ïîèñêîì, êîìïàêòíîñòü ïðåäñòàâëåíèÿ çàäà÷è, ðàñïðîñòðàíåíèå îãðàíè÷åíèé, áûñòðûå ìåòîäû íàõîæäåíèÿ ðåøåíèÿ, â òî âðåìÿ êàê èññëåäîâàíèå îïåðàöèé èíîãäà ÿâëÿåòñÿ åäèíñòâåííûì ñïîñîáîì ïîèñêà îïòèìàëüíîãî ðåøåíèÿ [194]. Ñëåäîâàòåëüíî, ðàçóìíî ïîïûòàòüñÿ âçÿòü ëó÷øåå ó êàæäîé èç óêàçàííûõ òåõíîëîãèé, ïóòåì ñîçäàíèÿ ãèáðèäíûõ ñèñòåì, èñïîëüçóþùèõ îáå òåõíîëîãèè äëÿ ðåøåíèÿ çàäà÷è [212], [213]. Ýòîò ïîäõîä óæå èñïîëüçîâàí â íåêîòîðûõ ñèñòåìàõ, òàêèõ êàê 2LP [270], [271], à òàêæå â ðàáîòå [127], â êîòîðîé ýòîò ãèáðèäíûé ïîäõîä èñïîëüçîâàí äëÿ ðåøåíèÿ êîíêðåòíûõ òåëåêîììóíèêàöèîííûõ çàäà÷. Çàìåòèì, ÷òî ðåøåíèå îïòèìèçàöèîííîé çàäà÷è ìîæåò áûòü ñâåäåíî ê ðåøåíèþ ïîñëåäîâàòåëüíîñòè çàäà÷ ÓÎ ñëåäóþùèì îáðàçîì. Íàõîäèòñÿ äîïóñòèìîå ðåøåíèå, ïîñëå ÷åãî äîáàâëÿåòñÿ îãðàíè÷åíèå, ñîîòâåòñòâóþùåå öåëåâîé ôóíêöèè, êîòîðîå âûðàæàåò óñëîâèå, ÷òî çíà÷åíèå öåëåâîé ôóíêöèè äîëæíî áûòü ëó÷øå, ÷åì äëÿ ýòîãî ðåøåíèÿ. Ïîñëåäîâàòåëüíûå êîððåêòèðîâêè ýòîãî ïîðîãîâîãî çíà÷åíèÿ, ïðîèçâîäèìûå äî òåõ ïîð, ïîêà çàäà÷à íå ñòàíåò íåðàçðåøèìîé, ïîçâîëÿþò íàéòè îïòèìàëüíîå ðåøåíèå. 1.2.1. Çàäà÷è âûïîëíèìîñòè è óäîâëåòâîðåíèå îãðàíè÷åíèé Çàäà÷à ÓÎ ìîæåò áûòü ïðåîáðàçîâàíà â çàäà÷ó âûïîëíèìîñòè SAT íåñêîëüêèìè ñïîñîáàìè (Walsh, 2000) [382] . Îäíèì èç íèõ ÿâëÿåòñÿ ïðÿìîå îòîáðàæåíèå. Äëÿ äàííîé ÇÓÎ P =< V, D, C >2 ïîñòðîèì çàäà÷ó âûïîëíèìîñòè SAT ñëåäóþùèì îáðàçîì. Ââåäåì |V | · |D| ïåðåìåííûõ xi,j . Ïåðåìåííûå xi,j ïðèíèìàþò çíà÷åíèå ¾èñòèíà¿ òîãäà è òîëüêî òîãäà, êîãäà ïåðåìåííîé Vi ïðèñâîåíî çíà÷åíèå j . Äëÿ êàæäîé ïåðåìåííîé äîáàâëÿåì êëàóçû (äèçúþíêòû) x̄i,j ∨ x̄i,k äëÿ âñåõ ïàð çíà÷åíèé îäíîé è òîé æå ïåðåìåííîé, ÷òîáû ãàðàíòèðîâàòü íåâîçìîæíîñòü îäíîâðåìåííîãî ïðèíÿòèÿ ïåðåìåííîé äâóõ ðàçëè÷íûõ çíà÷åíèé. Äîáàâëÿåì äèçúþíêò 2V = {x1 , . . . , xn } ìíîæåñòâî ïåðåìåííûõ, D = {D1 , . . . , Dn } ìíîæåñòâî äîìåíîâ ïåðåìåííûõ, C = {C1 , . . . , Cm } ìíîæåñòâî îãðàíè÷åíèé. 4 (xi,0 ∨ xi,1 ∨ . . . xi,d ), ÷òîáû ãàðàíòèðîâàòü, ÷òî ïåðåìåííîé ïðèñâîåíî õîòÿ áû îäíî çíà÷åíèå. 1.2.2. Ïðèìåðû Ïðèâåäåì ðÿä ïðèìåðîâ, èëëþñòðèðóþùèõ ïîñòàíîâêè çàäà÷ ÓÎ â äðóãèõ îáëàñòÿõ ìàòåìàòèêè [253]. Ïðèìåð 1. Íàèáîëåå òðèâèàëüíûì àëãåáðàè÷åñêèì ïðèìåðîì çàäà÷è ÓÎ ÿâëÿåòñÿ çàäà÷à ðåøåíèÿ ñèñòåìû óðàâíåíèé. Äàíà ñèñòåìà ëèíåéíûõ óðàâíåíèé íàä êîíå÷íûì ïîëåì F . Èìååò ëè îíà ðåøåíèå? ßñíî, ÷òî â ýòîì ïðèìåðå êàæäîå îòäåëüíîå óðàâíåíèå ÿâëÿåòñÿ îãðàíè÷åíèåì, ïðè÷åì ïåðåìåííûå óðàâíåíèÿ îáðàçóþò äèàïàçîí, à ìíîæåñòâî âñåõ êîðòåæåé, ñîîòâåòñòâóþùèõ ðåøåíèÿì óðàâíåíèÿ, îáðàçóåò îòíîøåíèå îãðàíè÷åíèÿ. Ïðèìåð 2. Çàäà÷à ñòàíäàðòíîé ïðîïîçèöèîíàëüíîé 3-âûïîëíèìîñòè (3-SAT) [3], [300] îïðåäåëÿåòñÿ çàäàíèåì ôîðìóëû ïðîïîçèöèîíàëüíîé ëîãèêè, ñîñòîÿùåé èç êîíúþíêöèè äèçúþíêòîâ, ïðè÷åì êàæäûé äèçúþíêò ñîäåðæèò 3 ëèòåðàëà (ëèòåðàë ýòî ïåðåìåííàÿ èëè åå îòðèöàíèå), è îòâåòîì íà âîïðîñ, èìåþòñÿ ëè çíà÷åíèÿ ïåðåìåííûõ, êîòîðûå äåëàþò ôîðìóëó èñòèííîé. Ïóñòü Φ = ϕ1 ∧· · ·∧ϕn òàêàÿ ôîðìóëà, ãäå ϕi äèçúþíêòû. Çàäà÷à âûïîëíèìîñòè äëÿ Φ ìîæåò áûòü âûðàæåíà â âèäå çàäà÷è ÓÎ (V, {0, 1}, C), ãäå V ìíîæåñòâî âñåõ ïåðåìåííûõ â ôîðìóëå, à C ìíîæåñòâî îãðàíè÷åíèé {(s1 , ρ1 ), . . . , (sn , ρn )}, ãäå êàæäîå îãðàíè÷åíèå (sk , ρk ) ïîñòðîåíî ñëåäóþùèì îáðàçîì: sk ñïèñîê ïåðåìåííûõ, âõîäÿùèõ â ϕk , à ρk ñîñòîèò èç âñåõ êîðòåæåé, êîòîðûå äåëàþò äèçúþíêò ϕk èñòèííûì (k = 1, . . . , n). Ðåøåíèÿ ýòîé çàäà÷è ÓÎ ïðèñâîåíèÿ çíà÷åíèé ïåðåìåííûì, êîòîðûå äåëàþò ôîðìóëó Φ èñòèííîé. Çíà÷èò, ëþáàÿ çàäà÷à 3-âûïîëíèìîñòè ìîæåò áûòü âûðàæåíà â âèäå çàäà÷è ÓÎ. Ïðèìåð 3. Ëþáàÿ êîíêðåòíàÿ çàäà÷à ÓÎ ìîæåò áûòü âûðàæåíà â ëîãè÷åñêîé ôîðìå. Äåéñòâèòåëüíî, èñïîëüçóÿ ñòàíäàðòíîå ñîîòâåòñòâèå ìåæäó îòíîøåíèÿìè è ïðåäèêàòàìè, ìîæíî ïåðåïèñàòü çàäà÷ó ÓÎ â âèäå ôîðìóëû ïåðâîãî ïîðÿäêà ρ1 (s1 ) ∧ · · · ∧ ρq (sq ), ãäå ρi ïðåäèêàòû íà D è ρi (si ) îçíà÷àåò ïðåäèêàò ρi , ïðèìåíåííûé ê êîðòåæó si ïåðåìåííûõ (1 ≤ i ≤ q). Âîïðîñ ñîñòîèò â òîì, ÿâëÿåòñÿ ëè ýòà ôîðìóëà âûïîëíèìîé [332]. Ýòà çàäà÷à îáû÷íî èñïîëüçóåòñÿ â òåîðèè áàç äàííûõ (ÁÄ), ïîñêîëüêó îíà ñîîòâåòñòâóåò îöåíêå êîíúþíêòèâíîãî çàïðîñà [247], êàê ïîêàçàíî â ñëåäóþùåì ïðèìåðå. Ïðèìåð 4. Ðåëÿöèîííàÿ ÁÄ ìîæåò áûòü ðàññìîòðåíà êàê êîíå÷íîå ìíîæåñòâî òàáëèö. Òàáëèöà ñîñòîèò èç ñõåìû è êîíêðåòíûõ äàííûõ, ãäå ñõåìà êîíå÷íîå ìíîæåñòâî àòðèáóòîâ, ïðè÷åì êàæäûé àòðèáóò èìååò ñîîòâåòñòâóþùåå åìó ìíîæåñòâî âîçìîæíûõ çíà÷åíèé, íàçûâàåìîå äîìåíîì. Êîíêðåòíûå äàííûå ýòî êîíå÷íîå ìíîæåñòâî ñòðîê, ãäå êàæäàÿ ñòðîêà îòîáðàæåíèå, ñòàâÿùåå â ñîîòâåòñòâèå êàæäîìó àòðèáóòó ñõåìû çíà÷åíèå èç åå äîìåíà. Ñòàíäàðòíîé çàäà÷åé äëÿ ðåëÿöèîííûõ ÁÄ ÿâëÿåòñÿ çàäà÷à îöåíêè êîíúþíêòèâíîãî çàïðîñà [196], [247], â êîòîðîé ñïðàøèâàåòñÿ, èìååò ëè ðåøåíèå êîíúþíêòèâíûé çàïðîñ, ò.å. çàïðîñ âèäà ρ1 ∧ · · · ∧ ρq , ãäå ρ1 , . . . , ρq àòîìàðíûå ôîðìóëû. Êîíúþíêòèâíûé çàïðîñ íàä 5 ðåëÿöèîííîé ÁÄ ñîîòâåòñòâóåò êîíêðåòíîìó ïðèìåðó çàäà÷è ÓÎ, ÷òî äîñòèãàåòñÿ ïðîñòîé çàìåíîé òåðìèíîâ: ¾àòðèáóòû¿ çàìåíÿþòñÿ íà ¾ïåðåìåííûå¿, ¾òàáëèöû¿ íà ¾îãðàíè÷åíèÿ¿, ¾ñõåìà¿ íà ¾äèàïàçîí¿, ¾êîíêðåòíûå äàííûå¿ íà ¾îòíîøåíèå îãðàíè÷åíèÿ¿, à ¾ñòðîêè¿ íà ¾êîðòåæè¿. Çíà÷èò, êîíúþíêòèâíûé çàïðîñ ýêâèâàëåíòåí êîíêðåòíîìó ïðèìåðó çàäà÷è ÓÎ, ïåðåìåííûå êîòîðîé ýòî àòðèáóòû çàïðîñà. Äëÿ êàæäîé àòîìàðíîé ôîðìóëû ρi â çàïðîñå íàéäåòñÿ îãðàíè÷åíèå C òàêîå, ÷òî äèàïàçîí îãðàíè÷åíèÿ C ýòî ñïèñîê ïåðåìåííûõ ôîðìóëû ρi è îòíîøåíèå îãðàíè÷åíèÿ C ýòî ìíîæåñòâî ìîäåëåé ρi . Ïðèìåð 5. Äðóãîé âàæíîé ïåðåôîðìóëèðîâêîé â âèäå çàäà÷è ÓÎ ÿâëÿåòñÿ çàäà÷à î ãîìîìîðôèçìå, ñîñòîÿùàÿ â îòâåòå íà âîïðîñ, ñóùåñòâóåò ëè ãîìîìîðôèçì ìåæäó äâóìÿ ñòðóêòóðàìè îòíîøåíèé (ñì. [161], [197], [247]). Ïóñòü τ = (R1 , . . . , Rk ) ñèãíàòóðà, ò.å. ñïèñîê èìåí îòíîøåíèé ñ ôèêñèðîâàííîé àðíîñòüþ, ïðèïèñàííîé êàæäîìó èìåíè. Ïóñòü A = (A; R1A , . . . , RkA ) è B = (B; R1B , . . . , RkB ) ñòðóêòóðû îòíîøåíèé èç ñèãíàòóðû τ . Îòîáðàæåíèå h : A → B íàçûâàåòñÿ ãîìîìîðôèçìîì èç A â B , åñëè äëÿ âñåõ 1 ≤ i ≤ k , (h(a1 ), . . . , h(am )) ∈ RiB âñÿêèé ðàç, êîãäà (a1 , . . . , am ) ∈ RiA . Â ýòîì ñëó÷àå ìû çàïèñûâàåì h : A → B . ×òîáû óáåäèòüñÿ, ÷òî çàäà÷à î ãîìîìîðôèçìå ñâîäèòñÿ ê çàäà÷å ÓÎ, ïðåäñòàâèì ýëåìåíòû A â âèäå ïåðåìåííûõ, ýëåìåíòû B â âèäå çíà÷åíèé, êîðòåæè â îòíîøåíèÿõ A â âèäå äèàïàçîíîâ îãðàíè÷åíèé, à îòíîøåíèÿ èç B â âèäå îòíîøåíèé îãðàíè÷åíèé. Î÷åâèäíî, ÷òî ðåøåíèÿ ýòîé çàäà÷è ÓÎ â òî÷íîñòè ñîîòâåòñòâóþò ãîìîìîðôèçìàì èç A â B . Ïðèìåð 6. Èíòåðâàëüíûå çàäà÷è ÓÎ. Îäíèì èç âèäîì çàäà÷ ÓÎ ñ áåñêîíå÷íûì ÷èñëîì çíà÷åíèé, èíòåíñèâíî èçó÷àåìûõ â ÈÈ, ÿâëÿþòñÿ çàäà÷è ÓÎ, â êîòîðûõ çíà÷åíèÿ, ïðèíèìàåìûå ïåðåìåííûìè, ÿâëÿþòñÿ èíòåðâàëàìè äåéñòâèòåëüíîé ïðÿìîé. Ýòè çàäà÷è èñïîëüçóþòñÿ äëÿ ìîäåëèðîâàíèÿ ïîâåäåíèÿ âî âðåìåíè ñèñòåì, ãäå èíòåðâàëû ïðåäñòàâëÿþò èíòåðâàëû âðåìåíè, â òå÷åíèå êîòîðûõ ïðîèñõîäÿò ñîáûòèÿ. Íàèáîëåå ïîïóëÿðíîé ôîðìàëüíîé ñòðóêòóðîé òàêîãî ðîäà ÿâëÿåòñÿ èíòåðâàëüíàÿ àëãåáðà Àëëåíà (ÈÀÀ), ââåäåííàÿ â [29], îïèñûâàþùàÿ áèíàðíûå êà÷åñòâåííûå îòíîøåíèÿ ìåæäó èíòåðâàëàìè. ÈÀÀ ñîäåðæèò 13 áàçîâûõ îòíîøåíèé, ñîîòâåòñòâóþùèõ 13 ðàçëè÷íûì ñïîñîáàì, ñîãëàñíî êîòîðûì ìîãóò ñîîòíîñèòüñÿ äâà èíòåðâàëà. Ïîëíàÿ ñèñòåìà îòíîøåíèé â ÈÀÀ ñîñòîèò èç 213 = 8192 âîçìîæíûõ îáúåäèíåíèé áàçîâûõ îòíîøåíèé. Èíòåðâàëüíàÿ àëãåáðà Àëëåíà èìååò òðè îïåðàöèè íàä îòíîøåíèÿìè: êîìïîçèöèÿ, ïåðåñå÷åíèå è èíâåðñèÿ. Ïðèìåð 7. Ìîäåëü óäîâëåòâîðåíèÿ îãðàíè÷åíèé äëÿ çàäà÷è ¾SEND MORE MONEY¿ Ñòóäåíò ïîñûëàåò äîìîé çàêîäèðîâàííóþ òåëåãðàììó: SEND + MORE MONEY Çàäà÷à ñîñòîèò â íàõîæäåíèè 8 ðàçëè÷íûõ öèôð, çàêîäèðîâàííûõ 8 áóêâàìè, èñ6 ïîëüçîâàííûìè â òåëåãðàììå, òàê, ÷òîáû ïîëó÷àëñÿ àðèôìåòè÷åñêè ïðàâèëüíûé ðåçóëüòàò. Çàìåòèì, ÷òî ýòà çàäà÷à ìîæåò áûòü çàïèñàíà ñ ïîìîùüþ ìîäåëè öåëî÷èñëåííîãî ïðîãðàììèðîâàíèÿ (ÖÏ), íî îíà ñîäåðæèò îêîëî ñîòíè öåëî÷èñëåííûõ ïåðåìåííûõ, ïðè÷åì ýòà ìîäåëü ÖÏ òðóäíà äëÿ ðåøåíèÿ ñ ïîìîùüþ êëàññè÷åñêèõ ïîäõîäîâ ÖÏ, òàêèõ, êàê àëãîðèòìû âåòâåé è ãðàíèö. Â òî æå âðåìÿ ëîãè÷åñêèé àíàëèç çàäà÷è ïîçâîëÿåò íàéòè åå ðåøåíèå. Òàê, çàìå÷àÿ, ÷òî íè S , íè M íå ìîãóò áûòü íóëÿìè, ïîëó÷èì: S 6= 0, M 6= 0. Äàëåå, åäèíñòâåííîé âîçìîæíîñòüþ äëÿ M ÿâëÿåòñÿ çíà÷åíèå 1, ò.å. M = 1. Çàòåì, èç ðàâåíñòâà M = 1 ïîëó÷àåì, ÷òî S ìîæåò áûòü ðàâíî òîëüêî 9. Ìîæíî ïîêàçàòü, ÷òî ñèìâîëó O ñîîòâåòñòâóåò öèôðà 0, ò.å. O = 0. Äàëåå, ïåðåõîäÿ ê ñîòíÿì, èç O = 0 è óñëîâèÿ, ÷òî E è N äîëæíû áûòü ðàçè÷íûìè, ïîëó÷èì N = E + 1. Ïðîäîëæàÿ àíàëîãè÷íî ëîãè÷åñêèé àíàëèç îãðàíè÷åíèé, íàéäåì ðåøåíèå çàäà÷è: E = 5, N = 6, D = 7, R = 8, Y = 2. Îñíîâíàÿ ñëîæíîñòü ýòîé çàäà÷è ñîñòîèò â íåîáõîäèìîñòè ó÷åòà îãðàíè÷åíèÿ ¾âñå öèôðû ðàçëè÷íû¿ (òàê íàçûâàåìîå ãëîáàëüíîå îãðàíè÷åíèå all-dierent). Çàïèøåì îãðàíè÷åíèÿ çàäà÷è. Ïåðâàÿ ôîðìóëèðîâêà. Ïåðåìåííûå: {S, E, N, D, M, O, R, Y } Äîìåíû: {0, 1, . . . , 9} Îãðàíè÷åíèÿ: C1 : ∀x, y ∈ {S, E, N, D, M, O, R, Y }, x 6= y C2 : M = 0 or M = 1 C3 : (1000 × S + 100 × E + 10 × N + D) + (1000 × M + 100 × O + 10 × R + E) = = (10000 × M + 1000 × O + 100 × N + 10 × E + Y ) Âòîðàÿ ôîðìóëèðîâêà. Ïåðåìåííûå: {S, E, N, D, M, O, R, Y } + {C1, C2, C3} Äîìåíû: ∀x ∈ S, E, N, D, M, O, R, Y , Dx = 0, . . . , 9 ∀y ∈ {C1, C2, C3}, Dy = 0, 1 Îãðàíè÷åíèÿ: C1 : ∀x, y ∈ {S, E, N, D, M, O, R, Y }, x 6= y C2 : M = 0 or M = 1 C3 : D + E = 10 × C1 + Y C4 : N + R + C1 = 10 × C2 + E C5 : E + O + C2 = 10 × C3 + N C6 : S + M + C3 = 10 × M + O Íèæå, â ðàçäåëå 5, ïðèâåäåíà ïðîãðàììà ðåøåíèÿ ýòîé çàäà÷è íà ÿçûêå PROLOG. Ïðèìåð 8. Çàäà÷à î N ôåðçÿõ Çàäà÷à î N ôåðçÿõ ñîñòîèò â ðàçìåùåíèè N ôåðçåé íà øàõìàòíîé äîñêå N ×N òàê, ÷òîáû îíè íå óãðîæàëè äðóã äðóãó (ñì. ðèñ. 1). Ìîäåëü óäîâëåòâîðåíèÿ îãðàíè÷åíèé: ôåðçè â âåðòèêàëüíûõ ñòîëáöàõ: ∀i r(i) ∈ {1, . . . , N } 7 îòñóòñòâèå êîíôëèêòîâ: ∀i 6= j, r(i) 6= r(j) & |i − j| 6= |r(i) − r(j)|. A B C D E F G H 1 2 3 4 5 6 7 8 Ðèñ. 1. Îäíî èç ðåøåíèé çàäà÷è î 8 ôåðçÿõ Îäíèì èç ïðèìåðîâ èñïîëüçîâàíèÿ ÓÎ ÿâëÿåòñÿ ðåøåíèå êðîññâîðäîâ. Ïðèìåð 9. Ïðèìåð ñîñòàâëåíèÿ êðîññâîðäà Ðàññìîòðèì ïðèìåð ñîñòàâëåíèÿ êðîññâîðäà èç [142]. Çàäà÷à ñîñòîèò â âåðòèêàëüíîé èëè ãîðèçîíòàëüíîé çàïèñè ñëîâ èç çàäàííîãî ìíîæåñòâà ñëîâ (ñëîâàðÿ) â òàáëèöó ñ ó÷åòîì íåêîòîðûõ îãðàíè÷åíèé. Åñëè ðàçðåøåíî âñòàâëÿòü êàæäîå ñëîâî íà ïóñòîå ìåñòî ñîîòâåòñòâóþùåé äëèíû, âîçìîæíàÿ ôîðìóëèðîâêà çàäà÷è î êðîññâîðäå â âèäå çàäà÷è ÓÎ âûãëÿäèò ñëåäóþùèì îáðàçîì. Äëÿ êàæäîãî êâàäðàòà êðîññâîðäà ââîäèòñÿ ïåðåìåííàÿ, ïðèíèìàþùàÿ áóêâåííûå çíà÷åíèÿ èç àëôàâèòà, ïðè÷åì ìîãóò áûòü çàäàíû âîçìîæíûå çíà÷åíèÿ äëÿ ãðóïï ñìåæíûõ ïåðåìåííûõ. Ïðèâåäåì ôîðìàëüíóþ ôîðìóëèðîâêó çàäà÷è î êðîññâîðäå â òåðìèíàõ 1 2 3 4 6 8 5 7 9 10 12 13 11 Ðèñ. 2. Êðîññâîðä. îãðàíè÷åíèé. Êàæäîìó êâàäðàòó i êðîññâîðäà, êîòîðûé äîëæåí áûòü çàïîëíåí 8 (ñì. ðèñ. 2) ïîñòàâèì â ñîîòâåòñòâèå ïåðåìåííóþ xi (i = 1, . . . , 13). Ïåðåìåííûå x1 , . . . , x13 â êà÷åñòâå äîìåíîâ èìåþò áóêâû àëôàâèòà, â êà÷åñòâå îãðàíè÷åíèé ñëóæàò äîïóñòèìûå ñëîâà. Íàïðèìåð, îãðàíè÷åíèÿ ìîãóò áûòü çàäàíû ñëåäóþùèì îáðàçîì [142]: R1,2,3,4,5 = {(H, O, S, E, S), (L, A, S, E, S), (S, H, E, E, T ), (S, N, A, I, L), (S, T, E, E, R)} R3,6,9,12 = {(A, L, S, O), (E, A, R, N ), (H, I, K, E), (I, R, O, N ), (S, A, M, E)}, R5,7,11 = {(E, A, T ), (L, E, T ), (R, U, N ), (S, U, N ), (T, E, N ), (Y, E, S)}, R8,9,10,11 = R3,6,9,12 , R10,13 = {(B, E), (I, T ), (N, O), (U, S)}, R12,13 = R10,13 . Ïðèìåð 10. Çàäà÷à ñîñòàâëåíèÿ ðàñïèñàíèÿ Âàæíîé äëÿ ïðàêòè÷åñêèõ ïðèëîæåíèé ÿâëÿåòñÿ çàäà÷à ñîñòàâëåíèÿ ðàñïèñàíèÿ, ñîñòîÿùàÿ â óïîðÿäî÷åíèè íàáîðà çàäàíèé (ðàáîò). Â ýòîé çàäà÷å çàäàíû ñïèñîê çàäàíèé è îãðàíè÷åíèÿ, êàêèå çàäà÷è ìîãóò áûòü âûïîëíåíû îäíîâðåìåííî, âûïîëíåíèå êàêèõ çàäàíèé äîëæíî ïðåäøåñòâîâàòü âûïîëíåíèþ äðóãèõ è ò.ä. Äëÿ ðåøåíèÿ ýòîé çàäà÷è íóæíî íàéòè íàçíà÷åíèå âðåìåí íà÷àë ðàáîò çàäàíèÿì òàê, ÷òîáû âñå îãðàíè÷åíèÿ óäîâëåòâîðÿëèñü [367]. Ðàññìîòðèì çàäà÷ó ñîñòàâëåíèÿ ðàñïèñàíèÿ äëÿ ïÿòè çàäàíèé T 1, T 2, T 3, T 4, T 5, êàæäîå èç êîòîðûõ ìîæåò áûòü âûïîëíåíî çà îäèí ÷àñ. Çàäàíèÿ ìîãóò íà÷èíàòüñÿ â 1:00, 2:00 èëè 3:00. Ëþáûå ðàáîòû ìîãóò âûïîëíÿòüñÿ îäíîâðåìåííî, ó÷èòûâàÿ îãðàíè÷åíèÿ íà òî, ÷òî T 1 ìîæåò íà÷èíàòüñÿ ïîñëå T 3, T 3 ìîæåò íà÷èíàòüñÿ äî T 4 è ïîñëå T 5, T 2 íå ìîæåò íà÷èíàòüñÿ â òî æå âðåìÿ, ÷òî T 1 èëè T 4, T 4 íå ìîæåò íà÷èíàòüñÿ â 2:00. Ìîæíî ïîñòðîèòü ìîäåëü ñîñòàâëåíèÿ ãðàôèêà, ââåäÿ ïÿòü ïåðåìåííûõ, ñîîòâåòñòâóþùèõ çàäàíèÿì ñ äîìåíàìè {1 : 00, 2 : 00, 3 : 00}. Ñîîòâåòñòâóþùèé ãðàô îãðàíè÷åíèé ïîêàçàí íà ðèñ. 3. Ïðèìåð 11. Ðàçìåòêà èçîáðàæåíèé Çàäà÷à ðàçìåòêè èçîáðàæåíèé [384] ÿâëÿåòñÿ, ïî-âèäèìîìó, îäíîé èç ïåðâûõ ôîðìàëèçîâàííûõ çàäà÷ ÓÎ. Öåëü ýòîé çàäà÷è ñîñòîèò â ðàñïîçíàâàíèè îáúåêòîâ 3-ìåðíîãî èçîáðàæåíèÿ ñ ïîìîùüþ èíòåðïðåòàöèè ëèíèé íà 2-ìåðíûõ ðèñóíêàõ. Ëèíèè èëè ðåáðà äîëæíû áûòü ïîìå÷åíû, ò.å. îíè ðàçáèòû íà íåñêîëüêî òèïîâ, à èìåííî íà âûïóêëûå (+), âîãíóòûå (-) è çàìûêàþùèå ðåáðà (<). Â áîëåå ñëîæíûõ ñèñòåìàõ ðàñïîçíàåòñÿ òàêæå òåíåâîé êðàé. Èìååòñÿ ìíîãî ñïîñîáîâ ðàçìåòêè èçîáðàæåíèÿ (â òî÷íîñòè 3n , ãäå n ÷èñëî ðåáåð), íî òîëüêî íåìíîãèå èç íèõ èìåþò ñìûñë äëÿ òðåõìåðíîãî èçîáðàæåíèÿ. Èäåÿ ðåøåíèÿ ýòîé êîìáèíàòîðíîé çàäà÷è ñîñòîèò â íàõîæäåíèè äîïóñòèìûõ ìåòîê äëÿ òî÷åê ïåðåñå÷åíèÿ, óäîâëåòâîðÿþùèõ îãðàíè÷åíèÿì, ÷òî ðåáðî èìååò òó æå ìåòêó íà îáîèõ êîíöàõ. Ýòî íàìíîãî ñíèæàåò ïåðåáîð, òàê êàê èìååòñÿ ëèøü îãðàíè÷åííîå ÷èñëî äîïóñòèìûõ ìåòîê äëÿ òî÷åê ïåðåñå÷åíèÿ. 9 ɍɧɚɪɧɨɟ ɨɝɪɚɧɢɱɟɧɢɟ DT4={(1:00, 3:00)}. T1 Ȼɢɧɚɪɧɵɟ ɨɝɪɚɧɢɱɟɧɢɹ RT2,T1={(1:00, 2:00), (1:00, 3:00), (2:00, 1:00), (2:00, 3:00), (3:00, 1:00), (3:00, 2:00)}, T2 T5 RT2,T4={(1:00, 2:00), (1:00, 3:00), (2:00, 1:00), (2:00, 3:00), (3:00, 1:00), (3:00, 2:00)}, RT1,T3={(2:00, 1:00), (3:00, 1:00), (3:00, 2:00)}, RT3,T4={(1:00, 2:00), (1:00, 3:00), (3:00, 2:00)}, T3 T4 RT3,T5={(2:00, 1:00), (3:00, 1:00), (3:00, 2:00)}. Ðèñ. 3. Ãðàô îãðàíè÷åíèé è îòíîøåíèÿ çàäà÷è ñîñòàâëåíèÿ ãðàôèêà. + + + + + + Ðèñ. 4. Ðàçìåòêà èçîáðàæåíèé. + < + < Figure 2 - Ðèñ. 5. Äîïóñòèìûå âàðèàíòû ðàçìåòêè ïðè ñîåäèíåíèè â âåðøèíå. 10 Îñíîâíûå àëãîðèòìû ðåøåíèÿ îïèñàííîé çàäà÷è (ïîäîáíî àëãîðèòìó ðàçìåòêè Waltz [384]) îòíîñÿòñÿ ê äîñòèæåíèþ íåêîòîðûõ ôîðì ñîâìåñòèìîñòè. Äðóãèå ïðàêòè÷åñêèå ïðèëîæåíèÿ ÓÎ è ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ ïðèâåäåíû íèæå, â ðàçäåëå 5. 2. Îñíîâíûå ïîíÿòèÿ òåîðèè óäîâëåòâîðåíèÿ îãðàíè÷åíèé 2.1. Áàçîâûå ïîíÿòèÿ 2.1.1. Çàäà÷à óäîâëåòâîðåíèÿ îãðàíè÷åíèé Â äàííîé ñòàòüå ðàññìàòðèâàþòñÿ çàäà÷è ÓÎ ñ äèñêðåòíûìè ïåðåìåííûìè. Çàäà÷è ÓÎ ïðîñòåéøåãî âèäà õàðàêòåðèçóþòñÿ òåì, ÷òî â íèõ èñïîëüçóþòñÿ äèñêðåòíûå ïåðåìåííûå, êîòîðûå èìåþò êîíå÷íûå äîìåíû (îáëàñòè îïðåäåëåíèÿ). Ê òàêîìó âèäó îòíîñèòñÿ çàäà÷à ðàñêðàñêè êàðòû. Çàäà÷à ñ âîñåìüþ ôåðçÿìè, îïèñàííàÿ â ãëàâå 1, òàêæå ìîæåò ðàññìàòðèâàòüñÿ êàê çàäà÷à ÓÎ ñ êîíå÷íîé îáëàñòüþ îïðåäåëåíèÿ, ãäå ïåðåìåííûå Q1 , . . . , Q8 ïðåäñòàâëÿþò ñîáîé ïîçèöèè êàæäîãî ôåðçÿ íà ñòîëáöàõ 1, . . . , 8, à êàæäàÿ ïåðåìåííàÿ èìååò îáëàñòü îïðåäåëåíèÿ {1, 2, 3, 4, 5, 6, 7, 8}. Ïî îöåíêå Bartak [47], âåðîÿòíî, áîëåå 95% ïðèëîæåíèé ÓÎ èñïîëüçóþò êîíå÷íûå äîìåíû. Åñëè ìàêñèìàëüíûé ðàçìåð äîìåíîâ ïåðåìåííûõ â çàäà÷å ÓÎ ðàâåí d, òî êîëè÷åñòâî âîçìîæíûõ ïîëíûõ ïðèñâàèâàíèé çíà÷åíèé ïåðåìåííûì èçìåðÿåòñÿ âåëè÷èíîé O(Dn ), ò.å. çàâèñèò ýêñïîíåíöèàëüíî îò êîëè÷åñòâà ïåðåìåííûõ. Ê êàòåãîðèè çàäà÷ ÓÎ ñ êîíå÷íîé îáëàñòüþ îïðåäåëåíèÿ îòíîñÿòñÿ áóëåâû çàäà÷è ÓÎ, â êîòîðûõ ïåðåìåííûå ìîãóò èìåòü çíà÷åíèÿ ëèáî èñòèíà, ëèáî ëîæü. Áóëåâû çàäà÷è ÓÎ âêëþ÷àþò â êà÷åñòâå ÷àñòíûõ ñëó÷àåâ íåêîòîðûå N P -ïîëíûå çàäà÷è, òàêèå êàê çàäà÷è 3-âûïîëíèìîñòè 3SAT . Äèñêðåòíûå ïåðåìåííûå ìîãóò èìåòü áåñêîíå÷íûå îáëàñòè îïðåäåëåíèÿ, íàïðèìåð, òàêèå, êàê ìíîæåñòâî âñåõ öåëûõ ÷èñåë èëè ìíîæåñòâî âñåõ ñòðîê. Â èññëåäîâàíèè îïåðàöèé ÷àñòî âñòðå÷àþòñÿ çàäà÷è ÓÎ ñ íåïðåðûâíûìè îáëàñòÿìè îïðåäåëåíèÿ. Â ÷àñòíîñòè, ïðè êàëåíäàðíîì ïëàíèðîâàíèè ñòðîèòåëüíûõ ðàáîò äàòà íà÷àëà êàæäîé ðàáîòû ïðåäñòàâëÿåò ñîáîé ïåðåìåííóþ, à åå âîçìîæíûìè çíà÷åíèÿìè ÿâëÿþòñÿ öåëî÷èñëåííûå èíòåðâàëû âðåìåíè â ñóòêàõ, îòñ÷èòûâàåìûå îò òåêóùåé äàòû. Ïðè ðåøåíèè çàäà÷ ñ áåñêîíå÷íûìè îáëàñòÿìè îïðåäåëåíèÿ áîëüøå íåò âîçìîæíîñòè îïèñûâàòü îãðàíè÷åíèÿ, ïåðå÷èñëÿÿ âñå äîïóñòèìûå êîìáèíàöèè çíà÷åíèé. Âìåñòî ýòîãî äîëæåí èñïîëüçîâàòüñÿ ñïåöèàëüíûé ÿçûê îãðàíè÷åíèé. Íàïðèìåð, åñëè ðàáîòà Job1 êîòîðàÿ çàíèìàåò ïÿòü äíåé, äîëæíà ïðåäøåñòâîâàòü ðàáîòå Job3 , òî äëÿ îïèñàíèÿ ýòîãî óñëîâèÿ ïîòðåáóåòñÿ ÿçûê îãðàíè÷åíèé, ïðåäñòàâëÿþùèé ýòè óñëîâèÿ â âèäå StartJob1 + 5 ≤ StartJob3 . Êðîìå òîãî, óæå íåò âîçìîæíîñòè ðåøàòü òàêèå îãðàíè÷åíèÿ, ïðîñòî ïåðå÷èñëÿÿ âñå 11 âîçìîæíûå ïðèñâàèâàíèÿ, ïîñêîëüêó èìååòñÿ áåñêîíå÷íî ìíîãî âîçìîæíûõ ïðèñâàèâàíèé. Äëÿ ëèíåéíûõ îãðàíè÷åíèé ñ öåëî÷èñëåííûìè ïåðåìåííûìè ñóùåñòâóþò ñïåöèàëüíûå àëãîðèòìû ïîèñêà ðåøåíèé. Â íåêîòîðûõ ñëó÷àÿõ çàäà÷è ñ öåëî÷èñëåííûìè îãðàíè÷åíèÿìè ìîæíî ñâåñòè ê çàäà÷àì ñ êîíå÷íîé îáëàñòüþ îïðåäåëåíèÿ, óñòàíàâëèâàÿ ïðåäåëû çíà÷åíèé ýòèõ ïåðåìåííûõ. Íàïðèìåð, â çàäà÷å ïëàíèðîâàíèÿ ìîæíî óñòàíîâèòü âåðõíèé ïðåäåë, ðàâíûé îáùåé ïðîäîëæèòåëüíîñòè âñåõ çàïëàíèðîâàííûõ ðàáîò. Ñåòü îãðàíè÷åíèé èëè çàäà÷à óäîâëåòâîðåíèÿ îãðàíè÷åíèé (ÓÎ) ñîñòîèò èç ìíîæåñòâà ïåðåìåííûõ V = {x1 , . . . , xn }, ìíîæåñòâà äîìåíîâ çíà÷åíèé Di äëÿ êàæäîé ïåðåìåííîé xi , è ìíîæåñòâà îãðàíè÷åíèé è îòíîøåíèé. Êàæäûé äîìåí çíà÷åíèé ÿâëÿåòñÿ êîíå÷íûì ìíîæåñòâîì çíà÷åíèé, êîòîðûå ìîæåò ïðèíèìàòü ñîòâåòñòâóþùàÿ ïåðåìåííàÿ. Îãðàíè÷åíèå RS íàä S ⊆ X ýòî ïîäìíîæåñòâî äåêàðòîâà ïðîèçâåäåíèÿ äîìåíîâ ïåðåìåííûõ â S . Åñëè S = {xi1 , . . . , xir }, òî RS ⊆ Di1 × · · · × Dir . S íàçûâàåòñÿ äèàïàçîíîì (scope) îãðàíè÷åíèÿ RS . Â áèíàðíîé ñåòè îãðàíè÷åíèé âñå îãðàíè÷åíèÿ çàäàíû íàä ïàðàìè ïåðåìåííûõ. Ãðàô îãðàíè÷åíèé ñòàâèò â ñîîòâåòñòâèå êàæäîé ïåðåìåííîé âåðøèíó, ïðè÷åì äâå âåðøèíû ñîåäèíÿþòñÿ ðåáðîì, åñëè ñîîòâåòñòâóþùèå ïåðåìåííûå èìåþòñÿ â äèàïàçîíå êàêîãî-ëèáî îãðàíè÷åíèÿ. Ïðè çàäàíèè îãðàíè÷åíèé èñïîëüçóþòñÿ îòíîøåíèÿ. Îïðåäåëåíèå. Äëÿ äàííûõ ìíîæåñòâà ïåðåìåííûõ V = {x1 , . . . , xn } è ñîîòâåò- ñòâóþùèõ èõ îáëàñòåé çíà÷åíèé D1 , . . . , Dn îòíîøåíèåì R íà ìíîæåñòâå ïåðåìåííûõ íàçûâàåòñÿ ëþáîå ïîäìíîæåñòâî äåêàðòîâà ïðîèçâåäåíèÿ èõ îáëàñòåé çíà÷åíèé. Ìíîæåñòâî ïåðåìåííûõ, íà êîòîðîì îïðåäåëåíî îòíîøåíèå R, íàçûâàåòñÿ äèàïàçîíîì îòíîøåíèÿ è îáîçíà÷àåòñÿ scope(R). Åñëè R = D1 × . . . × Dn , òî R íàçûâàåòñÿ óíèâåðñàëüíûì îòíîøåíèåì. Îòíîøåíèÿ ìîãóò çàäàâàòüñÿ ñ ïîìîùüþ òàáëèö, îïèñûâàþùèõ äîïóñòèìûå ñî÷åòàíèÿ çíà÷åíèé ïåðåìåííûõ. Îïðåäåëåíèå. Çàäà÷à ÓÎ (ÇÓÎ) îïðåäåëÿåòñÿ ìíîæåñòâîì äèñêðåòíûõ ïåðåìåí- íûõ V = {x1 , . . . , xn }, äëÿ êàæäîé èç êîòîðûõ çàäàíà îáëàñòü îïðåäåëåíèÿ èëè äî(p ) (1) ìåí Dj = {dj , . . . , dj j } (j = 1, . . . , n), è ìíîæåñòâîì îãðàíè÷åíèé. Îãðàíè÷åíèåì íàçûâàåòñÿ ïàðà (R, S), ãäå R îòíîøåíèå, îïðåäåëåííîå íà äèàïàçîíå S . ÇÓÎ ìîæåò ðàññìàòðèâàòüñÿ êàê òðîéêà (V, D, C), ãäå V = {x1 , . . . , xn } ìíîæåñòâî ïåðåìåííûõ, D = {D1 , . . . , Dn } ìíîæåñòâî äîìåíîâ ïåðåìåííûõ, C = {C1 , . . . , Cm } ìíîæåñòâî îãðàíè÷åíèé. Ðåøåíèåì ÇÓÎ íàçûâàåòñÿ ïðèñâîåíèå çíà÷åíèé âñåì ïåðåìåííûì, êîòîðîå óäîâëåòâîðÿåò âñåì îãðàíè÷åíèÿì. Öåëüþ ðåøåíèÿ ÇÓÎ ìîæåò áûòü íàõîæäåíèå îäíîãî èëè âñåõ ðåøåíèé. 12 2.1.2. Äåéñòâèÿ íàä îòíîøåíèÿìè Ïóñòü R1 , R2 äâà îòíîøåíèÿ ñ îäèíàêîâûì äèàïàçîíîì. Òîãäà ïåðåñå÷åíèåì R1 ∩ R2 íàçûâàåòñÿ îòíîøåíèå, ñîäåðæàùåå âñå íàáîðû çíà÷åíèé ïåðåìåííûõ, êîòîðûå èìåþòñÿ îäíîâðåìåííî â îáîèõ îòíîøåíèÿõ R1 è R2 . Îáúåäèíåíèå R1 ∪ R2 îòíîøåíèå, ñîäåðæàùåå âñå íàáîðû çíà÷åíèé ïåðåìåííûõ, êîòîðûå èìåþòñÿ ëèáî â R1 , ëèáî â R2 , èëè â îáîèõ îòíîøåíèÿõ. Ðàçíîñòüþ R1 − R2 íàçûâàåòñÿ îòíîøåíèå, ñîäåðæàùåå íàáîðû çíà÷åíèé ïåðåìåííûõ, ñîäåðæàùèåñÿ â R1 , íî íå ñîäåðæàùèåñÿ â R2 . Ïðîåêöèÿ ΠY (R) îòíîøåíèÿ R íà ìíîæåñòâî ïåðåìåííûõ Y ⊆ scope(R) ÿâëÿåòñÿ îòíîøåíèåì, ñîäåðæàùèì íàáîðû çíà÷åíèé òîëüêî ïåðåìåííûõ, ñîäåðæàùèõñÿ â Y . Â òàáëè÷íîì ïðåäñòàâëåíèè îòíîøåíèé ïðîåêöèÿ âûáèðàåò ïîäìíîæåñòâî ñòîëáöîâ, ñîîòâåòñòâóþùèõ ìíîæåñòâó Y . Ñîåäèíåíèå îòíîøåíèé. Îïåðàòîð ñîåäèíåíèÿ RS ./ RT èç äâóõ îòíîøåíèé RS ñ äèàïàçîíîì S è RT ñ äèàïàçîíîì T ñòðîèò íîâîå îòíîøåíèå, ñîñòîÿùåå èç èõ îáùèõ ïåðåìåííûõ â S è T . Íàáîð r èç ñîåäèíåíèÿ RS ./ RT îòíîøåíèé RS è RT ìîæíî ïîñòðîèòü, èñïîëüçóÿ ñëåäóþùèå øàãè: (1) âçÿòü íàáîð s èç RS ; (2) âûáðàòü íàáîð t èç RT òàêîé, ÷òî êîìïîíåíòû s è t ñîãëàñóþòñÿ ïî ïåðåìåííûì èç S ∩ T , îáùèì äëÿ RS è RT ; (3) îáðàçîâàòü íîâûé íàáîð r, êîìáèíèðóÿ êîìïîíåíòû s è t, ñîõðàíÿÿ ëèøü îäèí èç ïîëó÷åííûõ îäèíàêîâûõ íàáîðîâ. Äèàïàçîí ïîëó÷àþùåãîñÿ îòíîøåíèÿ S ∪ T . Ñîåäèíåíèå äâóõ îòíîøåíèé ñ îäèíàêîâûìè äèàïàçîíàìè ýêâèâàëåíòíî ïåðåñå÷åíèþ ýòèõ îòíîøåíèé. 2.2. Âèäû îãðàíè÷åíèé Ðàññìîòðèì ðàçëè÷íûå âèäû îãðàíè÷åíèé. Ïðîñòåéøèì òèïîì îãðàíè÷åíèÿ ÿâëÿåòñÿ óíàðíîå îãðàíè÷åíèå, êîòîðîå îãðàíè÷èâàåò çíà÷åíèå îäíîé ïåðåìåííîé. Ëþáîå óíàðíîå îãðàíè÷åíèå ìîæíî óñòðàíèòü, âûïîëíÿÿ ïðåäâàðèòåëüíóþ îáðàáîòêó îáëàñòè îïðåäåëåíèÿ ñîîòâåòñòâóþùåé ïåðåìåííîé, ÷òîáû óäàëèòü çíà÷åíèÿ, íàðóøàþùèå ýòî îãðàíè÷åíèå. Áèíàðíîå îãðàíè÷åíèå ñâÿçûâàåò ìåæäó ñîáîé äâå ïåðåìåííûå. Íàïðèìåð, áèíàðíûì îãðàíè÷åíèåì ÿâëÿåòñÿ x 6= y . Áèíàðíîé çàäà÷åé ÓÎ íàçûâàåòñÿ çàäà÷à, â êîòîðîé èìåþòñÿ òîëüêî áèíàðíûå îãðàíè÷åíèÿ; îíà ìîæåò áûòü ïðåäñòàâëåíà â âèäå ãðàôà îãðàíè÷åíèé. Â îãðàíè÷åíèÿõ âûñîêîãî ïîðÿäêà ó÷àñòâóþò òðè èëè áîëüøå ïåðåìåííûõ. Îäíèì èç èçâåñòíûõ ïðèìåðîâ òàêèõ çàäà÷ ÿâëÿþòñÿ êðèïòîàðèôìåòè÷åñêèå ãîëîâîëîìêè, íàçûâàåìûå òàêæå ÷èñëîâûìè ðåáóñàìè (ñì. ðàçäåë 1). Îáû÷íî íàêëàäûâàåòñÿ òðåáîâàíèå, ÷òîáû êàæäàÿ áóêâà â êðèïòîàðèôìåòè÷åñêîé ãîëîâîëîìêå ïðåäñòàâëÿëà îòäåëüíóþ öèôðó. Â ñëó÷àå çàäà÷è SEND+MORE=MONEY èç ïðèìåðà 7 ðàçäåëà 1, òàêîå òðåáîâàíèå ìîæåò áûòü âûðàæåíî ñ ïîìîùüþ îãðàíè÷åíèÿ ñ âîñåìüþ ïåðåìåííûìè all − dif f erent(S, E, N, D, M, O, R, Y ). Èíûì îáðàçîì ýòî òðåáîâàíèå ìîæåò áûòü ïðåäñòàâëåíî â âèäå íàáîðà áèíàðíûõ îãðàíè÷åíèé, òàêèõ êàê S 6= E . 13 2.2.1. Ãëîáàëüíûå îãðàíè÷åíèÿ Ðÿä èññëåäîâàíèé ïî óäîâëåòâîðåíèþ îãðàíè÷åíèé ñ êîíå÷íûìè äîìåíàìè èñïîëüçóåò ãëîáàëüíûå îãðàíè÷åíèÿ [325]. Ïîíÿòèå ãëîáàëüíîãî îãðàíè÷åíèÿ ñîñòîèò â ñëåäóþùåì: èìåþòñÿ ñëó÷àè, êîãäà íàëè÷èå îäíîãî áîëüøåãî îãðàíè÷åíèÿ (ïî ÷èñëó ïåðåìåííûõ â íåì) ìîæåò áûòü ëó÷øå, ÷åì íàëè÷èå ìíîæåñòâà ìåíüøèõ îãðàíè÷åíèé. Òèïè÷íûì ïðèìåðîì ñëóæèò ìíîæåñòâî îãðàíè÷åíèé-íåðàâåíñòâ, êîòîðîå ðåäêî ïîçâîëÿåò ïðèìåíèòü óñïåøíîå ðàñïðîñòðàíåíèå îãðàíè÷åíèé. Â ïîäîáíûõ ñëó÷àÿõ íåîáõîäèìî èñïîëüçîâàòü ïîëíûé ïîèñê ïî äîìåíàì âñåõ ïåðåìåííûõ äëÿ íàõîæäåíèÿ îäíîãî ðåøåíèÿ. Ìîæíî èçáåæàòü ýòîãî áåñïîëåçíîãî ïîèñêà, èñïîëüçóÿ îãðàíè÷åíèå all-dierent, â êîòîðîå âõîäÿò âñå ïåðåìåííûå. Äåéñòâèòåëüíî, èñïîëüçóÿ îäíî òàêîå îãðàíè÷åíèå, çàäà÷à ïîèñêà ðåøåíèÿ ìîæåò áûòü ïåðåôîðìóëèðîâàíà â âèäå çàäà÷è ïàðîñî÷åòàíèÿ â äâóäîëüíîì ãðàôå (bipartite matching) ìåæäó ïåðåìåííûìè è çíà÷åíèÿìè, äëÿ ðåøåíèÿ êîòîðîé ìîãóò áûòü èñïîëüçîâàíû êëàññè÷åñêèå ãðàôîâûå àëãîðèòìû, ÷òî ìîæåò ñíèçèòü ïåðåáîð. Â îáùåì, ââåäåíèå ãëîáàëüíûõ ïåðåìåííûõ ðàáîòàåò ñëåäóþùèì îáðàçîì: âìåñòî çàïèñè çàäà÷è ñ ïîìîùüþ ìíîãèõ èñõîäíûõ áèíàðíûõ îãðàíè÷åíèé, ìû îïèñûâàåì åå ñ ïîìîùüþ íåáîëüøîãî ÷èñëà ãëîáàëüíûõ îãðàíè÷åíèé äëÿ ìíîæåñòâ ïåðåìåííûõ. Òàêèì îáðàçîì, ìû ìîæåì ïðîùå ìîäåëèðîâàòü çàäà÷ó ñ ìåíüøèì ÷èñëîì îãðàíè÷åíèé, ïðè ýòîì ìîãóò áûòü èñïîëüçîâàíû áîëåå ñîâåðøåííûå àëãîðèòìû. Òèïè÷íûìè ïðèìåðàìè ñëóæàò îãðàíè÷åíèÿ all-dierent è atmost, óñòàíàâëèâàþùåå ëèìèò íà ÷èñëî ïåðåìåííûõ ñ îïðåäåëåííûì çíà÷åíèåì, à òàêæå êóìóëÿòèâíîå îãðàíè÷åíèå. Íåñìîòðÿ íà òî, ÷òî ãëîáàëüíûå îãðàíè÷åíèÿ ÿâëÿþòñÿ âàæíûì àñïåêòîì òåîðèè ÓÎ, äîñòàòî÷íî ÷åòêîãî èõ îïðåäåëåíèÿ íåò. Ãëîáàëüíîå îãðàíè÷åíèå ýòî îãðàíè÷åíèå íàä ïîäìíîæåñòâîì ïåðåìåííûõ. Ó ãëîáàëüíûõ îãðàíè÷åíèé åñòü äâà ïðåèìóùåñòâà. Âî-ïåðâûõ, ãëîáàëüíûå îãðàíè÷åíèÿ îáëåã÷àþò çàäà÷ó ìîäåëèðîâàíèÿ ïðèêëàäíîé ïðîáëåìû â âèäå ÇÓÎ. Âî-âòîðûõ, ìîæíî ðàçðàáîòàòü àëãîðèòì ðàñïðîñòðàíåíèÿ îãðàíè÷åíèé ñïåöèàëüíîãî âèäà, ó÷èòûâàþùèé îñîáåííîñòè îãðàíè÷åíèÿ è ïîýòîìó íàìíîãî áîëåå ýôôåêòèâíûé. Íàïîìíèì, ÷òî âûíóæäåíèå äóãîâîé ñîâìåñòèìîñòè íà ïðîèçâîëüíîé çàäà÷å èìååò âðåìåííóþ ñëîæíîñòü O(rdr ) â íàèõóäøåì ñëó÷àå, ãäå r àðíîñòü îãðàíè÷åíèÿ, à d ðàçìåð äîìåíîâ ïåðåìåííûõ. Â òî æå âðåìÿ all-dierent-îãðàíè÷åíèå ìîæåò áûòü ñäåëàíî âåðøèííî-ñîâìåñòíûì çà âðåìÿ O(r2 d) â õóäøåì ñëó÷àå [311]. Êëàññè÷åñêèì ïðèìåðîì ãëîáàëüíîãî îãðàíè÷åíèÿ ñëóæèò all-dierent-îãðàíè÷åíèå, êîòîðîå îïðåäåëÿåò, ÷òî ïåðåìåííûå äîëæíû áûòü ïîïàðíî ðàçëè÷íûìè. all-dierent-îãðàíè÷åíèå ÷àñòî èñïîëüçóåòñÿ íà ïðàêòèêå è â ñâÿçè ñ ýòèì ïðåäëàãàåòñÿ êàê âñòðîåííîå îãðàíè÷åíèå âî ìíîãèõ îñíîâíûõ êîììåð÷åñêèõ è èññëåäîâàòåëüñêèõ ñèñòåìàõ ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ. Íà÷èíàÿ ñ ïåðâûõ ãëîáàëüíûõ îãðàíè÷åíèé â ñèñòåìå ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ CHIP [27], ïîçäíåå áûëè ïðåäëîæåíû è âíåäðåíû ñîòíè ãëîáàëüíûõ îãðàíè÷åíèé (ñì. [58]). Äðóãèìè ïðèìåðàìè ãëîáàëüíûõ îãðàíè÷åíèé ÿâëÿþòñÿ ãëîáàëüíîå êàðäèíàëüíîå îãðàíè÷åíèå (global cardinality constraint (gcc)) [312] è êóìóëÿòèâíîå îãðàíè14 ÷åíèå [27]. Ãëîáàëüíîå êàðäèíàëüíîå îãðàíè÷åíèå íàä ìíîæåñòâîì ïåðåìåííûõ è çíà÷åíèé îïðåäåëÿåò óñëîâèå, ÷òî ÷èñëî ïåðåìåííûõ, êîòîðûì ïðèñâîåíû çíà÷åíèÿ, äîëæíî áûòü ìåæäó çàäàííûìè âåðõíåé è íèæíåé ãðàíèöàìè, ïðè÷åì ýòè ãðàíèöû ìîãóò áûòü ðàçëè÷íû äëÿ êàæäîãî çíà÷åíèÿ. Êóìóëÿòèâíîå îãðàíè÷åíèå íàä ìíîæåñòâîì ïåðåìåííûõ, ïðåäñòàâëÿþùèõ âðåìåíà âûïîëíåíèÿ ðàçëè÷íûõ ðàáîò, ãàðàíòèðóåò, ÷òî ðàáîòû óïîðÿäî÷åíû òàê, ÷òî íå ïðåâûøàþòñÿ èìåþùèåñÿ â íàëè÷èè âåëè÷èíû ðåñóðñîâ, èñïîëüçóåìûå â ëþáûå ïåðèîäû âðåìåíè. Îãðàíè÷åíèÿ îáîèõ óêàçàííûõ òèïîâ îáû÷íî èñïîëüçóþòñÿ ïðè ñîñòàâëåíèè ðàñïèñàíèé, óïîðÿäî÷åíèè ðàáîò è â êàëåíäàðíîì ïëàíèðîâàíèè. 2.2.2. Ìÿãêèå îãðàíè÷åíèÿ Ñëåäóåò çàìåòèòü, ÷òî îãðàíè÷åíèé ñ êîíå÷íûìè äîìåíàìè íåäîñòàòî÷íî äëÿ òî÷íîãî ìîäåëèðîâàíèÿ ðåàëüíûõ çàäà÷. Ïðîáëåìà ñîñòîèò â òîì, ÷òî ðåàëüíûå çàäà÷è îáû÷íî íå ìîãóò áûòü îïèñàíû òîëüêî ñ ïîìîùüþ êëàññè÷åñêèõ îãðàíè÷åíèé, êîòîðûå ìîãóò áûòü èñòèííû ëèáî ëîæíû, ïîñêîëüêó îíè èìåþò òàêèå õàðàêòåðèñòèêè, êàê ïðåäïî÷òåíèÿ, âåðîÿòíîñòè, ñòîèìîñòè, à òàêæå ñòåïåíü íåîïðåäåëåííîñòè. Ìíîãèå ïðèêëàäíûå çàäà÷è ÓÎ îáû÷íî ÿâëÿþòñÿ ïåðåîãðàíè÷åííûìè (overconstrained) [223]: ïîëüçîâàòåëü çàäàåò ñëèøêîì ìíîãî îãðàíè÷åíèé, òàê ÷òî âñå èõ îäíîâðåìåííî óäîâëåòâîðèòü íåâîçìîæíî. Â òàêèõ ñëó÷àÿõ, âíà÷àëå íóæíî âûÿñíèòü, èìåþòñÿ ëè ðåøåíèÿ çàäà÷è ÓÎ âîîáùå (äëÿ ýòîãî ìîæåò ïîòðåáîâàòüñÿ ìíîãî âðåìåíè!), à çàòåì íóæíî ðåøèòü, êàêèå èç îãðàíè÷åíèé îñëàáèòü, ÷òîáû çàäà÷à ñòàëà ðàçðåøèìîé. Â ñâÿçè ñ ýòèì áûëî ââåäåíî ïîíÿòèå ìÿãêîãî îãðàíè÷åíèÿ (soft constraint) [50]: ìÿãêîå îãðàíè÷åíèå ýòî îáû÷íîå îãðàíè÷åíèå ïëþñ åãî äîïîëíèòåëüíàÿ õàðàêòåðèñòèêà (êðèòåðèé), êîòîðàÿ ìîæåò èíòåðïðåòèðîâàòüñÿ êàê ñòîèìîñòü, óðîâåíü âàæíîñòè, ñòåïåíü íåîïðåäåëåííîñòè è ò.ï. Íàõîæäåíèå ðåøåíèÿ ÇÓÎ ñ ìÿãêèìè îãðàíè÷åíèÿìè íå îçíà÷àåò íàõîæäåíèå òàêîâîãî, óäîâëåòâîðÿþùåãî âñåì îãðàíè÷åíèÿì, íî ïîëó÷åíèå çíà÷åíèé ïåðåìåííûõ, äëÿ êîòîðûõ äîñòèãàåòñÿ ¾íàèëó÷øåå¿ çíà÷åíèå äëÿ âûáðàííîãî êðèòåðèÿ. Èññëåäîâàíèÿ â ýòîé îáëàñòè íà÷àëîñü ñ èåðàðõè÷åñêîé ñèñòåìû ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ [79], ÿçûêà ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ, â êîòîðîì êàæäîå îãðàíè÷åíèå èìååò óðîâåíü âàæíîñòè è ðåøåíèå çàäà÷è ÓÎ íàõîäèòñÿ ñ ó÷åòîì èåðàõèè îãðàíè÷åíèé. Äëÿ ôîðìàëèçàöèè ýòîé íîâîé ìîäåëè è èññëåäîâàíèÿ ýôôåêòèâíîñòè àëãîðèòìîâ åå ðåøåíèÿ â [154], [328], [160], [333] áûëè ðàçðàáîòàíû íå÷åòêèå îãðàíè÷åíèÿ, ó êîòîðûõ êàæäîìó îãðàíè÷åíèþ ïðèñâîåíî çíà÷åíèå ìåæäó 0 è 1. Â [235] ðàññìàòðèâàåòñÿ êîìáèíàöèÿ èåðàðõèè îãðàíè÷åíèé è íå÷åòêèõ îãðàíè÷åíèé. Ïåðåîãðàíè÷åííûå çàäà÷è ðåøàëèñü òàêæå â [172] ñ èñïîëüçîâàíèåì ïîíÿòèÿ ÷àñòè÷íîãî ÓÎ, ïðè ýòîì â ÇÓÎ ââîäèëàñü ìåòðèêà, ÷òî ïîçâîëèëî íàõîäèòü ðåøåíèå, óäîâëåòâîðÿþùåå âîçìîæíî áîëüøåìó ÷èñëó îãðàíè÷åíèé. Áûëè ïðîâåäåíû òåîðåòè÷åñêèå èññëåäîâàíèÿ ïî ðàçðàáîòêå áîëåå îáùèõ ñõåì äëÿ ìÿãêèõ îãðàíè÷åíèé, èñïîëüçóÿ àïïàðàò ïîëóêîëåö [68], [69], [70] (ãäå êîðòåæó êàæäîãî îãðàíè÷åíèÿ ïîñòàâëåí â ñîîòâåòñòâèå ýëåìåíò ïîëóêîëüöà), à òàêæå àïïàðàò îãðàíè÷åíèé ñ îöåíêàìè (valued constraint), â êîòîðîì êàæäîìó îãðàíè÷åíèþ ïîñòàâëåí â ñîîòâåòñòâèå ýëåìåíò èç âïîëíå óïîðÿäî÷åííîãî ìíîæåñòâà 15 x1, D1={4, 5} a) > x2 x2, D2={3, 4, 5} b) > x2 = x4 x1 > x2 x2 x2 = x2 > x3 x3, D3={1, 2} x4, D4={4, 5} Ðèñ. 6. a) Ãðàô îãðàíè÷åíèé è b) äâîéñòâåííûé ãðàô. [98], [143], [334]. Äëÿ ýòèõ îáùèõ ñõåì ïîëó÷åíû îáîáùåíèÿ è ïðîèçâåäåíà àäàïòàöèÿ ðàñïðîñòðàíåíèÿ îãðàíè÷åíèé è ìåòîäîâ ðåøåíèÿ, îáû÷íî èñïîëüçóåìûõ äëÿ îãðàíè÷åíèé ñ êîíå÷íûìè äîìåíàìè, è èçó÷åíû èõ ñâîéñòâà. 2.2.3. Ïðèâåäåíèå çàäà÷ ÓÎ âûñîêîãî ïîðÿäêà ê áèíàðíûì çàäà÷àì ÓÎ Êàæäîå îãðàíè÷åíèå âûñîêîãî ïîðÿäêà ñ êîíå÷íûìè äîìåíàìè ìîæíî ñâåñòè ê ìíîæåñòâó áèíàðíûõ îãðàíè÷åíèé, ââåäÿ äîñòàòî÷íîå êîëè÷åñòâî âñïîìîãàòåëüíûõ ïåðåìåííûõ. Íåòðóäíî çàìåòèòü, ÷òî îãðàíè÷åíèå all-dierent ìîæåò áûòü ðàçáèòî íà áèíàðíûå îãðàíè÷åíèÿ S 6= E, S 6= N è ò.ä. Äâîéñòâåííîå ïðåîáðàçîâàíèå áûëî ïåðåíåñåíî èç òåîðèè ÁÄ â òåîðèþ ÓÎ Dechter & Pearl [141]. Ïåðâûå ïîïûòêè ïðèâåäåíèÿ çàäà÷ ÓÎ âûñîêîãî ïîðÿäêà ê ÷èñòî áèíàðíûì çàäà÷àì ÓÎ ñ ïîìîùüþ âñïîìîãàòåëüíûõ ïåðåìåííûõ áûëè âïåðâûå ïðåäïðèíÿòû â XIX âåêå ëîãèêîì ×àðëüçîì Ñàíäåðñîì Ïèðñîì (Peirce). Îñíîâó ìåòîäà ñêðûòîãî ïðåîáðàçîâàíèÿ ñîñòàâëÿåò ìåòîä Peirce [302] ïðåäñòàâëåíèÿ íåáèíàðíûõ îòíîøåíèé ñ ïîìîùüþ ñèñòåìû áèíàðíûõ îòíîøåíèé. Rossi et al. [323] íà îñíîâå ìåòîäà Peirce [302] âïåðâûå ïîêàçàëè, ÷òî áèíàðíûå îòíîøåíèÿ èìåþò òå æå âûðàçèòåëüíûå âîçìîæíîñòè, ÷òî è íåáèíàðíûå îòíîøåíèÿ. Ñîîòâåòñòâóþùèå ìåòîäû áûëè ââåäåíû â òåìàòèêó ÓÎ â ðàáîòå Dechter (1990) [134] è äîðàáîòàíû Bacchus & van Beek [39]. 2.3. Ãðàôû îãðàíè÷åíèé 2.3.1. Ãðàô îãðàíè÷åíèé, äâîéñòâåííûé ãðàô îãðàíè÷åíèé, ãèïåðãðàô îãðàíè÷åíèé Ñòðóêòóðà èëè òîïîëîãèÿ çàäà÷ ÓÎ ìîæåò îïèñûâàòüñÿ ñ ïîìîùüþ ðàçëè÷íûõ ãðàôîâûõ ñòðóêòóð: (ïåðâè÷íîãî) ãðàôà îãðàíè÷åíèé, ãèïåðãðàôà îãðàíè÷åíèé, äâîéñòâåííîãî ãðàôà îãðàíè÷åíèé. 16 Îïðåäåëåíèå. Ïåðâè÷íûé ãðàô îãðàíè÷åíèé ÇÓÎ (V, D, C) ýòî íåîðèåíòèðîâàííûé ãðàô G = (V, E), âåðøèíû V êîòîðîãî ñîîòâåòñòâóþò ïåðåìåííûì ÇÓÎ, ïðè÷åì äâå âåðøèíû ñîåäèíÿþòñÿ ðåáðîì â ãðàôå G, åñëè ñîîòâåòñòâóþùèå ïåðåìåííûå èìåþòñÿ â îäíîì è òîì æå îãðàíè÷åíèè (ò.å. â îäíîì è òîì æå äèàïàçîíå êàêîãî-òî îòíîøåíèÿ). Ãðàô îãðàíè÷åíèé îïðåäåëåí êàê äëÿ áèíàðíûõ, òàê è äëÿ íåáèíàðíûõ îãðàíè÷åíèé. Îäíàêî äëÿ íåáèíàðíîãî ñëó÷àÿ ñòðóêòóðà îãðàíè÷åíèé ìîæåò áûòü áîëåå òî÷íî ïåðåäàíà ñ ïîìîùüþ ãèïåðãðàôà îãðàíè÷åíèé. Ãèïåðãðàô îãðàíè÷åíèé ÇÓÎ ýòî ãèïåðãðàô GP , âåðøèíû êîòîðîãî ñîîòâåòñòâóþò ïåðåìåííûì çàäà÷è ÓÎ, à ãèïåð-ðåáðà îãðàíè÷åíèÿì C , âåðíåå, ïîäìíîæåñòâàì ïåðåìåííûõ èç äèàïàçîíîâ îãðàíè÷åíèé scope(C). Åñëè àðíîñòü êàæäîãî îãðàíè÷åíèÿ â C íå áîëåå 2, òî çàäà÷à ÓÎ íàçûâàåòñÿ áèíàðíîé çàäà÷åé ÓÎ. Ãèïåðãðàô, ñîîòâåòñòâóþùèé áèíàðíîé ÇÓÎ, ÿâëÿåòñÿ îáû÷íûì ãðàôîì. Ãèïåðãðàô, ñîîòâåòñòâóþùèé íåáèíàðíîé ÇÓÎ, ìîæåò áûòü ïðåîáðàçîâàí ê îáû÷íîìó ãðàôó îãðàíè÷åíèé ïóòåì çàìåíû êàæäîãî ãèïåððåáðà êëèêîé, ñîñòîÿùåé èç âåðøèí ãèïåðãðàôà è ñîåäèíÿþùèõ èõ ðåáåð [239]. Â ðÿäå ñëó÷àåâ óäîáíî èñïîëüçîâàòü äâîéñòâåííûé ãðàô îãðàíè÷åíèé 3 , âåðøèíû êîòîðîãî ñîîòâåòñòâóþò äèàïàçîíàì îãðàíè÷åíèé, à ðåáðà ñîåäèíÿþò âåðøèíû ïðè íàëè÷èè â íèõ îáùèõ ïåðåìåííûõ. Ïðè ýòîì ðåáðà ïîìå÷àþòñÿ ìíîæåñòâàìè îáùèõ ïåðåìåííûõ. Çàìåòèì, ÷òî äâîéñòâåííûé ãðàô îãðàíè÷åíèé òåñíî ñâÿçàí ñ äâîéñòâåííîé ÇÓÎ, â êîòîðîé ïåðåìåííûìè (òàê íàçûâàåìûìè c-ïåðåìåííûìè ) ÿâëÿþòñÿ îãðàíè÷åíèÿ èñõîäíîé çàäà÷è. Îãðàíè÷åíèÿ äâîéñòâåííîé çàäà÷è âûíóæäàþò îáùèå ïåðåìåííûå èç îãðàíè÷åíèé ïðèíèìàòü îäíè è òå æå çíà÷åíèÿ, ò.å. äâîéñòâåííûå îãðàíè÷åíèÿ áèíàðíû. Îòñþäà âèäíî, ÷òî ñ ïîìîùüþ äâîéñòâåííûõ çàäà÷ ÓÎ ìîæíî ëþáóþ íåáèíàðíóþ ÇÓÎ ñâåñòè ê áèíàðíîé è ðåøèòü åå ñ ïîìîùüþ ìåòîäîâ ðåøåíèÿ áèíàðíûõ ÓÎ. 2.3.2. Èíäóöèðîâàííûé ãðàô è èíäóöèðîâàííàÿ øèðèíà çàäà÷è ÓÎ Ââåäåì ïîíÿòèÿ èíäóöèðîâàííîãî ãðàôà è èíäóöèðîâàííîé øèðèíû (induced width) çàäà÷è ÓÎ, âàæíûå äëÿ ïðèìåíåíèÿ ñòðóêòóðíûõ ìåòîäîâ ðåøåíèÿ çàäà÷ ÓÎ. Èíäóöèðîâàííàÿ øèðèíà îïðåäåëÿåòñÿ â òåðìèíàõ ãðàôà îãðàíè÷åíèé çàäà÷è ÓÎ. Ïîïóòíî ââåäåì ïîíÿòèå óïîðÿäî÷åíèÿ ãðàôà èëè çàäà÷è ÓÎ. Îïðåäåëåíèå. Ðàññìîòðèì çàäà÷ó ÓÎ (V, D, C), ãäå |V | = n, è ñîîòâåòñòâóþùèé ãðàô îãðàíè÷åíèé G = (V, E). Óïîðÿäî÷åíèåì äëÿ çàäà÷è ÓÎ è ãðàôà G = (V, E) íàçûâàåòñÿ áèåêöèÿ h : V ↔ {1, 2, . . . , n}, êîòîðàÿ ñîîòâåòñòâóåò âåðøèíàì èëè ïåðåìåííûì, ïðîñìàòðèâàåìûì â ïîðÿäêå (h−1 (1), h−1 (2), . . . , h−1 (n)). Åñëè äâå ïåðåìåííûå ñîåäèíåíû â ãðàôå ðåáðîì, èíîãäà ïîëåçíî ïðåäñòàâèòü ïåðâóþ 3 Äâîéñòâåííûé ãðàô îãðàíè÷åíèé ïî ñóòè ñîâïàäàåò ñ ðåáåðíûì ãðàôîì ãèïåðãðàôà : ðåáåðíûì ãðàôîì ãèïåðãðàôà H = (V, E) ñîãëàñíî [4] íàçûâàåòñÿ ãðàô L(H) = (E, E) ìíîæåñòâî âåðøèí êîòîðîãî ñîâïàäàåò ñ ìíîæåñòâîì ðåáåð E ãèïåðãðàôà H , ïðè ýòîì äâå âåðøèíû ãðàôà L(H) ñìåæíû òîãäà è òîëüêî òîãäà, êîãäà ñìåæíû ñîîòâåòñòâóþùèå èì ðåáðà ãèïåðãðàôà H . Òàêèì îáðàçîì, L(H) ãðàô ïåðåñå÷åíèé ðåáåð ãèïåðãðàôà H . 17 ïåðåìåííóþ, êîòîðàÿ ñòîèò ðàíüøå â óïîðÿäî÷åíèè, â âèäå ¾ðîäèòåëÿ¿, à âòîðóþ êàê ¾ñûíà¿ ïåðâîé âåðøèíû. Ïåðåä îïðåäåëåíèåì èíäóöèðîâàííîé øèðèíû ââåäåì áîëåå ïðîñòîå ïîíÿòèå øèðèíû ãðàôà. Îïðåäåëåíèå. Ïóñòü G = (V, E) ãðàô, à h íåêîòîðîå óïîðÿäî÷åíèå âåðøèí ýòîãî ãðàôà. Øèðèíîé âåðøèíû v äëÿ ýòîãî óïîðÿäî÷åíèÿ h íàçûâàåòñÿ ÷èñëî âåðøèí, ñîåäèíåííûõ ñ v è ïðåäøåñòâóþùèõ åé â ýòîì óïîðÿäî÷åíèè (ò.å. ÷èñëî ðîäèòåëåé âåðøèíû v ). Øèðèíîé ãðàôà äëÿ óïîðÿäî÷åíèÿ h íàçûâàåòñÿ ìàêñèìàëüíàÿ øèðèíà âñåõ âåðøèí äëÿ ýòîãî óïîðÿäî÷åíèÿ. È íàêîíåö, øèðèíîé ãðàôà íàçûâàåòñÿ ìèíèìàëüíàÿ øèðèíà äëÿ âñåõ óïîðÿäî÷åíèé. Íàïðèìåð, äåðåâî èìååò øèðèíó 1, à ïîëíûé ãðàô ñ n âåðøèíàìè èìååò øèðèíó n − 1. Ââåäåì äàëåå ñëåäóþùèå îïðåäåëåíèÿ: Îïðåäåëåíèå. Øèðèíîé çàäà÷è ÓÎ äëÿ äàííîãî óïîðÿäî÷åíèÿ íàçûâàåòñÿ øèðèíà åãî ãðàôà îãðàíè÷åíèé äëÿ ýòîãî óïîðÿäî÷åíèÿ, à øèðèíà ÇÓÎ îïðåäåëÿåòñÿ êàê øèðèíà åå ãðàôà îãðàíè÷åíèé. Ââåäåì ñîãëàñíî Dechter & Pearl [140] ïîíÿòèå èíäóöèðîâàííîãî ãðàôà ïî îòíîøåíèþ ê äàííîìó óïîðÿäî÷åíèþ. Îïðåäåëåíèå. Äëÿ äàííîãî ãðàôà G = (V, E) è óïîðÿäî÷åíèÿ h, èíäóöèðîâàííûé ãðàô G∗ = (V, E ∗ ) îïðåäåëÿåòñÿ êàê ãðàô ñ ìèíèìàëüíûì ìíîæåñòâîì ðåáåð E ∗ , òàêèõ ÷òî E ∗ ⊇ E , è åñëè {x, v} ∈ E ∗ , {y, v} ∈ E ∗ , h(x) < h(v), h(y) < h(v), è x 6= y , òî {x, y} ∈ E ∗ . Èíäóöèðîâàííûé ãðàô ìîæåò áûòü ïîñòðîåí çà îäèí ïðîõîä, îáðàáàòûâàÿ âåðøèíû èñõîäíîãî ãðàôà â îáðàòíîì ïîðÿäêå; òî åñòü âíà÷àëå ñîåäèíÿåì ðîäèòåëåé ïîñëåäíåé âåðøèíû, çàòåì ðîäèòåëåé ïðåäïîñëåäíåé âåðøèíû è ò.ä. Îïðåäåëåíèå. Èíäóöèðîâàííàÿ øèðèíà w(h) ãðàôà G = (V, E) ïî îòíîøåíèþ ê óïîðÿäî÷åíèþ h ýòî øèðèíà èíäóöèðîâàííîãî ãðàôà (G∗ , h) ïî îòíîøåíèþ ê ýòîìó óïîðÿäî÷åíèþ. Èíäóöèðîâàííàÿ øèðèíà w∗ ãðàôà ýòî åãî ìèíèìàëüíàÿ èíäóöèðîâàííàÿ øèðèíà ïî âñåì óïîðÿäî÷åíèÿì. Èíäóöèðîâàííàÿ øèðèíà çàäà÷è ÓÎ ïî îòíîøåíèþ ê óïîðÿäî÷åíèþ h ýòî èíäóöèðîâàííàÿ øèðèíà åå ãðàôà îãðàíè÷åíèé ïî îòíîøåíèþ ê ýòîìó óïîðÿäî÷åíèþ, à èíäóöèðîâàííàÿ øèðèíà çàäà÷è ÓÎ ýòî èíäóöèðîâàííàÿ øèðèíà åå ãðàôà îãðàíè÷åíèé. Ñèíîíèì èíäóöèðîâàííîé øèðèíû äðåâîâèäíàÿ øèðèíà [23]. Ðîäñòâåííûì ÿâëÿåòñÿ ïîíÿòèå òàêæå ÷àñòè÷íîãî k -äåðåâà4 . Dechter & Pearl [140] ïîêàçàëè, ÷òî äàííàÿ çàäà÷à ÓÎ ïðè çàäàííîì óïîðÿäî÷åíèè ìîæåò áûòü ðåøåíà çà âðåìÿ, ýêñïîíåíöèàëüíîå îò èíäóöèðîâàííîé øèðèíû ïî îòíîøåíèþ ê ýòîìó óïîðÿäî÷åíèþ. 2.3.3. Õîðäàëüíûå ãðàôû Âû÷èñëåíèå èíäóöèðîâàííîé øèðèíû äîñòàòî÷íî íåñëîæíî äëÿ õîðäàëüíûõ ãðàôîâ. Íàïîìíèì, ÷òî ãðàô íàçûâàåòñÿ õîðäàëüíûì, åñëè â íåì êàæäûé öèêë äëèíû 4 èìååò õîðäó. 4 ×àñòè÷íîå k -äåðåâî ýòî ãðàô, èíäóöèðîâàííàÿ øèðèíà êîòîðîãî íå ïðåâûøàåò k [38]. 18 Ìíîãèå òðóäíûå ãðàôîâûå çàäà÷è ëåãêî ðåøàþòñÿ íà õîðäàëüíûõ ãðàôàõ. Íàïðèìåð, íàõîæäåíèå âñåõ ìàêñèìàëüíûõ êëèê â ãðàôå, ÷òî ÿâëÿåòñÿ NP-ïîëíîé çàäà÷åé íà ãðàôàõ îáùåãî âèäà, ëåãêî ïðîèçâåñòè äëÿ õîðäàëüíûõ ãðàôîâ. Ýòà çàäà÷à (ïîèñê ìàêñèìàëüíûõ êëèê â õîðäàëüíûõ ãðàôàõ) îáëåã÷àåòñÿ çà ñ÷åò èñïîëüçîâàíèÿ ïðîöåäóðû óïîðÿäî÷åíèÿ max-cardinality 5 . Óïîðÿäî÷åíèå max-cardinality ãðàôà óïîðÿäî÷èâàåò âåðøèíû, íà÷èíàÿ îò ïåðâîé, ñîãëàñíî ñëåäóþùåìó ïðàâèëó: ïåðâàÿ âåðøèíà âûáèðàåòñÿ ïðîèçâîëüíî. Ïîñëå ýòîãî, ñëåäóþùåé âûáèðàåòñÿ âåðøèíà, ñîåäèíåííàÿ ñ ìàêñèìàëüíûì ÷èñëîì óæå óïîðÿäî÷åííûõ âåðøèí è ò.ä. Óïîðÿäî÷åíèå max-cardinality ìîæåò áûòü èñïîëüçîâàíî äëÿ ðàñïîçíàâàíèÿ õîðäàëüíûõ ãðàôîâ. Èìåííî, ãðàô õîðäàëüíûé, òîãäà è òîëüêî òîãäà, êîãäà ïðè óïîðÿäî÷åíèè max-cardinality êàæäàÿ âåðøèíà âìåñòå ñ åå ðîäèòåëÿìè îáðàçóåò êëèêó. Ìîæíî òàêèì îáðàçîì ïåðå÷èñëèòü âñå ìàêñèìàëüíûå êëèêè, ñîîòâåòñòâóþùèå êàæäîé âåðøèíå (çàïèñûâàÿ ìíîæåñòâà, ñîñòîÿùèå èç êàæäîé âåðøèíû è åå ðîäèòåëåé, è îïðåäåëÿÿ çàòåì ìàêñèìàëüíûé ðàçìåð êëèêè). Çàìåòèì, ÷òî èìååòñÿ íå áîëåå n êëèê, ñîñòîÿùèõ èç âåðøèí è èõ ðîäèòåëåé. Êðîìå òîãî, ïðè èñïîëüçîâàíèè óïîðÿäî÷åíèÿ max-cardinality õîðäàëüíîãî ãðàôà, óïîðÿäî÷åííûé ãðàô èäåíòè÷åí åãî èíäóöèðîâàííîìó ãðàôó è, çíà÷èò, åãî øèðèíà ðàâíà åãî èíäóöèðîâàííîé øèðèíå. Ñïðàâåäëèâî Ïðåäëîæåíèå. Åñëè G∗ èíäóöèðîâàííûé ãðàô ãðàôà G äëÿ íåêîòîðîãî óïîðÿäî÷åíèÿ, òîãäà ãðàô G∗ õîðäàëüíûé. 3. Ìåòîäû ïîèñêà ðåøåíèÿ ÇÓÎ 3.1. Îñíîâíûå ìåòîäû ðåøåíèÿ Ìåòîäû ïîñòðîåíèÿ ðåøåíèÿ çàäà÷è ÓÎ ìîãóò áûòü ðàçáèòû íà òðè êëàññà [329]: 1) Ïåðâûé êëàññ ñîäåðæèò âàðèàíòû ïîèñêà ñ âîçâðàòàìè. Ýòè àëãîðèòìû ñòðîÿò ðåøåíèå ñ ïîìîùüþ ðàñøèðåíèÿ ÷àñòè÷íîãî ðåøåíèÿ øàã çà øàãîì, èñïîëüçóÿ ðàçëè÷íûå ýâðèñòèêè è ïðèìåíÿÿ ðàçóìíûå ñòðàòåãèè âîçâðàòà èç òóïèêîâûõ âåðøèí. Ñíèæåíèå ðàçìåðà çàäà÷è ïîçâîëÿåò óìåíüøèòü ðàçìåðû ïðîñòðàíñòâà ïîèñêà. 2) Àëãîðèòìû ðàñïðîñòðàíåíèÿ îãðàíè÷åíèé èñêëþ÷àþò íåêîòîðûå ýëåìåíòû, íå âõîäÿùèå â ðåøåíèÿ, èç ïðîñòðàíñòâà ïîèñêà. Â îáùåì, ýòè àëãîðèòìû íå èñêëþ÷àþò âñå ýëåìåíòû, íå âõîäÿùèå â ðåøåíèÿ è, ñëåäîâàòåëüíî, îíè íå ñòðîÿò ñàìè ïî ñåáå ðåøåíèÿ, à èñïîëüçóþòñÿ ëèáî äëÿ ïðåïðîöåññèíãà çàäà÷è äî èñïîëüçîâàíèÿ àëãîðèòìà äðóãîãî òèïà, èëè ïåðåìåæàþòñÿ ñ øàãàìè àëãîðèòìà äðóãîãî òèïà íàïðèìåð, ïîèñêà ñ âîçâðàòàìè äëÿ ïîâûøåíèÿ ïðîèçâîäèòåëüíîñòè ïîñëåäíåãî. 3) Íàêîíåö, ñòðóêòóðíûå àëãîðèòìû èñïîëüçóþò èíôîðìàöèþ î ñòðóêòóðå ïåðâè÷íîãî èëè äâîéñòâåííîãî ãðàôà îãðàíè÷åíèé çàäà÷è. Èìåþòñÿ ðàçëè÷íûå àëãîðèòìû ýòîãî êëàññà, ïðè ýòîì íåêîòîðûå ïðîèçâîäÿò äåêîìïîçèöèþ èñõîäíîé çàäà÷è ÓÎ íà ñëàáî ñâÿçàííûå ïîäçàäà÷è, êîòîðûå ìîãóò áûòü ðåøåíû ñ ïîìîùüþ ìåòîäîâ èç ïðåäûäóùèõ äâóõ êëàññîâ. Òàêèì îáðàçîì, 5 max-cardinality ìàêñèìàëüíîå êîëè÷åñòâî ýëåìåíòîâ. 19 ñòðóêòóðíûå àëãîðèòìû ìîãóò òàêæå èñïîëüçîâàòüñÿ â ñî÷åòàíèè ñ àëãîðèòìàìè äðóãèõ òèïîâ. Âñå àëãîðèòìû èç óêàçàííûõ âûøå òðåõ êëàññîâ ñèñòåìàòè÷åñêè èññëåäóþò ïðîñòðàíñòâî ðåøåíèé. Ýòè àëãîðèòìû: • êîððåêòíû, òî åñòü îíè çàêàí÷èâàþò ðàáîòó ñ ïðèñâîåíèåì çíà÷åíèé âñåì ïåðåìåííûì, êîòîðîå ÿâëÿåòñÿ ðåøåíèåì; • ïîëíû, ò.å. îíè ñïîñîáíû èññëåäîâàòü âñå ïðîñòðàíñòâî ïîèñêà è íàéòè âñå ðåøåíèÿ. Îáñóäèì ìåòîäû ïîèñêà ñ âîçâðàòàìè. 3.2. Ïîèñê ñ âîçâðàòàìè (Backtrack Search) è åãî ïðîèçâîäíûå 3.2.1. Ïîèñê ñ âîçâðàòàìè Ôîðìóëèðîâêà çàäà÷è ÓÎ â âèäå çàäà÷ ïîèñêà ïîçâîëÿåò ðåøàòü çàäà÷è ÓÎ ñ ïîìîùüþ àëãîðèòìîâ ïîèñêà. Ïîèñê â ãëóáèíó, â êîòîðîì êàæäûé ðàç âûáèðàþòñÿ çíà÷åíèÿ äëÿ îäíîé ïåðåìåííîé è âûïîëíÿåòñÿ âîçâðàò, åñëè áîëüøå íå îñòàåòñÿ äîïóñòèìûõ çíà÷åíèé, êîòîðûå ìîæíî áûëî áû ïðèñâîèòü ïåðåìåííîé, íàçûâàåòñÿ ïîèñêîì ñ âîçâðàòàìè (backtracking)6 . Äåðåâî ïðèñâîåíèé çíà÷åíèé ïåðåìåííûì ýòî äåðåâî, â êîòîðîì êàæäàÿ âåðøèíà ñîîòâåòñòâóåò ìíîæåñòâó ïðèñâîåíèé. Êîðåíü äåðåâà îòâå÷àåò ïóñòîìó ìíîæåñòâó ïðèñâîåíèé. Â êàæäîé âåðøèíå v âûáèðàåòñÿ ïåðåìåííàÿ, êîòîðîé â v íå áûëî ïðèñâîåíî çíà÷åíèå. Àëãîðèòìû ïîèñêà ñ âîçâðàòàìè âûïîëíÿþò ïîèñê â ãëóáèíó â ýòèõ äåðåâüÿõ ïðèñâîåíèé çíà÷åíèé. Ïðîñòåéøèì ìåòîäîì ðåøåíèÿ çàäà÷ ÓÎ ÿâëÿåòñÿ ìåòîä Generate and Test, êîòîðûé ïîðîæäàåò êàæäîå âîçìîæíîå ðåøåíèå ïóòåì ïðèñâîåíèÿ âñåõ âîçìîæíûõ çíà÷åíèé äëÿ êàæäîé ïåðåìåííîé è ïðîâåðÿåò, óäîâëåòâîðÿåò ëè ýòî ðåøåíèå âñåì îãðàíè÷åíèÿì. Â íàèõóäøåì ñëó÷àå ÷èñëî âñåõ âîçìîæíûõ ïðîâåðåííûõ ðåøåíèé ðàâíî ðàçìåðó äåêàðòîâà ïðîèçâåäåíèÿ äîìåíîâ âñåõ ïåðåìåííûõ7 . Ïîíÿòíî, ÷òî ìåòîä ïîëíîãî ïåðåáîðà ìîæåò òðåáîâàòü áîëüøèõ çàòðàò âðåìåíè ïðè ðåøåíèè çàäà÷. Àëãîðèòìû ïîèñêà ñ âîçâðàòàìè ýòî àëãîðèòìû ñèñòåìàòè÷åñêîãî ïîèñêà äëÿ çàäà÷ ÓÎ, êîòîðûå èñïîëüçóþò ÷àñòè÷íûå ðåøåíèÿ, êîòîðûå ñòðîÿòñÿ ïóòåì ïîî÷åðåäíîãî ïðèñâàèâàíèÿ çíà÷åíèé ïåðåìåííûì. Åñëè àëãîðèòì îáíàðóæèâàåò òóïèêîâóþ âåðøèíó, â êîòîðîì ÷àñòè÷íîå ðåøåíèå íå èìååò ñîâìåñòíîãî ðàñøèðåíèÿ, òî âûáîð ïîñëåäíåãî ïðèñâîåíèÿ îòìåíÿåòñÿ è äåëàåòñÿ ïîïûòêà ïðèñâîåíèÿ äðóãîãî çíà÷åíèÿ. Ýòîò ïðîöåññ ïîâòîðÿåòñÿ äî òåõ ïîð, ïîêà íå áóäóò èñ÷åðïàíû âñå âîçìîæíîñòè è/èëè íå áóäåò íàéäåíî ðåøåíèå. Àëãîðèòìû ïîèñêà ñ âîçâðàòàìè 6 Èñïîëüçîâàíèå ïîèñêà ñ âîçâðàòàìè äëÿ óäîâëåòâîðåíèÿ îãðàíè÷åíèé áûëî ïðåäëîæåíî Bitner & Reingold (1975) â [71]. 7 Åñëè äîìåíû âñåõ n ïåðåìåííûõ èìåþò ðàçìåð d, òî ýòà âåëè÷èíà ñîñòàâëÿåò dn . 20 Procedure BT(i) Foreach Val In D[i] Assignments[i]:= Val Consistent:=True For h:=1 To i-1 While Consistent Consistent:=Test(i,h) If Consistent If i=n Show - Solution() Else BT(i+1) Return False End Procedure Ðèñ. 7. Ïîèñê ñ âîçâðàòàìè. ÿâëÿþòñÿ ïîëíûìè â òîì ñìûñëå, ÷òî îíè íàõîäÿò ðåøåíèå, åñëè îíî ñóùåñòâóåò. Ðàçëè÷íûå àëãîðèòìû ïîèñêà ñ âîçâðàòàìè ðàçëè÷àþòñÿ ïî ñêîðîñòè îáíàðóæåíèÿ òóïèêîâûõ âåðøèí, à òàêæå ïî òîìó, íàñêîëüêî äàëåêî îíè ñïîñîáíû äåëàòü âîçâðàò. Àëãîðèòì ïîèñêà ñ âîçâðàòàìè õàðàêòåðèçóåòñÿ ýêñïîíåíöèàëüíîé âðåìåííîé ñëîæíîñòüþ, íî ÿâëÿåòñÿ ëèíåéíûì ïî èñïîëüçóåìîé ïàìÿòè (Mackworth, 1977) [262]. Â õðîíîëîãè÷åñêîì ïîèñêå ñ âîçâðàòàìè (chronological backtracking) ïðîöåññ ïîèñêà ðàññìàòðèâàåòñÿ êàê ïîñòåïåííîå ðàñøèðåíèå ÷àñòè÷íûõ ðåøåíèé (Bitner & Reingold, 1975) [71]. Ïîðÿäîê çàäàíèÿ çíà÷åíèé ýòî ïîðÿäîê, â êîòîðîì ïðèñâàèâàþòñÿ çíà÷åíèÿ ïåðåìåííûì âî âðåìÿ ïîèñêà. Íà óðîâíå i ïîðÿäêà ïðèñâàèâàíèÿ ïðîøëûå ïåðåìåííûå èìåþò èíäåêñû, ìåíüøèå ÷åì i, ïðè÷åì èì óæå ïðèñâîåíû çíà÷åíèÿ. Áóäóùèå ïåðåìåííûå èìåþò èíäåêñû, áîëüøèå ÷åì i è èì åùå íå ïðèñâîåíû çíà÷åíèÿ. Íà ðèñ. 7 îïèñàí àëãîðèòì ïîèñêà ñ âîçâðàòàìè. Äëÿ ïðîñòîòû, çàïèñàííûé çäåñü ïñåâäîêîä íå ðàçëè÷àåò ñëó÷àåâ, êîãäà íå íàéäåíî ðåøåíèé èëè êîãäà íå íàéäåíî áîëüøå ðåøåíèé, çíà÷åíèå ëîæü âîçâðàùàåòñÿ â îáîèõ óêàçàííûõ ñëó÷àÿõ. Àëãîðèòì îïèñàí äëÿ ñëó÷àÿ áèíàðíûõ îãðàíè÷åíèé c(xi , xj ). Èñïîëüçóþòñÿ ñëåäóþùèå îáîçíà÷åíèÿ: • n ÷èñëî ïåðåìåííûõ çàäà÷è. • Assignments [ ]. Ýòîò ìàññèâ ðàçìåðíîñòè n çàïîìèíàåò çíà÷åíèå, ïðèñâîåííîå â íàñòîÿùèé ìîìåíò êàæäîé ïåðåìåííîé. • D[ ]. i-é ýëåìåíò ýòîãî ìàññèâà ðàçìåðíîñòè n ñîäåðæèò ïîñëåäîâàòåëüíîñòü ýëåìåíòîâ äîìåíà, ñîîòâåòñòâóþùåãî ïåðåìåííîé xi . Ôóíêöèÿ T est(i, j) ïðèíèìàåò çíà÷åíèå èñòèíà, åñëè îãðàíè÷åíèå c(xi , xj ) óäîâëåòâîðÿåòñÿ òåêóùèìè ïðèñâîåíèÿìè çíà÷åíèé ïåðåìåííûì xi è xj . Åñëè îãðàíè÷å21 - «ɦɟɬɚɧɢɹ» x1 5 4 4 5 3 2 2 1 1 4 5 4 4 3 5 4 5 4 x2 1 x3 4 x4 5 Ðèñ. 8. Ïîèñê ñ âîçâðàòàìè êàê îáõîä äåðåâà. íèÿ c(xi , xj ) íå ñóùåñòâóåò, Test( ) ïðèíèìàåò èñòèíà (ýòî íå ñ÷èòàåòñÿ ïðîâåðêîé îãðàíè÷åíèÿ). Êàæäûé ýëåìåíò äîìåíà Di ïîî÷åðåäíî ïðèñâàèâàåòñÿ ïåðåìåííîé xi , ðàñøèðÿÿ ïðèñâîåíèå çíà÷åíèé ïåðåìåííûì x1 , . . . , xi−1 . Åñëè ïðîâåðêà îãðàíè÷åíèÿ äëÿ ïðåäûäóùåé ïåðåìåííîé íå ïðîõîäèò, ýòî ÷àñòè÷íîå ïðèñâîåíèå áîëüøå íå ðàñøèðÿåòñÿ. Åñëè äîìåí Di èñ÷åðïàí (òóïèê èëè ïîëíîå óíè÷òîæåíèå äîìåíà), àëãîðèòì äåëàåò âîçâðàò äî óðîâíÿ i − 1 ñ öåëüþ íàõîæäåíèÿ äðóãîãî ñîâìåñòíîãî ïðèñâîåíèÿ äëÿ xi−1 . Åñëè æå òàêîâîãî íåò, äåëàåòñÿ âîçâðàò ê xi−2 , è ò.ä. ïîêà íå áóäåò íàéäåíà ïåðåìåííàÿ ñ äðóãèì ñîâìåñòíûì ïðèñâîåíèåì èëè áóäåò ïîêàçàíî, ÷òî ÇÓÎ íå èìååò ðåøåíèÿ. Äëÿ äàííîãî ñîâìåñòíîãî ïðèñâîåíèÿ (âñå ïðîâåðêè îãðàíè÷åíèé áûëè óñïåøíû), äåëàåòñÿ ðåêóðñèâíûé âûçîâ BT(i + 1) èëè, åñëè íå îñòàåòñÿ ñâîáîäíûõ ïåðåìåííûõ, ïîëó÷åíî ðåøåíèå. Ïîèñê ñ âîçâðàòàìè ìîæåò ðàññìàòðèâàòüñÿ êàê îáõîä âåðøèí äåðåâà. Íà ðèñ. 8 ïîêàçàíî, êàê ïîèñê ñ âîçâðàòàìè ðåøàåò çàäà÷ó, îïèñàííóþ íà íà ðèñ. 6. Êàæäàÿ âåòâü ñîîòâåòñòâóåò ÷àñòè÷íîìó ïðèñâîåíèþ (êîòîðîå ñòàíîâèòñÿ ïîëíûì, êîãäà âåòâü äîõîäèò äî n-é ïåðåìåííîé) è óðîâåíü i äåðåâà ïðåäñòàâëÿåò âûáîðû çíà÷åíèé, ñäåëàííûå äëÿ i-é (ñîãëàñíî óïîðÿäî÷åíèþ) ïåðåìåííîé. ×åðíûå è áåëûå âåðøèíû ïðåäñòàâëÿþò ñîîòâåòñòâåííî óñïåøíûå è íåóäà÷íûå ïðèñâîåíèÿ â äàííîé òî÷êå ïîèñêà. Âåðøèíû ñíàáæåíû îïèñàíèåì çíà÷åíèé äîìåíîâ, ïðèïèñàííûõ äàííîé ïåðåìåííîé â ðàññìàòðèâàåìîé òî÷êå ïîèñêà. Ïóòü ïîèñêà ïîêàçàí ïóíêòèðíîé íàïðàâëåííîé ëèíèåé. Ïîèñê ñ âîçâðàòàìè íåýôôåêòèâåí â òîì ñìûñëå, ÷òî òðåáóåò âûïîëíåíèÿ áîëüøåãî ÷èñëà îïåðàöèé, ÷åì ýòî íóæíî äëÿ íàõîæäåíèÿ ðåøåíèÿ. 3.2.2. Ïðîáëåìà ìå÷óùåãîñÿ ïîâåäåíèÿ àëãîðèòìà ñ âîçâðàòàìè Ìåòàíèÿ (Gaschnig, 1979) [181] ÿâëÿþòñÿ ãëàâíûì ôàêòîðîì ïëîõîãî ïîâåäåíèÿ ïîèñêà ñ âîçâðàòàìè: ïîèñê áûâàåò áåçóñïåøåí ïîâòîðíî ïî òîé æå ïðè÷èíå. Íàïðèìåð, ïóñòü c(xg , xi ) óñëîâèå òîãî, ÷òî êîíêðåòíîå ïðèñâîåíèå ïåðåìåííîé xg çàïðåùàåò âñå ïîòåíöèàëüíûå çíà÷åíèÿ äëÿ xi . Ïîèñê ñ âîçâðàòàìè áåçóñïåøåí 22 íà óðîâíå i äëÿ êàæäîãî ýëåìåíòà Di , ïðè÷åì ýòà íåóäà÷à ïîâòîðÿåòñÿ äëÿ êàæäîé êîìáèíàöèè ïðèñâîåíèé íà ïðîìåæóòî÷íûõ óðîâíÿõ ïåðåìåííîé xh , ãäå g < h < i. Ìåòàíèå íàáëþäàåòñÿ â ïîïûòêå ïîèñêà ñ âîçâðàòàìè ðåøèòü çàäà÷ó íà ðèñ. 6. Ðèñ. 8 ïîêàçûâàåò ïîâòîðÿþùèåñÿ íåóäà÷íûå ïîïûòêè ïîèñêà â ñâÿçè ñ óñëîâèåì c(x2 , x4 ). Ìåòàíèé ñòàíîâèòñÿ áîëüøå ïðè óâåëè÷åíèè ÷èñëà ïðîìåæóòî÷íûõ ïåðåìåííûõ (è âåëè÷èí èõ äîìåíîâ). 3.2.3. Èíòåëëåêòóàëüíûé ïîèñê ñ âîçâðàòàìè Èíòåëëåêòóàëüíûé ïîèñê ñ âîçâðàòàìè (intelligent backtracking) [42], [86] ýòî ñîáèðàòåëüíûé òåðìèí, îáúåäèíÿþùèé àëãîðèòìû ïîèñêà ñ âîçâðàòàìè, ïîçâîëÿþùèå ðåøàòåëþ îáíàðóæèòü, ÷òî êîíêðåòíàÿ òóïèêîâàÿ âåðøèíà äåðåâà ïðèñâîåíèÿ çíà÷åíèé íå ñâÿçàíà ñ íåêîòîðûìè ïðèñâîåíèÿìè. À èìåííî, åñëè òóïèêîâàÿ âåðøèíà áûëà îáíàðóæåíà íà óðîâíå l è àëãîðèòì ìîæåò âûÿñíèòü, ÷òî ïðèñâîåíèÿ, ñäåëàííûå íà óðîâíÿõ j ∈ {k + 1, . . . , l} íå îòíîñÿòñÿ ê ýòîé òóïèêîâîé âåðøèíå, îí ìîæåò ïåðåéòè îáðàòíî ê óðîâíþ k , ïîñêîëüêó ïîïûòêà ðàçëè÷íûõ ïðèñâîåíèé ýòèì ïåðåìåííûì ïîñòîÿííî ïðèâîäèëà áû ê òóïèêîâîé âåðøèíå. Òàêèì îáðàçîì, âìåñòî õðîíîëîãè÷åñêîãî ïîèñêà ñ âîçâðàòàìè íà óðîâíå l − 1, ýòîò àëãîðèòì ïîçâîëÿåò èçáåæàòü ëèøíåé ðàáîòû, ñâÿçàííîé ñ ïîïûòêîé ïðèñâàèâàíèÿ ðàçëè÷íûõ ïðèñâîåíèé ïåðåìåííûì íà ýòèõ óðîâíÿõ, è âìåñòî ýòîãî ïðè âîçâðàòå ïðÿìî ïûòàåòñÿ ïðèñâîèòü äðóãîå çíà÷åíèå íà áîëåå ðàííåì óðîâíå k . Ê ðàçëè÷íûì ìåòîäàì èíòåëëåêòóàëüíîãî ïîèñêà ñ âîçâðàòàìè îòíîñÿòñÿ îáðàòíûé ïåðåõîä (backjumping), îáðàòíûé ïåðåõîä, óïðàâëÿåìûé êîíôëèêòàìè (conict-directed backjumping), îáðàòíàÿ ïðîâåðêà (backchecking), ïðîñòàâëåíèå îáðàòíûõ îòìåòîê (backmarking). Îñíîâíîé ìåòîä îáðàòíîãî ïåðåõîäà (backjumping) áûë ðàçðàáîòàí â ðàáîòàõ (Gaschnig, 1977, 1979) [180], [181]. Ìåòîäû îáðàòíîãî ïåðåõîäà ÿâëÿþòñÿ îäíèì èç ãëàâíûõ ñðåäñòâ äëÿ ïðåîäîëåíèÿ òåíäåíöèé ïîèñêà ñ âîçâðàòàìè ê ïîâòîðíîìó íàõîæäåíèþ îäíèõ è òåõ æå òóïèêîâûõ âåðøèí. Òóïèêîâàÿ âåðøèíà âñòðå÷àåòñÿ, êîãäà ó xi íå îñòàåòñÿ ñîâìåñòíûõ çíà÷åíèé, â ýòîì ñëó÷àå àëãîðèòì ïîèñêà ñ âîçâðàòàìè âîçâðàùàåòñÿ â âåðøèíó xi−1 . Ïóñòü ñóùåñòâóåò íîâîå çíà÷åíèå äëÿ xi−1 , íî íåò îãðàíè÷åíèé ìåæäó xi è xi−1 . Òóïèêîâàÿ âåðøèíà áóäåò â xi äëÿ ëþáîãî çíà÷åíèÿ ïåðåìåííîé xi−1 , ïîêà íå áóäóò èñ÷åðïàíû âñå çíà÷åíèÿ ïåðåìåííîé xi−1 . Ìû ìîæåì îáîéòè ýòó ñèòóàöèþ ïóòåì îïðåäåëåíèÿ ïåðåìåííîé âèíîâíèöû çà òóïèê è ïîñëåäóþùèì íåìåäëåííûì îáðàòíûì ïåðåõîäîì ê ïðèñâîåíèþ çíà÷åíèÿ ïåðåìåííîé-âèíîâíèöû òóïèêà (êîíôëèêòà), âìåñòî ïîâòîðíîãî ôèêñèðîâàíèÿ çíà÷åíèÿ õðîíîëîãè÷åñêè ïðåäøåñòâóþùåé ïåðåìåííîé. Îïðåäåëåíèå ïåðåìåííîé-âèíîâíèöû òóïèêà â àëãîðèòìå ïîèñêà ñ âîçâðàòàìè îñíîâàíî íà ïîíÿòèè êîíôëèêòíîãî ìíîæåñòâà. Äëÿ ïðîñòîòû èçëîæåíèÿ áóäåì ïðåäïîëàãàòü, ÷òî ïåðåìåííûå óïîðÿäî÷åíû ñëåäóþùèì îáðàçîì: h = {x1 , . . . , xn }. − Âìåñòî òîãî, ÷òîáû äîæèäàòüñÿ òóïèêîâîé âåðøèíû → a i 8 , ìåòîä îáðàòíîãî ïåðåõîäà Gaschnig'à [181] çàïîìèíàåò íåêîòîðóþ èíôîðìàöèþ â ïðîöåññå ïîñòðîåíèÿ − 8→ a i òóïèêîâûé êîðòåæ, ïðåäñòàâëÿþùèé ñîáîé íàáîð ïðèñâîåíèé çíà÷åíèé ïåðåìåííûì. 23 4 5 3 3 1 4 1 Ɉɛɪɚɬɧɵɣ ɩɟɪɟɯɨɞ 5 x1 5 4 4 5 Ɉɛɪɚɬɧɵɣ ɩɟɪɟɯɨɞ 4 x2 1 x3 4 x4 Ðèñ. 9. Ìåòîä îáðàòíîãî ïåðåõîäà. − → a i , è èñïîëüçóåò ýòó èíôîðìàöèþ äëÿ îïðåäåëåíèÿ ïåðåìåííîé xb âèíîâíèöû òóïèêîâîé âåðøèíû (ñì. ðèñ. 9). Àëãîðèòì èñïîëüçóåò ìåòîä ïðîñòàâëåíèÿ îòìåòîê, ïîñðåäñòâîì êîòîðîãî äëÿ êàæäîé ïåðåìåííîé èñïîëüçóåòñÿ óêàçàòåëü íà ïîñëåäíåãî ïðåäøåñòâåííèêà, êîòîðûé îêàçàëñÿ íåñîâìåñòíûì ñ ëþáûì èç çíà÷å− íèé ïåðåìåííîé. Ïðè ïîñòðîåíèè êîðòåæåé → a íà ïðÿìîì øàãå àëãîðèòì ñîõðàíÿåò óêàçàòåëü highi äëÿ êàæäîé ïåðåìåííîé xi . Ýòîò óêàçàòåëü îïðåäåëÿåò ïîñëåäíþþ ïåðåìåííóþ, ïðîâåðåííóþ íà ñîâìåñòíîñòü ñ xi , åñëè áûëî îáíàðóæåíî, ÷òî îíà ÿâ0 ëÿåòñÿ ñàìîé ïåðâîé ïåðåìåííîé, êîíôëèêòóþùåé ñî çíà÷åíèåì â Di . Íàïðèìåð, â ñëó÷àå, êîãäà íåò ñðàâíèìûõ çíà÷åíèé äëÿ xi è åñëè highi = 3, ýòî ïîêàçûâàåò, → → ÷òî − a 3 êîíôëèêòíîå ìíîæåñòâî ïåðåìåííîé xi . Åñëè − a i èìååò ñîâìåñòíîå çíà÷åíèå, òî highi ïðèïèñûâàåòñÿ çíà÷åíèå i − 1. Àëãîðèòì ïåðåõîäèò èç òóïèêîâîãî → ëèñòà − a i , êîòîðûé íåñîâìåñòåí ñ xi+1 , îáðàòíî ê ïåðåìåííîé xhighi+1 âèíîâíèöå êîíôëèêòà, òàê êàê òóïèêîâàÿ ïåðåìåííàÿ xi+1 . Äâå èäåè: îáðàòíûé ïåðåõîä ê ïåðåìåííîé, êîòîðàÿ, áóäó÷è çàôèêñèðîâàíà, íàõîäèòñÿ â êîíôëèêòå ñ òåêóùåé ïåðåìåííîé, è îáðàòíûé ïåðåõîä ê âíóòðåííèì òóïèêàì, èíòåãðèðîâàíû â îäíîì àëãîðèòìå îáðàòíîãî ïåðåõîäà, óïðàâëÿåìîãî êîíôëèêòàìè (Prosser, 1993) [309]. Ýòîò àëãîðèòì èñïîëüçóåò ñõåìó îáðàòíîãî ïåðåõîäà, èñïîëüçóÿ èíôîðìàöèþ, ñîáðàííóþ â ïðîöåññå ïîèñêà. Äëÿ êàæäîé ïåðåìåííîé àëãîðèòì ïîääåðæèâàåò èíäóöèðîâàííîå ìíîæåñòâî îáðàòíûõ ïåðåõîäîâ. → Äëÿ äàííîãî òóïèêîâîãî êîðòåæà − a i , îïðåäåëèì ìíîæåñòâî îáðàòíûõ ïåðåõîäîâ − → äëÿ a i (èëè äëÿ xi+1 ) êàê ìíîæåñòâî ïåðåìåííûõ, ó÷àñòâóþùèõ â ñàìîì ðàííåì → ìèíèìàëüíîì êîíôëèêòíîì ìíîæåñòâå − a i . Îáðàòíûé ïåðåõîä, óïðàâëÿåìûé êîíôëèêòàìè, âêëþ÷àåò âî ìíîæåñòâî ïåðåõîäîâ ïåðåìåííóþ, åñëè åå òåêóùåå çíà÷åíèå êîíôëèêòóåò ñî çíà÷åíèåì òåêóùåé ïåðåìåííîé, êîòîðàÿ íå áûëà â êîíôëèêòå íè ñ îäíèì áîëåå ðàííèì ïðèñâîåíèåì ïåðåìåííîé. Ïðåäëîæåííûé Ñòàëìàíîì è Çþññìàíîì (Stallman & Sussman, 1977) [355] ìåòîä ïîèñêà ñ âîçâðàòàìè óïðàâëÿåìîãî çàâèñèìîñòÿìè, ÿâëÿþùèéñÿ îáùåé è ìîùíîé ôîðìîé èíòåëëåêòóàëüíîãî ïîèñêà ñ âîçâðàòàìè, ïðèâåë ê ðàçðàáîòêå 24 ñèñòåì îáåñïå÷åíèÿ èñòèííîñòè (truth maintenance systems) (Doyle, 1979) [152]. Ñâÿçü ìåæäó ýòèìè äâóìÿ íàïðàâëåíèÿìè ïðîàíàëèçèðîâàíà â (Kleer, 1989) [130]. Â óêàçàííîé ðàáîòå [355] áûëà òàêæå ïðåäëîæåíà èäåÿ ðåãèñòðàöèè îãðàíè÷åíèÿ (constraint recording), â ñîîòâåòñòâèè ñ êîòîðîé ÷àñòè÷íûå ðåçóëüòàòû, ïîëó÷åííûå â õîäå ïîèñêà, ìîæíî ñîõðàíÿòü è ïîâòîðíî èñïîëüçîâàòü íà ïîñëåäóþùèõ ýòàïàõ ýòîãî ïîèñêà. Äàííàÿ èäåÿ áûëà ââåäåíà ôîðìàëüíî â ïîèñê ñ âîçâðàòàìè â ðàáîòå (Dechter, 1990 a) [133]. Îñîáåííî ïðîñòûì ìåòîäîì ÿâëÿåòñÿ ïðîñòàâëåíèå îáðàòíûõ îòìåòîê (backmarking) (Gaschnig, 1979) [181], â êîòîðîì äëÿ ïðåäîòâðàùåíèÿ ïîâòîðíîé ïðîâåðêè îãðàíè÷åíèé ñîõðàíÿþòñÿ è èñïîëüçóþòñÿ ñîâìåñòíûå è íåñîâìåñòíûå ïîïàðíûå ïðèñâàèâàíèÿ. Ïðîñòàâëåíèå îáðàòíûõ îòìåòîê [181] ýòî àëãîðèòì òèïà êýøèðîâàíèÿ, ñíèæàþùèé ÷èñëî ïðîâåðîê íà ñîâìåñòíîñòü, íåîáõîäèìûõ äëÿ ïîñòðîåíèÿ êàæäîé âåðøèíû, íå ïûòàÿñü îòñå÷ü ÷àñòü ïðîñòðàíñòâà ïîèñêà ñàìîãî ïî ñåáå. Ñîõðàíÿÿ èíôîðìàöèþ î òîì, ãäå ïðîâåðêà íà ñîâìåñòíîñòü íå ïðîøëà ðàíåå, ïðîñòàâëåíèå îáðàòíûõ îòìåòîê ìîæåò èñêëþ÷èòü íåîáõîäèìîñòü èçëèøíå ïîâòîðÿòü âûïîëíåííûå ðàíåå ïðîâåðêè. Îïèøåì ýòîò àëãîðèòì êàê óñîâåðøåíñòâîâàíèå íàèâíîãî àëãîðèòìà ïîèñêà ñ âîçâðàòàìè. Íàïîìíèì, ÷òî àëãîðèòì ïîèñêà ñ âîçâðàòàìè äâèæåòñÿ ëèáî âïåðåä, ëèáî íàçàä â ïðîñòðàíñòâå ïîèñêà. Äîïóñòèì, ÷òî òåêóùàÿ ïåðåìåííàÿ xi , à xp ñàìàÿ ïåðâàÿ ïåðåìåííàÿ (ñîãëàñíî óïîðÿäî÷åíèþ), êîòîðàÿ èçìåíèëà ñâîå çíà÷åíèå ïîñëå ïîñëåäíåãî çàõîäà â xi . Ïîíÿòíî, ÷òî ëþáàÿ ïðîâåðêà çíà÷åíèé xi ïî îòíîøåíèþ ê ïåðåìåííûì, ïðåäøåñòâóþùèì xp äàñò òå æå ðåçóëüòàòû: åñëè îíà áûëà íåóäà÷íîé ïî îòíîøåíèþ ê áîëåå ðàííèì ïðèñâîåíèÿì, òî îíà áóäåò íåóäà÷íà ñíîâà; åñëè îíà áûëà óñïåøíîé, òî îíà îïÿòü áóäåò óñïåøíîé. Ñëåäîâàòåëüíî, ïðè ñîõðàíåíèè ïðàâèëüíîé èíôîðìàöèè èç áîëåå ðàííèõ ÷àñòåé ïîèñêà, çíà÷åíèÿ â äîìåíå ïåðåìåííîé xi ìîãóò áûòü ëèáî íåìåäëåííî ðàñïîçíàíû, êàê íåñîâìåñòíûå, ëèáî åùå ïðîâåðåíû òîëüêî ïî îòíîøåíèþ ê ïðåäøåñòâóþùèì ïðèñâîåíèÿì, íà÷èíàÿ ñ xp . Ïðîñòàâëåíèå îáðàòíûõ îòìåòîê ðåàëèçóåò ýòó èäåþ ñ ïîìîùüþ èñïîëüçîâàíèÿ äâóõ íîâûõ òàáëèö. Âíà÷àëå äëÿ êàæäîé ïåðåìåííîé xi è äëÿ êàæäîãî èç åå çíà÷åíèé av , ïðîñòàâëåíèå îáðàòíûõ îòìåòîê çàïîìèíàåò ñàìóþ ïåðâóþ ïðåäøåñòâóþùóþ ïåðåìåííóþ xp , òàêóþ, ÷òî òåêóùåå ÷àñòè÷íîå → ïðèñâàèâàíèå − a p êîíôëèêòóåò ñ xi = av . Ýòà èíôîðìàöèÿ ñîäåðæèòñÿ â òàáëèöå ñ ýëåìåíòàìè Mi,v . (Çàìåòèì, ÷òî ýòî ïðåäïîëàãàåò, ÷òî îãðàíè÷åíèÿ, âêëþ÷àþùèå áîëåå ðàííèå ïåðåìåííûå, ïðîòåñòèðîâàíû äî îãðàíè÷åíèé, âêëþ÷àþùèõ áîëåå ïîçäíèå ïåðåìåííûå). Åñëè xi = av ñîâìåñòíî ñî âñåìè ðàíåå âûïîëíåííûìè → ÷àñòè÷íûìè ïðèñâîåíèÿìè − a j , j < i, òî Mi,v = i. Íàïðèìåð, M10,2 = 4 îçíà÷àåò, − → → → ÷òî a 4 îêàçàëîñü íåñîâìåñòíûì ñ x10 = a2 ïîñëå çàäàíèÿ çíà÷åíèé è ÷òî − a 1, − a 2, − → a 3 íå êîíôëèêòîâàëè ñ x10 = a2 . Âòîðàÿ òàáëèöà ñ ýëåìåíòàìè lowi çàïîìèíàåò ñàìûé ïåðâûé ýëåìåíò, èçìåíèâøèé çíà÷åíèå ïîñëå òîãî, êàê ïåðåìåííîé xi â ïîñëåäíèé ðàç áûëî ïðèñâîåíî çíà÷åíèå. Ýòà èíôîðìàöèÿ âêëþ÷åíà â èñïîëüçîâàíèå íà êàæäîì øàãå ïîðîæäåíèÿ âåðøèíû. Åñëè Mi,v ìåíüøå lowi , òî àëãîðèòì çíàåò, ÷òî ïåðåìåííàÿ, íà êîòîðóþ óêàçûâàåò Mi,v , íå èçìåíÿëàñü è ÷òî xi = av → áóäåò ïðèâîäèòü ê òóïèêó ñíîâà ïðè ïðîâåðêå ñ − a Mi,v , òàê ÷òî íèêàêîé äàëüíåé25 øåé ïðîâåðêè â ýòîé âåðøèíå íà ñîâìåñòíîñòü íå òðåáóåòñÿ. Åñëè Mi,v íå ìåíüøå → lowi , òî xi = av ñîâìåñòíî ñ − a j äëÿ âñåõ j < lowi , è ýòè ïðîâåðêè ìîãóò áûòü ïðîïóùåíû. Ïðîñòàâëåíèå îáðàòíûõ îòìåòîê ìîæíî êîìáèíèðîâàòü ñ îáðàòíûì ïåðåõîäîì, óïðàâëÿåìûì êîíôëèêòàìè; â [248] ïðåäëîæåí ãèáðèäíûé àëãîðèòì, êîòîðûé ïðåâîñõîäèò ëþáîé èç ýòèõ îòäåëüíî âçÿòûõ ìåòîäîâ (÷òî ïîäòâåðæäàåòñÿ ôîðìàëüíûì äîêàçàòåëüñòâîì). Ïðèìåíåíèå ëîêàëüíîãî ïîèñêà ïðè ðåøåíèè çàäà÷ óäîâëåòâîðåíèÿ îãðàíè÷åíèé áûëî ïîêàçàíî (Kirkpatrick et al., 1983) â [243], ãäå îïèñàí ìåòîä ýìóëÿöèè îòæèãà. Ýâðèñòèêà ñ ìèíèìàëüíûìè êîíôëèêòàìè (minimal conicts) áûëà âïåðâûå ïðåäëîæåíà (Gu, 1989) â [200] è íåçàâèñèìî ðàçðàáîòàíà â [285] (Minton et al., 1992). Â (Sosic & Gu, 1994) [354] çàäà÷à ñ 3 000 000 ôåðçåé áûëà ðåøåíà áûñòðåå, ÷åì çà ìèíóòó. Ýòî ïðèâåëî ê ïåðåîöåíêå õàðàêòåðà è ðàñïðîñòðàíåííîñòè ¾ëåãêèõ¿ è ¾òðóäíûõ¿ çàäà÷. Â (Cheeseman et al., 1991) [104] èññëåäîâàëàñü ñëîæíîñòü çàäà÷ ÓÎ, ñôîðìèðîâàííûõ ñëó÷àéíûì îáðàçîì, è áûëî îáíàðóæåíî, ÷òî ïî÷òè âñå òàêèå çàäà÷è ÿâëÿþòñÿ ëèáî òðèâèàëüíî ëåãêèìè, ëèáî íå èìåþò ðåøåíèé. ¾Òðóäíûå¿ ýêçåìïëÿðû çàäà÷ âñòðå÷àþòñÿ, òîëüêî åñëè ïàðàìåòðû ãåíåðàòîðà çàäà÷ óñòàíàâëèâàþòñÿ â íåêîòîðîì óçêîì äèàïàçîíå, â ïðåäåëàõ êîòîðîãî ëèøü ïðèìåðíî ïîëîâèíà çàäà÷ ÿâëÿåòñÿ ðàçðåøèìîé. Äëÿ ðåøåíèÿ çàäà÷ ñ îãðàíè÷åíèÿìè âåñüìà óñïåøíî ìîæåò áûòü èñïîëüçîâàí ëîêàëüíûé ïîèñê íà îñíîâå ýâðèñòèêè ñ ìèíèìàëüíûìè êîíôëèêòàìè. Â ðàáîòå [88] ïîêàçàíî, êàê ìîæåò áûòü óñîâåðøåíñòâîâàíà ïðîãðàììà õðîíîëîãè÷åñêîãî ïîèñêà ñ âîçâðàòàìè íà ÿçûêå Prolog, äëÿ ðàáîòû ñ èíòåëëåêòóàëüíûì ïîèñêîì ñ âîçâðàòàìè. 3.2.4. Çàïîìèíàíèå êîíôëèêòîâ Â òóïèêîâîé âåðøèíå ïðèñâîåíèÿ ïåðåìåííûõ x1 , . . . , xi−1 îáðàçóþò êîíôëèêòíîå ìíîæåñòâî (ÊÌ) (conict set) ñ xi . Ýòà èíôîðìàöèÿ èñïîëüçóåòñÿ äëÿ çàïîìèíàíèÿ îäíîãî èëè íåñêîëüêèõ îãðàíè÷åíèé (èëè ìíîæåñòâ íåïåðñïåêòèâíûõ íàáîðîâ çíà÷åíèé (ÌÍÍÇ) (nogoods) ñì. íèæå ïàðàãðàô 3.2.5), êîòîðûå ïðè äàëüíåéøåì ïîèñêå íå äîïóñêàþò íåñîâìåñòíîñòåé, ïðèâîäÿùèõ ê ýòîé òóïèêîâîé âåðøèíå. Ðàññìîòðèì ðèñ. 6 è äåðåâî ïîèñêà, ïîñòðîåííîå ïîèñêîì ñ âîçâðàòàìè (ðèñ. 8). Â ïåðâîé òóïèêîâîé âåðøèíå ÊÌ {x1 = 4, x2 = 3, x3 = 1} êîíôëèêòóåò ñ x4 . Çàïîìèíàíèå ÊÌ ïîëåçíî ëèøü â ñëó÷àå, åñëè çàäà÷à ÓÎ áóäåò ïîâòîðíî ðåøàòüñÿ: òî÷íî òàêîå æå ìíîæåñòâî ïðèñâîåíèé íå âñòðåòèòñÿ âî âðåìÿ ýòîãî ïîèñêà. Îäíàêî, ÊÌ âîçìîæíî ñîäåðæèò ïîäìíîæåñòâà â êîíôëèêòå ñ xi . Ìèíèìàëüíûå ÊÌ (Bruynooghe, 1985) [87] íå ñîäåðæàò íèêàêèõ äðóãèõ ÊÌ è îòâå÷àþò çà òåêóùèé êîíôëèêò. Ïðîöåññ ïîèñêà âûèãðûâàåò, åñëè òàêèå ìèíèìàëüíûå ÊÌ áîëüøå íå âñòðå÷àþòñÿ. Çàïîìèíàíèå êîíôëèêòîâ ìîæåò ðàññìàòðèâàòüñÿ êàê îáó÷åíèå (ãëóáîêîå èëè ïîâåðõíîñòíîå) (Dechter, 1990) [133]: ïîëåçíàÿ èíôîðìàöèÿ, âñêðûòàÿ ðåøàòåëåì çàäà÷è, çàïîìèíàåòñÿ äëÿ äàëüíåéøåãî èñïîëüçîâàíèÿ. Ãëóáîêîå îáó÷åíèå íàõîäèò âñå ìèíèìàëüíûå êîíôëèêòû â òóïèêîâîé âåðøèíå, ðåàëèçîâàííûå â ìåòîäå ïîèñ26 êà ñ âîçâðàòàìè, óïðàâëÿåìîãî çàâèñèìîñòÿìè (Dependency Directed Backtracking) (Stallman & Sussman, 1977 [355]) è îáû÷íî èñïîëüçóåòñÿ â ñèñòåìàõ îáåñïå÷åíèÿ èñòèííîñòè (truth-maintenance systems) (Doyle, 1979 [152]). Ýòî òðåáóåò çàòðàò ïàìÿòè: êàæäàÿ òóïèêîâàÿ âåðøèíà ïðåäñòàâëÿåò íîâóþ âîçìîæíîñòü äîáàâëåíèÿ îãðàíè÷åíèé. Â ñâÿçè ñ ýòèì, â ïàìÿòè ñîõðàíÿåòñÿ áîëüøàÿ ÷àñòü ïðîñòðàíñòâà ïîèñêà. Ïîâåðõíîñòíîå îáó÷åíèå [133] îãðàíè÷èâàåò îáúåì ðàáîòû, âûïîëíÿåìîé â òóïèêîâîé âåðøèíå xi ñ ïîìîùüþ óäàëåíèÿ âñåõ ïðèñâîåíèé ïåðåìåííûõ èç êîíôëèêòíîãî ìíîæåñòâà, êîòîðûå íå ðåëåâàíòíû (ò.å. ñîâìåñòíû ñî âñåìè âîçìîæíûìè ïðèñâîåíèÿìè) ê xi . Ãðàôîâîå ïîâåðõíîñòíîå îáó÷åíèå èñïîëüçóåò ãðàô îãðàíè÷åíèé äëÿ (íåïîëíîé) ïðîâåðêè íà íåðåëåâàíòíîñòü: xh íå ðåëåâàíòåí ê xi , åñëè ïåðåìåííûå íå ñìåæíû â ãðàôå. Â âûøåîïèñàííîì ïðèìåðå, ýòî âûâîäèò {x1 = 4, x3 = 1}, îñòàâëÿÿ {x2 = 3}. Ýòîò êîíôëèêò çàïîìèíàåòñÿ ñ óäàëåíèåì çíàâ÷åíèÿ 3 èç D2 . Äîáàâëåíèå íîâûõ îãðàíè÷åíèé, îäíàêî, çàòðóäíÿåò ïîèñê èç-çà óâåëè÷åíèÿ ÷èñëà ïðîâåðîê îãðàíè÷åíèé. Çàïîìíåííûå êîíôëèêòû ìîãóò âåñüìà âåðîÿòíî îêàçàòüñÿ ïîëåçíûìè â ïðîöåññå ïîèñêà. Dechter (1990) [133] ââîäèò îãðàíè÷åíèå íà ÷èñëî çàïîìèíàåìûõ êîíôëèêòîâ. Îáó÷åíèå ïåðâîãî ïîðÿäêà çàïîìèíàåò ëèøü îäíó ïåðåìåííóþ. Îáó÷åíèå âòîðîãî ïîðÿäêà äîáàâëÿåò îãðàíè÷åíèÿ, ñîäåðæàùèå äî äâóõ ïåðåìåííûõ. Dechter (1990) [133] ïðîèçâåëà ñðàâíåíèå ïîâåðõíîñòíîãî è ãëóáîêîãî îáó÷åíèÿ ïåðâîãî è âòîðîãî ïîðÿäêà. Äëÿ ëåãêèõ çàäà÷ îáó÷åíèå äàåò íåáîëüøîé ýôôåêò, íî äëÿ áîëåå ñëîæíûõ çàäà÷ íàáëþäàåòñÿ áîëüøèé ýôôåêò íà ïðîèçâîäèòåëüíîñòü àëãîðèòìà. Íå âñåãäà âåðíî òî, ÷òî íàèáîëåå ñèëüíàÿ ôîðìà îáó÷åíèÿ äàåò ìàêñèìàëüíûé ýôôåêò, òàê êàê â íåêîòîðûõ ñëó÷àÿõ çàòðàòû ìîãóò ïðåâûøàòü âûãîäó. Bitner & Reingold âïåðâûå ïðåäëîæèëè èñïîëüçîâàòü ýâðèñòèêó MRV, íàçâàííóþ èìè ýâðèñòèêîé ñ íàèáîëåå îãðàíè÷åííîé ïåðåìåííîé (most-constrained-variable). Brelaz (1979) [82] èñïîëüçîâàë ñòåïåííóþ ýâðèñòèêó (degree heuristic) äëÿ óñòðàíåíèÿ íåîïðåäåëåííîñòè, âîçíèêàþùåé ïîñëå ïðèìåíåíèÿ ýâðèñòèêè MRV. Ïîëó÷åííûé â èòîãå àëãîðèòì, íåñìîòðÿ íà åãî ïðîñòîòó, äî íàñòîÿùåãî âðåìåíè îñòàåòñÿ íàèëó÷øèì ìåòîäîì ðàñêðàñêè ïðîèçâîëüíûõ ãðàôîâ â k öâåòîâ. Haralick & Elliot (1980) [203] ïðåäëîæèëè ýâðèñòèêó ñ íàèìåíåå îãðàíè÷èòåëüíûì çíà÷åíèåì. Â ñòàòüå [248] ( Kondrak & van Beek, 1997) ïðèâåäåí àíàëèòè÷åñêèé îáçîð àëãîðèòìîâ ïîèñêà ñ âîçâðàòàìè. 3.2.5. Ìíîæåñòâà íåïåðñïåêòèâíûõ íàáîðîâ çíà÷åíèé Îïðåäåëåíèå (Dechter, 1990) [133]. Ìíîæåñòâîì íåïåðñïåêòèâíûõ íàáîðîâ çíà÷åíèé (ÌÍÍÇ) (nogood) [238] Ng èëè êîíôëèêòíûì ìíîæåñòâîì (conict set) äëÿ çàäà÷è ÓÎ P íàçûâàåòñÿ ìíîæåñòâî íàáîðîâ çíà÷åíèé, êîòîðûå íå ìîãóò áûòü ÷àñòüþ íèêàêîãî ðåøåíèÿ çàäà÷è P . Ïî-äðóãîìó, ÌÍÍÇ ÷àñòè÷íîå ïðèñâîåíèå çíà÷åíèé ïåðåìåííûì, êîòîðîå íå ïðèâîäèò ê ðåøåíèþ [66]. Òàêèì îáðàçîì, ÌÍÍÇ íå ìîæåò áûòü äîïîëíåíî äî ðåøåíèÿ çàäà÷è ÓÎ. Íàïðèìåð, ëþáîå ïðèñâîåíèå çíà÷åíèé, íàðóøàþùåå îãðàíè÷åíèå, ÿâëÿåòñÿ ÌÍÍÇ. ÇÓÎ íå èìååò ðåøåíèÿ òîãäà è òîëüêî òîãäà, êîãäà 27 ïóñòîå ìíîæåñòâî ïðèñâîåíèé ÿâëÿåòñÿ ÌÍÍÇ. Ìíîæåñòâî ïðèñâîåíèé A ÿâëÿåòñÿ ÌÍÍÇ, åñëè èìååòñÿ íå ïðîñìîòðåííîå ðåøåíèå, ñîäåðæàùåå A. Ëþáîå ñóïåðìíîæåñòâî ÌÍÍÇ ÿâëÿåòñÿ òàêæå ÌÍÍÇ. Èçó÷åíèå ÌÍÍÇ ñòàíäàðòíûé ìåòîä óñîâåðøåíñòâîâàíèÿ ïîèñêà ñ âîçâðàòàìè (Dechter 1990) [133]. Çàìåòèì, ÷òî ìåòîä îáó÷àþùèõ ÌÍÍÇ áûë ïåðåíåñåí â ðåøàòåëè çàäà÷è âûïîëíèìîñòè SAT â (Bayardo & Schrag 1997) [54]. Â êîíòåêñòå çàäà÷ âûïîëíèìîñòè SAT ÌÍÍÇ ñîîòâåòñòâóåò êëàóçå (äèçúþíêòó), ïðè÷åì èñïîëüçîâàíèå îáó÷åíèÿ è èñïîëüçîâàíèÿ áîëüøîãî ÷èñëà êëàóç âî âðåìÿ ïîèñêà (Moskewicz et al. 2001) [291] îêàçàëîñü âåñüìà ýôôåêòèâíûì äëÿ ðåøåíèÿ çàäà÷ âûïîëíèìîñòè. Â òî æå âðåìÿ, ÌÍÍÇ íå èìåëè ñòîëü áîëüøîãî âëèÿíèÿ íà ðåøàòåëè ÇÓÎ. Òàê, áîëüøèíñòâî êîììåð÷åñêèõ ðåøàòåëåé çàäà÷ ÓÎ íå èñïîëüçóþò îáó÷åíèÿ ñ ÌÍÍÇ. Ïîä îáó÷åíèåì (learning), êàê è âûøå, çäåñü ïîíèìàåòñÿ çàïîìèíàíèå ïîòåíöèàëüíî ïîëåçíîé èíôîðìàöèè, êîòîðàÿ ñòàíîâèòñÿ èçâåñòíîé ðåøàòåëþ çàäà÷è âî âðåìÿ ïðîöåññà ðåøåíèÿ [138]. Àëãîðèòìû ïîèñêà ñ âîçâðàòàìè èìåþò ðàçëè÷íûå âîçìîæíîñòè äëÿ íàõîæäåíèÿ ÌÍÍÇ â ïðîöåññå ïîèñêà, ÷òî è èñïîëüçóåòñÿ ïðè ðàçðàáîòêå ðàçëè÷íûõ àëãîðèòìîâ ïîèñêà ñ âîçâðàòàìè. Â [309] ÌÍÍÇ, íàçûâàåìûå êîíôëèêòàìè, èñïîëüçóþòñÿ äëÿ áîëåå ìîùíûõ îáðàòíûõ ïåðåõîäîâ â àëãîðèòìå îáðàòíîãî ïåðåõîäà, óïðàâëÿåìîãî êîíôëèêòàìè. Â ðàáîòå [4] ÌÍÍÇ, íàçâàííûå òàì èñêëþ÷àþùèìè îáúÿñíåíèÿìè (eliminating explanations), èñïîëüçóþòñÿ äëÿ óïðàâëåíèÿ äèíàìè÷åñêèì ïåðåóïîðÿäî÷åíèåì âåòâåé äåðåâà ïîèñêà â àëãîðèòìå äèíàìè÷åñêîãî ïîèñêà ñ âîçâðàòàìè. Â [5] ÌÍÍÇ èçó÷àþòñÿ â òóïèêàõ è èñïîëüçóþòñÿ äëÿ óëó÷øåíèÿ ïîñëåäóþùåãî ïîèñêà. Ðàáîòû [42], [130], [286], [355], òàêæå ñîäåðæàò ïîëåçíûå èäåè î ñâÿçè ìåæäó ÌÍÍÇ è ïîèñêîì ñ âîçâðàòàìè. Îòìå÷åíî, ÷òî ïî ñóòè äåëà ëþáîé àëãîðèòì ïîèñêà ñ âîçâðàòàìè èñïîëüçóåò ÌÍÍÇ äëÿ óïðàâëåíèÿ ïîèñêîì, ÿâíî èëè íåÿâíî. Â ðàáîòå [238] ïðåäëîæåíî ïîíÿòèå îáîáùåííîãî ÌÍÍÇ, êîòîðîå ñóùåñòâåííî ïîâûøàåò ðåçóëüòàòèâíîñòü ïîèñêà ñ âîçâðàòàìè. 3.2.6. Äèíàìè÷åñêèé ïîèñê ñ âîçâðàòàìè Îáðàòíûé ïåðåõîä íà óðîâåíü âûøå óäàëÿåò ðåçóëüòàòû ïðîäåëàííîé ðåøàòåëåì ðàáîòû ïî îïðåäåëåíèþ ñîâìåñòíîãî ïðèñâîåíèÿ íà ïðåäûäóùåì óðîâíå. Âî ìíîãèõ ñëó÷àÿõ óðîâíè, ñ êîòîðûõ äåëàåòñÿ âîçâðàò, íå èìåþò íè÷åãî îáùåãî ñ íåñîâìåñòíîñòüþ, êîòîðóþ àëãîðèòì ïûòàåòñÿ ðàçðåøèòü (ïðèìåðîì ìîæåò ñëóæèòü ìå÷óùååñÿ ïîâåäåíèå, îïèñàííîå âûøå). Â ðåçóëüòàòå, ïîòåíöèàëüíî ïîëåçíàÿ èíôîðìàöèÿ áåññìûñëåííî âûáðàñûâàåòñÿ ïðè ïîèñêå ðåàëüíîé ïðè÷èíû íåñîâìåñòíîñòè. Â ìåòîäå äèíàìè÷åñêîãî ïîèñêà ñ âîçâðàòàìè (ÄÏÂ) (dynamic backtracking) (Ginsberg, 1993) [188], [189] ñîõðàíÿþòñÿ óñïåøíûå ÷àñòè÷íûå ïðèñâàèâàíèÿ èç ïîëó÷åííûõ ïîçäíåå ïîäìíîæåñòâ ïåðåìåííûõ, ïðè÷åì ÄÏÂ îáõîäèòñÿ áåç ôèêñèðîâàííîãî ïîðÿäêà ïðèñâîåíèÿ çíà÷åíèé, âìåñòî ýòîãî âûáèðàÿ ïåðåìåííûå äëÿ ïðèñâîåíèÿ çíà÷åíèé äèíàìè÷åñêèì îáðàçîì. ÄÏÂ ýôôåêòèâíûì îáðàçîì ìåíÿåò ïîðÿäîê ïðèñâàèâàíèÿ çíà÷åíèé òàê, ÷òî óðîâåíü, îòâåòñòâåííûé çà íåñîâìåñò28 íîñòü, ñòàíîâèòñÿ ñàìûì ãëóáîêèì óðîâíåì, îñòàâëÿÿ íå çàòðîíóòûìè âñå íåðåëåâàíòíûå ïðèñâîåíèÿ ïåðåìåííûõ. Äëÿ êàæäîãî dia ∈ Di ÄÏÂ ñîõðàíÿåò ìíîæåñòâî îáúÿñíåíèé ïðè÷èí ýëèìèíàöèè, â êîòîðîì çàïîìèíàåòñÿ ïðè÷èíà òîãî, ïî÷åìó dia â íàñòîÿùåå âðåìÿ èñêëþ÷åíî èç ðàññìîòðåíèÿ. Ýòè îáúÿñíåíèÿ îïðåäåëåíû â òåðìèíàõ òåêóùèõ ïðèñâîåíèé ðàíåå ðàññìîòðåííûì ïåðåìåííûì. Êîãäà íåîáõîäèì ïåðåõîä îò xi ê xh , ÄÏÂ ñîõðàíÿåò òåêóùåå ïðèñâîåíèå xi è âñå åãî ìíîæåñòâà îáúÿñíåíèé ïðè÷èí ýëèìèíàöèè, êðîìå òåõ, êîòîðûå âêëþ÷àþò xh (ïðèñâîåíèå êîòîðîãî äîëæíî ìåíÿòüñÿ). Èñïîëüçîâàíèå ìíîæåñòâ îáúÿñíåíèé ïðè÷èí ýëèìèíàöèè àíàëîãè÷íî ÌÍÍÇ, çàïîìèíàþùèõ ïðè÷èíû êîíôëèêòîâ. Îáúÿñíåíèÿ ïðè÷èí ýëèìèíàöèè õðàíÿòñÿ ëèøü äëÿ òåêóùèõ ïðèñâîåíèé äðóãèì ïåðåìåííûì; êàê òîëüêî ïðèñâàèâàíèå èçìåíèëîñü, âñå îáúÿñíåíèÿ, âêëþ÷àþùèå åãî, èñêëþ÷àþòñÿ. Èòàê, ÄÏÂ ïðåäñòàâëÿåò ñîáîé êîìïðîìèññ ìåæäó åìêîñòíîé è âðåìåííîé ñëîæíîñòÿìè. ÄÏÂ, îäíàêî,íå ðàáîòàåò âñåãäà ëó÷øå ñóùåñòâóþùèõ àëãîðèòìîâ. Ñðàâíèâàÿ ýôôåêòèâíîñòü ÄÏÂ â ñðàâíåíèè ñ äðóãèìè àëãîðèòìàìè, èñïîëüçóÿ ñòàíäàðòíûå ýâðèñòèêè (Baker, 1994) [42] ïîêàçàë, ÷òî ÄÏÂ ðàáîòàåò õóæå, ÷åì àëãîðèòì îáðàòíîãî ïåðåõîäà íà ìíîæèòåëü, ýêñïîíåíöèàëüíî çàâèñÿùèé îò ðàçìåðà çàäà÷è. Ïðîáëåìà çäåñü ñîñòîèò â òîì, ÷òî ïðèñâîåíèÿ ïåðåìåííûõ, êîòîðûå ÄÏÂ íå îòáðîñèë, ÷àñòî íå èìåþò íà îñíîâå ýâðèñòèê ïîäòâåðæäåíèÿ òîãî, ÷òî íàéäåíà íåñîâìåñòíîñòü. Õðàíåíèå òàêèõ ïðèñâîåíèé ïåðåìåííûõ óõóäøàåò ðåçóëüòàò ïðèìåíåíèÿ ýâðèñòèê. 3.3. Ðàñïðîñòðàíåíèå îãðàíè÷åíèé 3.3.1. Î ðàñïðîñòðàíåíèè îãðàíè÷åíèé Ñðåäè ðàçëè÷íûõ âèäîâ ìåòîäîâ äëÿ ïðåîäîëåíèÿ ïðèñóùåé çàäà÷àì ÓÎ òðóäíîé ðàçðåøèìîñòè âûäåëÿþòñÿ àëãîðèòìû ðàñïðîñòðàíåíèÿ îãðàíè÷åíèé [35]. Ïðè óäàëåíèè èçáûòî÷íûõ çíà÷åíèé èç äîìåíîâ ïåðåìåííûõ ðàçìåð ïðîñòðàíñòâà ðåøåíèé ñíèæàåòñÿ. Àëãîðèòìû óìåíüøåíèÿ ïðîñòðàíñòâà ðåøåíèé èñêëþ÷àþò çíà÷åíèÿ ñ ïîìîùüþ ðàñïðîñòðàíåíèÿ îãðàíè÷åíèé, íàçûâàåìîãî òàêæå ñóæåíèåì [116]. Óìåíüøåíèå ïðîñòðàíñòâà ðåøåíèé çàäà÷è ìîæåò áûòü âûïîëíåíî ëèáî îäíîêðàòíî â âèäå ýòàïà ïðåïðîöåññèíãà ïåðåä äðóãèì àëãîðèòìîì, ëèáî ìíîãîêðàòíî, øàã çà øàãîì, ïåðåìåæàÿñü ñ èññëåäîâàíèåì ïðîñòðàíñòâà ðåøåíèé ñ ïîìîùüþ àëãîðèòìà ïîèñêà. Â ïîñëåäíåì ñëó÷àå îòñåêàþòñÿ ïîäìíîæåñòâà ïðîñòðàíñòâà ðåøåíèé, ýêîíîìÿ âðåìÿ, êîòîðîå àëãîðèòì ïîèñêà çàòðàòèë áû íà ñèñòåìàòè÷åñêîå èññëåäîâàíèå îòáðîøåííûõ ýëåìåíòîâ. Åñëè â ðåçóëüòàòå óìåíüøåíèÿ ïðîñòðàíñòâà ðåøåíèé êàêîé-ëèáî äîìåí ñòàë ïóñòûì, îòñþäà ñëåäóåò, ÷òî ðàññìàòðèâàåìàÿ çàäà÷à ÓÎ íå èìååò ðåøåíèé. Îáúåì ðàñïðîñòðàíåíèÿ îãðàíè÷åíèé õàðàêòåðèçóåòñÿ óðîâíåì ñîâìåñòíîñòè çàäà÷è, ïîýòîìó ýòè àëãîðèòìû íàçûâàþòñÿ òàêæå àëãîðèòìàìè ñîâìåñòíîñòè. Ðàñïðîñòðàíåíèå îãðàíè÷åíèé [48] èìååò äàâíþþ òðàäèöèþ â ÓÎ è ÿâëÿåòñÿ î÷åíü îáùèì ïîíÿòèåì, êîòîðîå ïîÿâëÿëîñü ïîä ðàçíûìè íàçâàíèÿìè: ðåëàêñàöèÿ 29 îãðàíè÷åíèÿ (constraint relaxation), ôèëüòðóþùèå àëãîðèòìû (ltering algorithms), ñóæàþùèå àëãîðèòìû (narrowing algorithms), âûâîä îãðàíè÷åíèé (constraint inference), àëãîðèòìû óïðîùåíèÿ (simplication algorithms) è äð. Íèæå áóäóò ðàññìîòðåíû íàèáîëåå õîðîøî èçâåñòíûå è ÷àñòî èñïîëüçóåìûå àëãîðèòìû. Ðàñïðîñòðàíåíèå îãðàíè÷åíèé, ïðåäëîæåííîå âíà÷àëå äëÿ ðàçìåòêè èçîáðàæåíèé â ðàáîòàõ [289], [384], áûëî çàòåì ðàñïðîñòðàíåíî è ïðèñïîñîáëåíî ê ðåøåíèþ ìíîãèõ äðóãèõ çàäà÷, ïîðîäèâ áîëüøîå ÷èñëî àëãîðèòìîâ ðàñïðîñòðàíåíèÿ îãðàíè÷åíèé, ïðè ýòîì â ÷èñëå íàèáîëåå çíà÷èìûõ ñëåäóåò îòìåòèòü ðàáîòû [136], [140], [255], [262], [263], [265], [289]. Ñ ïîìîùüþ ðàñïðîñòðàíåíèÿ (èíôîðìàöèè) âîçäåéñòâèå ïðåäøåñòâóþùèõ ïðèñâîåíèé ìîæåò óìåíüøèòü ðàçìåð ïðîñòðàíñòâà ïîèñêà. Îñíîâíûì ìåòîäîì çäåñü ÿâëÿåòñÿ óñå÷åíèå äîìåíà, êîòîðîå óäàëÿåò íåêîòîðûå çíà÷åíèÿ èç äîìåíîâ ïåðåìåííûõ, êîòîðûå áóäóò ðàññìîòðåíû ïîçäíåå. Êàê òîëüêî ïðèñâîåíèå áóäåò îñóùåñòâëåíî, ìîæåò áûòü èñïîëüçîâàí àëãîðèòì ðàñïðîñòðàíåíèÿ îãðàíè÷åíèé äëÿ íàõîæäåíèÿ ïîñëåäñòâèé ïðåäøåñòâóþùèõ ïðèñâîåíèé íà äîìåíû ïåðåìåííûõ, ïðîñìàòðèâàåìûõ ïîçäíåå. Èññëåäîâàíèÿ ïî ðàñïðîñòðàíåíèþ îãðàíè÷åíèé è àëãîðèòìàì ñîâìåñòíîñòè áûëè èíèöèèðîâàíû â ðàáîòå Waltz [383], â êîòîðîé áûë ïðåäëîæåí àëãîðèòì ðàñïðîñòðàíåíèÿ îãðàíè÷åíèé (ïîçäíåå ïîëó÷èâøèé íàçâàíèå äóãîâîé ñîâìåñòíîñòè) äëÿ çàäà÷ ïîëèýäðàëüíîé ëèíåéíîé ðàçìåòêè äëÿ ñèñòåì êîìïüþòåðíîãî âèäåíèÿ. Îí ïîêàçàë, ÷òî äëÿ íåêîòîðûõ ìíîãîãðàííûõ ôèãóð äîñòàòî÷íî ïðèìåíèòü îñíîâíûå àëãîðèòìû ðàñïðîñòðàíåíèÿ äëÿ ðåøåíèÿ ýòèõ çàäà÷. Ïðåïðîöåññèíã èëè ïðåäâàðèòåëüíàÿ îáðàáîòêà â çàäà÷àõ ÓÎ ñîçäàåò ýêâèâàëåíòíóþ, ñ òåì æå ìíîæåñòâîì ðåøåíèé, íî áîëåå ïðîñòóþ çàäà÷ó. Ïîèñê íà äåðåâå ïîñëå ïðåïðîöåññèíãà ìîæåò ðåøèòü ïîëó÷åííóþ çàäà÷ó ñ ãîðàçäî ìåíüøèìè çàòðàòàìè. Èñõîäíàÿ çàäà÷à ÓÎ ïðåîáðàçóåòñÿ ñëåäóþùèì îáðàçîì. Âíà÷àëå äîìåíû ïåðåìåííûõ ìîãóò ôèëüòðîâàòüñÿ äëÿ óäàëåíèÿ ýëåìåíòîâ, êîòîðûå íå ìîãóò áûòü ÷àñòüþ íè îäíîãî ðåøåíèÿ. Äàëåå ìíîæåñòâî îãðàíè÷åíèé ìîæåò áûòü èçìåíåíî ñ öåëüþ íåäîïóùåíèÿ íåñîâìåñòíûõ êîìáèíàöèé ïðèñâîåíèé. Èçìåíåíèÿ ïðîèçâîäÿòñÿ ñ ïîìîùüþ ðàñïðîñòðàíåíèÿ îãðàíè÷åíèé: èñïîëüçóåòñÿ ëîêàëüíàÿ ãðóïïà îãðàíè÷åíèé äëÿ âûâîäà èíôîðìàöèè, êîòîðàÿ çàïîìèíàåòñÿ â âèäå èçìåíåííîãî äîìåíà îãðàíè÷åíèÿ èëè ïåðåìåííîé. Ýòî ëîêàëüíîå èçìåíåíèå ÿâëÿåòñÿ îñíîâîé äëÿ äàëüíåéøèõ âûâîäîâ, ïîýòîìó ðåçóëüòàò ëþáîãî èçìåíåíèÿ ïîñòåïåííî ðàñïðîñòðàíÿåòñÿ ïî âñåé çàäà÷å. Õîòÿ ñ öåëüþ ïðîñòîòû èçëîæåíèÿ ïðåäñòàâëåííûå çäåñü ìåòîäû îïèñàíû äëÿ áèíàðíûõ çàäà÷ ÓÎ, îíè ïðèìåíèìû ê n-àðíîé çàäà÷å ÓÎ, èñïîëüçóÿ, íàïðèìåð, äâîéñòâåííóþ ãðàôîâóþ èíòåðïðåòàöèþ. Ðàññìîòðèì äàëåå ïîíÿòèå ëîêàëüíîé ñîâìåñòíîñòè. 3.3.2. Ëîêàëüíàÿ ñîâìåñòíîñòü Îäíèì èç íàèáîëåå âàæíûõ ïîíÿòèé â ÓÎ ÿâëÿåòñÿ ïîíÿòèå ëîêàëüíîé ñîâìåñòíîñòè. Ëîêàëüíàÿ íåñîâìåñòíîñòü ýòî ïðèñâîåíèå çíà÷åíèé íåêîòîðûì ïåðåìåííûì, óäîâëåòâîðÿþùåå ñîîòâåòñòâóþùèì îãðàíè÷åíèÿì, íî êîòîðîå íå ìîæåò 30 áûòü ðàñøèðåíî íà îäíó èëè áîëåå ïåðåìåííûõ è ïîýòîìó íå ìîæåò áûòü ÷àñòüþ íèêàêîãî ðåøåíèÿ. Åñëè äëÿ ïîèñêà ðåøåíèÿ ÇÓÎ èñïîëüçóåòñÿ ïîèñê ñ âîçâðàòàìè, òî ïîäîáíàÿ íåñîâìåñòíîñòü ìîæåò áûòü ïðè÷èíîé ìíîãèõ òóïèêîâûõ âåðøèí è ìîæåò ñòàòü ïðè÷èíîé ìíîãèõ áåñïîëåçíûõ ïîïûòîê ïîèñêà. Ýòî ïðèâîäèò ê (a) îïðåäåëåíèþ óñëîâèé, õàðàêòåðèçóþùèõ óðîâåíü ëîêàëüíîé ñîâìåñòíîñòè çàäà÷è ÓÎ ([167], [262], [289]), (b) ðàçðàáîòêå àëãîðèòìîâ ðàñïðîñòðàíåíèÿ îãðàíè÷åíèé àëãîðèòìîâ, êîòîðûå âûíóæäàþò ýòè óðîâíè ê ëîêàëüíîé ñîâìåñòíîñòè ñ ïîìîùüþ óäàëåíèÿ íåñîâìåñòíîñòåé èç ÇÓÎ ([262], [289]), è (c) ðàçðàáîòêå ýôôåêòèâíûõ àëãîðèòìîâ ïîèñêà ñ âîçâðàòàìè äëÿ ïîèñêà ðåøåíèé ÇÓÎ, ïîääåðæèâàþùèõ óðîâåíü ëîêàëüíîé ñîâìåñòíîñòè âî âðåìÿ ïîèñêà ([128], [179], [203]). Â îáùåì, èìååòñÿ êîìïðîìèññ ìåæäó çàòðàòàìè íà ðàñïðîñòðàíåíèå îãðàíè÷åíèé, âûïîëíÿåìîå â êàæäîé âåðøèíå äåðåâà ïîèñêà è âåëè÷èíîé óñå÷åíèÿ äîìåíîâ. Îäèí èç ñïîñîáîâ óìåíüøåíèÿ çàòðàò íà ðàñïðîñòðàíåíèå îãðàíè÷åíèé ýòî ðàññìîòðåíèå áîëåå îãðàíè÷åííûõ ëîêàëüíûõ ñîâìåñòíîñòåé, îäíèì èç ïðèìåðîâ êîòîðûõ ñëóæèò ñîâìåñòíîñòü ãðàíèö. Ïðåäïîëîæèì, ÷òî äîìåíû ïåðåìåííûõ âåëèêè è óïîðÿäî÷åíû, ïðè÷åì îíè ìîãóò áûòü ïðåäñòàâëåíû èíòåðâàëàìè (çàäàâàåìûìè ìèíèìàëüíûì è ìàêñèìàëüíûì çíà÷åíèåì äîìåíà). Èñïîëüçóÿ ñîâìåñòíîñòü ãðàíèö, âìåñòî âûÿñíåíèÿ òîãî, ÷òî êàæäîå çíà÷åíèå äîìåíà èìååò ïîääåðæêó â îãðàíè÷åíèè, âûÿñíÿåòñÿ ëèøü òî, èìåþò ëè ìèíèìàëüíîå è ìàêñèìàëüíîå çíà÷åíèå ïîääåðæêó â îãðàíè÷åíèè. Õîòÿ ñîâìåñòíîñòü ãðàíèö ñëàáåå, ÷åì äóãîâàÿ ñîâìåñòíîñòü, îíà ïîëåçíà äëÿ àðèôìåòè÷åñêèõ îãðàíè÷åíèé è ãëîáàëüíûõ îãðàíè÷åíèé, òàê êàê îíà ìîæåò áûòü âûíóæäåíà áîëåå ýôôåêòèâíî. Çàäà÷à, âêëþ÷àþùàÿ áèíàðíûå îãðàíè÷åíèÿ, ïóòå-ñîâìåñòíà, åñëè êàæäàÿ ñîâìåñòíàÿ ïàðà çíà÷åíèé äâóõ ïåðåìåííûõ ìîæåò áûòü ðàñøèðåíà äî ëþáîé òðåòüåé ïåðåìåííîé. Äëÿ íåêîòîðûõ âèäîâ çàäà÷, ïîäîáíûõ òåìïîðàëüíûì îãðàíè÷åíèÿì, âîçìîæíî èìååò ñìûñë âûíóæäåíèå áîëåå ñèëüíûõ óðîâíåé ñîâìåñòíîñòè, ÷åì ïóòåâàÿ ñîâìåñòíîñòü [262]. 3.3.3. Âåðøèííàÿ è äóãîâàÿ ñîâìåñòíîñòü Ïðîñòåéøèì ìåòîäîì ïðåïðîöåññèíãà ÿâëÿåòñÿ âåðøèííàÿ ñîâìåñòíîñòü, êîòîðàÿ óäàëÿåò òå çíà÷åíèÿ èç äîìåíà ïåðåìåííîé, êîòîðûå íåñîâìåñòíû ñ óíàðíûìè îãðàíè÷åíèÿìè íà ñîîòâåòñòâóþùóþ ïåðåìåííóþ: åñëè c(xi ) óíàðíîå îãðàíè÷åíèå äëÿ âåðøèíû xi , òîãäà âåëè÷èíû, íå óäîâëåòâîðÿþùèå c(xi ), èñêëþ÷àþòñÿ èç äîìåíà Di . Ïðÿìîé àëãîðèòì âåðøèííîé ñîâìåñòíîñòè (node-consistency algorithm (NC)), êîòîðûé óäàëÿåò èçëèøíèå ýëåìåíòû äîìåíîâ ïåðåìåííûõ ñ ïîìîùüþ ïðîâåðêè äîìåíîâ îäíîãî çà äðóãèì, èìååò âðåìåííóþ ñëîæíîñòü O(dn), ãäå d ìàêñèìàëüíûé ðàçìåð äîìåíîâ, à n ÷èñëî ïåðåìåííûõ. Äóãîâàÿ ñîâìåñòíîñòü ñîñòîèò â ðàññìîòðåíèè êàæäîé äóãè9 , ñîîòâåòñòâóþùåé áèíàðíîìó îãðàíè÷åíèþ c(xi , xj ), è âûïîëíåíèè ñóæåíèÿ äîìåíîâ. Âåëè÷èíû, íå óäîâëåòâîðÿþùèå c(xi , xj ), èñêëþ÷àþòñÿ èç äîìåíîâ Di è Dj . Ïðîâåðêó ñîâìåñòíîñòè äóã ìîæíî èñïîëüçîâàòü ëèáî â êà÷åñòâå ýòàïà ïðåäâà9Â äàííîì ñëó÷àå òåðìèí ¾äóãà¿ îáîçíà÷àåò îðèåíòèðîâàííîå ðåáðî â ãðàôå îãðàíè÷åíèé. 31 ðèòåëüíîé îáðàáîòêè ïåðåä íà÷àëîì ïðîöåññà ïîèñêà, ëèáî â êà÷åñòâå ýòàïà ðàñïðîñòðàíåíèÿ îãðàíè÷åíèÿ (àíàëîãè÷íî ïðåäâàðèòåëüíîé ïðîâåðêå) ïîñëå êàæäîãî ïðèñâàèâàíèÿ âî âðåìÿ ïîèñêà. Ïîèñê êîìïðîìèññà ìåæäó çàòðàòàìè íà îáåñïå÷åíèå ñîâìåñòíîñòè è äîñòèãàåìûìè çà ñ÷åò ýòîãî ïðåèìóùåñòâàìè ñ òî÷êè çðåíèÿ ñîêðàùåíèÿ îáúåìà ïîèñêà ïðèâåë ê íîâûì àëãîðèòìàì. Òàê, (Haralick & Elliot, 1980) [203] ïðåäëîæèëè àëãîðèòì ïðåäâàðèòåëüíîé ïðîâåðêè (forward checking), îïèñàííûé â (McGregor, 1979) [273], à (Gaschnig, 1979) [181] ïðåäëîæèë ïðèìåíÿòü ïîëíóþ ïðîâåðêó ñîâìåñòíîñòè äóã ïîñëå ïðèñâàèâàíèÿ çíà÷åíèÿ êàæäîé ïåðåìåííîé; â äàëüíåéøåì ñîîòâåòñòâóþùèé àëãîðèòì â ðàáîòå (Sabin & Freuder, 1994) [330] ïîëó÷èë íàçâàíèå MAC (Maintaining Arc Consistency). Â ïîñëåäíåé ñòàòüå ïîêàçàíî, ÷òî ïðè ðåøåíèè áîëåå òðóäíûõ çàäà÷ ÓÎ ïîëíàÿ ïðîâåðêà ñîâìåñòíîñòè äóã âïîëíå îêóïàåòñÿ. Â ðàáîòå Mackworth & Freuder [265] áûëî çàìå÷åíî, ÷òî äóãîâàÿ ñîâìåñòíîñòü ìîæåò ðåøàòü äðåâîâèäíûå çàäà÷è ÓÎ. Äëÿ äàííîãî îãðàíè÷åíèÿ ãîâîðÿò, ÷òî çíà÷åíèå äëÿ ïåðåìåííîé â îãðàíè÷åíèè èìååò ïîääåðæêó (support), åñëè ñóùåñòâóþò òàêèå çíà÷åíèÿ äëÿ äðóãèõ ïåðåìåííûõ â ýòîì îãðàíè÷åíèè, ÷òî ýòî îãðàíè÷åíèå óäîâëåòâîðÿåòñÿ. Îãðàíè÷åíèå äóãî-ñîâìåñòíî, åñëè êàæäîå çíà÷åíèå èç äîìåíîâ ïåðåìåííûõ èìååò ïîääåðæêó. Äëÿ äîñòèæåíèÿ äóãîâîé ñîâìåñòíîñòè íà arc(i, j), äëÿ êàæäîãî dia ∈ Di , èùåòñÿ ýëåìåíò djb ∈ Dj , òàêîé ÷òî âûïîëíÿåòñÿ óñëîâèå äóãîâîé ñîâìåñòíîñòè; djb ïîääåðæèâàåò dia . Âñå ýëåìåíòû äîìåíà áåç ïîääåðæêè óäàëÿþòñÿ. Óäàëåíèå çíà÷åíèå áåç ïîääåðæêè íàçûâàåòñÿ óñå÷åíèåì (ñóæåíèåì) äîìåíîâ. Äëÿ îãðàíè÷åíèé, ñîäåðæàùèõ áîëåå äâóõ ïåðåìåííûõ, äóãîâàÿ ñîâìåñòíîñòü ÷àñòî íàçûâàåòñÿ ãèïåð-äóãîâîé ñîâìåñòíîñòüþ èëè îáîáùåííîé äóãîâîé ñîâìåñòíîñòüþ. Íàïðèìåð, ïóñòü äîìåíû ïåðåìåííûõ x è y {0, 1, 2} è ðàññìîòðèì îãðàíè÷åíèå x+y = 1. Âûíóæäåíèå äóãîâîé ñîâìåñòíîñòè íà ýòîì îãðàíè÷åíèè ñóçèëî áû äîìåíû îáåèõ ïåðåìåííûõ äî {0, 1}. Âåëè÷èíû, óäàëåííûå èç äîìåíîâ îáåèõ ïåðåìåííûõ, ëîêàëüíî íåñîâìåñòíû îíè íå ïðèíàäëåæàò íè îäíîìó íàáîðó ïðèñâîåíèé ïåðåìåííûõ, óäîâëåòâîðÿþùèõ îãðàíè÷åíèþ, è ïîýòîìó íå ìîãóò áûòü ÷àñòüþ êàêîãî-ëèáî ðåøåíèÿ âñåé çàäà÷è ÓÎ. Äëÿ âûíóæäåíèÿ äóãîâîé ñîâìåñòíîñòè â çàäà÷å ÓÎ òðåáóåòñÿ ïîâòîðíî óäàëÿòü âåëè÷èíû èç äîìåíîâ, ïîêà íå äîñòèãíåì óñòîé÷èâîé òî÷êè. Îñíîâûâàÿñü íà ðàáîòàõ [383], [289], Mackworth [262] âûäåëèë òðè âèäà ëîêàëüíîé ñîâìåñòíîñòè: âåðøèííóþ, äóãîâóþ è ïóòåâóþ ñîâìåñòíîñòü. Èì áûëè ïðåäëîæåíû àëãîðèòìû (AC-1, AC-2, AC-3) äëÿ âûíóæäåíèÿ äóãîâîé ñîâìåñòíîñòè, à òàêæå àëãîðèòìû PC-1, PC-2, PC-3 äëÿ âûíóæäåíèÿ ïóòåâîé ñîâìåñòíîñòè. Âû÷èñëèòåëüíàÿ ñëîæíîñòü óêàçàííûõ àëãîðèòìîâ áûëà èññëåäîâàíà äàëåå â ðàáîòå Mackworth & Freuder [265], ïðè÷åì òàì æå áûëî òàêæå çàìå÷åíî, ÷òî äóãîâàÿ ñîâìåñòíîñòü ìîæåò ðåøàòü äðåâîâèäíûå çàäà÷è ÓÎ. Àëãîðèòìû âûíóæäåíèÿ äóãîâîé ñîâìåñòíîñòè èíòåíñèâíî èññëåäîâàëèñü è ñîâåðøåíñòâîâàëèñü (ñì. [65], [262] è ññûëêè íà ëèòåðàòóðó â ýòèõ ðàáîòàõ). Îïòèìàëüíûé àëãîðèòì äëÿ ïðîèçâîëüíîãî îãðàíè÷åíèÿ õàðàêòåðèçóåòñÿ âðåìåííîé 32 ñëîæíîñòüþ â õóäøåì ñëó÷àå, ðàâíîé O(r · dr ), ãäå r àðíîñòü îãðàíè÷åíèÿ, à d ðàçìåð äîìåíîâ ïåðåìåííûõ [288]. Ïðîñòåéøèì ìåòîäîì äîñòèæåíèÿ ãëîáàëüíîé äóãîâîé ñîâìåñòíîñòè ÿâëÿåòñÿ ïîâòîðåíèå ïðîõîäîâ ïî ïðîâåðêå êàæäîé äóãè çàäà÷è äî òåõ ïîð, ïîêà íå áóäåò íèêàêèõ èçìåíåíèé ãðàôà îãðàíè÷åíèé ïîñëå î÷åðåäíîãî ïðîõîäà. Ýòîò ïîäõîä ðåàëèçîâàí â âèäå àëãîðèòìà AC-1 (èç (Machworth, 1977) [262]. Âñëåä çà èñòîðè÷åñêè ïåðâûì àëãîðèòìîì AC-1 áûëè ðàçðàáîòàíû àëãîðèòìû AC-2, ..., AC-7, êàæäûé èç êîòîðûõ óëó÷øàåò õðàíåíèå èíôîðìàöèè îá èçìåíåííûõ äîìåíàõ è óïðàâëåíèå èòåðàöèÿìè. Âû÷èñëèòåëüíàÿ ñëîæíîñòü àëãîðèòìîâ AC-1, AC-2, AC-3 áûëà èññëåäîâàíà äàëåå â ðàáîòå Mackworth & Freuder [265]. Âðåìåííàÿ ñëîæíîñòü àëãîðèòìà AC3 ñîñòàâëÿåò O(d3 e), åìêîñòíàÿ ñëîæíîñòü O(e + nd), ãäå e ÷èñëî áèíàðíûõ îãðàíè÷åíèé. Àëãîðèòì AC-4 [287] óëó÷øàåò AC-3 è èìååò âðåìåííóþ è åìêîñòíóþ ñëîæíîñòü O(d2 e). Äðóãèì óëó÷øåíèåì àëãîðèòìà AC-3 ÿâëÿåòñÿ àëãîðèòì AC-5 [369], èñïîëüçóþùèé ñåìàíòèêó ñïåöèàëüíûõ âèäîâ îãðàíè÷åíèé. Àëãîðèòì AC-7 èñïîëüçóåò ñèììåòðèþ áèíàðíûõ îãðàíè÷åíèé. Ñëîæíîñòü ïðîâåðêè ñîâìåñòíîñòè äóã ìîæíî ïðîàíàëèçèðîâàòü ñëåäóþùèì îáðàçîì: ëþáàÿ áèíàðíàÿ çàäà÷à ÓÎ èìååò ñàìîå áîëüøåå O(n2 ) äóã; êàæäàÿ äóãà (Xi , Xj ) ìîæåò áûòü ¾âíåñåíà â ïîâåñòêó äíÿ¿ òîëüêî d ðàç, ïîñêîëüêó îáëàñòü îïðåäåëåíèÿ Xi èìååò ñàìîå áîëüøåå d çíà÷åíèé, äîñòóïíûõ äëÿ óäàëåíèÿ; ïðîâåðêà ñîâìåñòíîñòè ëþáîé äóãè ìîæåò áûòü âûïîëíåíà çà âðåìÿ O(d2 ), ïîýòîìó â íàèõóäøåì ñëó÷àå çàòðàòû âðåìåíè ñîñòàâëÿþò O(n2 · d3 ). 3.3.4. Íàïðàâëåííàÿ äóãîâàÿ ñîâìåñòíîñòü Îïðåäåëåíèå. Çàäà÷à ÓÎ ÿâëÿåòñÿ íàïðàâëåííî-äóãî-ñîâìåñòíîé ïî îòíîøåíèþ ê óïîðÿäî÷åíèþ h = {x1 , . . . , xn }, òîãäà è òîëüêî òîãäà, êîãäà êàæäàÿ ïåðåìåííàÿ xi äóãî-ñîâìåñòíà ïî îòíîøåíèþ ê êàæäîé ïåðåìåííîé xj , òàêîé ÷òî i ≤ j . Ïðîöåäóðà ðàñïðîñòðàíåíèÿ îãðàíè÷åíèé DAC ðàáîòàåò ñ çàäà÷åé ÓÎ (V, D, C) è óïîðÿäî÷åíèåì h = {x1 , . . . , xn } äëÿ çàäà÷è ÓÎ. Ïðîöåäóðà îáðàáàòûâàåò ïåðåìåííûå â îáðàòíîì ïîðÿäêå (xn , . . . , x1 ). Äëÿ êàæäîé ïåðåìåííîé x îíà ïðîâåðÿåò ðîäèòåëåé x10 . Ïðè îáðàáîòêå xi , ðàññìàòðèâàþòñÿ âñå áèíàðíûå îãðàíè÷åíèÿ, èíöèäåíòíûå xi , Rki , ïðè k ≤ i, è ñóæàþòñÿ ñîîòâåòñòâóþùèå äîìåíû Dk ïåðåìåííûõ xk . Ïîñëå âûïîëíåíèÿ ýòîé ïðåäâàðèòåëüíîé îáðàáîòêè âûÿñíÿåòñÿ íàïðàâëåííàÿ äóãîâàÿ ñîâìåñòíîñòü [140]. Ýòî îçíà÷àåò, ÷òî äëÿ ëþáîãî çíà÷åíèÿ äîìåíà ïåðåìåííîé è äëÿ ëþáîãî ñûíà ýòîé ïåðåìåííîé íàéäåòñÿ ïî êðàéíåé ìåðå îäíî çíà÷åíèå èç äîìåíà ñûíà, ñîâìåñòíîå ñî çíà÷åíèåì ðîäèòåëÿ. Ãëîáàëüíî ñîâìåñòíûå ÇÓÎ îáëàäàþò ñâîéñòâîì, ÷òî ëþáîå ñîâìåñòíîå ïðèñâîåíèå çíà÷åíèé ïîäìíîæåñòâó ïåðåìåííûõ ìîæåò áûòü ðàñøèðåíî äî ñîâìåñòíîãî ïðèñâîåíèÿ çíà÷åíèé âñåì ïåðåìåííûì áåç ïîëó÷åíèÿ òóïèêîâ. 10 ïåðåìåííûå, ñîåäèíåííûå ñ x, êîòîðûå ïðåäøåñòâóþò x â óïîðÿäî÷åíèè. 33 3.3.5. Ïóòåâàÿ ñîâìåñòíîñòü Äëÿ áèíàðíûõ çàäà÷ ÓÎ èìååòñÿ åùå îäèí âèä ñîâìåñòíîñòè ïóòåâàÿ ñîâìåñòíîñòü, êîòîðóþ Montanari ïðåäëîæèë â [289]. Îïðåäåëåíèå Áèíàðíàÿ çàäà÷à ÓÎ ÿâëÿåòñÿ ïóòå-ñîâìåñòíîé (path-consistent), åñëè äëÿ ëþáîãî ïóòè â åå ãðàôå îãðàíè÷åíèé ñïðàâåäëèâî ñëåäóþùåå: åñëè ïðèñâîåíèÿ çíà÷åíèé íà÷àëüíîé è êîíå÷íîé ïåðåìåííîé ïóòè ñîâìåñòíû, òî ýòî ìîæåò áûòü ðàñøèðåíî íà ñîâìåñòíîå ÷àñòè÷íîå ïðèñâîåíèå çíà÷åíèé îñòàâøèìñÿ ïåðåìåííûõ íà ïóòè. Àíàëîãè÷íî ñåìåéñòâó àëãîðèòìîâ äóãîâîé ñîâìåñòíîñòè, èìååòñÿ ñåìåéñòâî àëãîðèòìîâ äîñòèæåíèÿ ïóòåâîé ñîâìåñòíîñòè áèíàðíîé çàäà÷è ÓÎ. Ýòè àëãîðèòìû òàêæå ãàðàíòèðóþò âåðøèííóþ è äóãîâóþ ñîâìåñòíîñòü. Íàèëó÷øèé èç ýòèõ àëãîðèòìîâ PC-4, àíàëîãè÷íûé AC-4, õàðàêòåðèçóåòñÿ âðåìåííîé è åìêîñòíîé ñëîæíîñòüþ O(d3 n3 ). Ëîêàëüíàÿ ïóòåâàÿ ñîâìåñòíîñòü âûïîëíÿåòñÿ, åñëè äëÿ äàííîãî ïóòè äëèíû l ÷åðåç âåðøèíû (x1 , . . . , xl ), äëÿ âñåõ çíà÷åíèé d1a ∈ D1 è dlb ∈ Dl òàêèõ, ÷òî âûïîëíÿþòñÿ óíàðíûå îãðàíè÷åíèÿ c(x1 ), c(xl ), è áèíàðíûå îãðàíè÷åíèÿ c(x1 , xl ), èìååòñÿ ïîñëåäîâàòåëüíîñòü çíà÷åíèé: d2c ∈ D2 , . . . , dl−1c ∈ Dl , òàêàÿ, ÷òî c(x2 ), . . . , c(xl−1 ), c(x1 , x2 ), c(x2 , x3 ), . . . , c(xl−1 , xl ) âûïîëíÿþòñÿ. Íà ðèñ. 10 ïîêàçàí ïðèìåð äëÿ ïóòè (x1 , x2 , x3 ). Íà÷àëüíîå ñîñòî- x1, D1={4, 5} x1, D1={4, 5} > > a) b) x2, D2={3, 4} x2, D2={3, 4} > > x3, D3={2, 3} x3, D3={2, 3} Ɋɚɡɪɟɲɟɧɧɵɟ ɩɚɪɵ {(4, 2), (5, 2), (5, 3)} Ðèñ. 10. Ýôôåêò âûíóæäåíèÿ ïóòåâîé ñîâìåñòíîñòè [283]. ÿíèå ïîêàçàíî íà ðèñ. 10 a. Äîáàâëåíî íîâîå îãðàíè÷åíèå (ðèñ. 10 b), êîòîðîå íå ðàçðåøàåò ïàðó ïðèñâîåíèé x1 = 4, x3 = 3: äëÿ ýòèõ ïðèñâîåíèé íå ñóùåñòâóåò ïðèñâîåíèÿ äëÿ x2 , óäîâëåòâîðÿþùåãî áèíàðíûì îãðàíè÷åíèÿì c(x1 , x2 ), c(x2 , x3 ). Âûíóæäåíèå äóãîâîé ñîâìåñòíîñòè íå ðàçðåøàåò ýòîé ïàðû ïðèñâîåíèé. 34 Ãëîáàëüíàÿ ïóòåâàÿ ñîâìåñòíîñòü âûïîëíÿåòñÿ, åñëè ëþáàÿ ïàðà ïðèñâîåíèé çíà÷åíèé, êîòîðûå ñîâìåñòíû ñ ïðÿìûì îãðàíè÷åíèåì ìåæäó äâóìÿ ïåðåìåííûìè, òàêæå ñîâìåñòíà ñî âñåìè ïóòÿìè ìåæäó äâóìÿ âåðøèíàìè. Åñëè ëþáîé ïóòü äëèíû 2 â ïîëíîì ãðàôå îãðàíè÷åíèé ÿâëÿåòñÿ ïóòå-ñîâìåñòíûì, òî ïóòåâàÿ ñîâìåñòíîñòü âûïîëíÿåòñÿ ãëîáàëüíî (Montanari, 1974) [289]. Ïóòåâàÿ ñîâìåñòíîñòü âûíóæäàåòñÿ ñ ïîìîùüþ èçìåíåíèÿ ïðÿìîãî îãðàíè÷åíèÿ ìåæäó íà÷àëüíîé è êîíå÷íîé òî÷êîé ïóòè äëÿ òîãî, ÷òîáû çàïðåòèòü âñå ïàðû íàçíà÷åíèé, êîòîðûå íàðóøàþò òðåáóåìîå óñëîâèå. Îãðàíè÷åíèÿ ìîãóò áûòü îáíîâëåíû ïóòåì äîáàâëåíèÿ íîâîãî ÿâíîãî îãðàíè÷åíèÿ ê çàäà÷å (ñì. ðèñ. 10). Ïåðâûé àëãîðèòì ïóòåâîé ñîâìåñòíîñòè áûë ïðåäëîæåí â (Montanari, 1974) [289], ïðè÷åì îí ýêâèâàëåíòåí àëãîðèòìó PC-1 (Mackworth, 1977) [262]. Mackworth [262] óñîâåðøåíñòâîâàë ýôôåêòèâíîñòü PC-1 è ïðåäëîæèë PC-2. PC-3 (Mohr & Henderson, 1986) [287] óëó÷øàåò ýôôåêòèâíîñòü êàê â AC-4. PC-3 íå ïîëíîñòüþ êîððåêòåí. PC-4 (Han & Lee, 1988) [202] ýòî êîððåêòíàÿ âåðñèÿ. Äàëüíåéøèå óñîâåðøåíñòâîâàíèÿ àëãîðèòìà ïóòåâîé ñîâìåñòíîñòè ìîãóò áûòü íàéäåíû â (Chmeiss & Jegou, 1996) [106]. Mohr & Henderson [287] ïðåäëîæèëè îïòèìàëüíûå àëãîðèòìû äëÿ äóãîâîé è ïóòåâîé ñîâìåñòíîñòè, ñîîòâåòñòâåííî AC-4 è PC-4, êîòîðûå áûëè îñíîâàíû íà ðàáîòå Lauriere [258]. 3.3.6. k -ñîâìåñòíîñòü Äóãîâàÿ ñîâìåñòíîñòü ìîæåò òàêæå ïîíèìàòüñÿ êàê èíôîðìàöèÿ î òîì, íàñêîëüêî äàëåêî ÷àñòè÷íîå ðåøåíèå ìîæåò âñåãäà áûòü ðàñøèðåíî. À èìåííî, ëþáîå ÷àñòè÷íîå ðåøåíèå, ñîäåðæàùåå ëèøü îäíó çàôèêñèðîâàííóþ ïåðåìåííóþ, ìîæåò áûòü ðàñøèðåíî ñ ïîìîùüþ ôèêñàöèè ëþáîé äðóãîé ïåðåìåííîé íà ñîîòâåòñòâóþùèì îáðàçîì âûáðàííîì çíà÷åíèè. Ïðèìåíåíèå òîãî æå ïðèíöèïà äëÿ áîëüøåãî ÷èñëà ïåðåìåííûõ ïðèâîäèò ê ïîíÿòèþ k -ñîâìåñòíîñòè, êîòîðîå ââåë Freuder [167], [168], [169], èññëåäîâàâøèé è åãî ñâÿçü ñî ñëîæíîñòüþ ðåøåíèÿ çàäà÷ ÓÎ. Ìîæíî îïðåäåëèòü k -ñîâìåñòíîñòü, ðàññìàòðèâàÿ îãðàíè÷åíèÿ ñ k ïåðåìåííûìè (äëÿ k = 3 ïóòåâàÿ ñîâìåñòíîñòü, äëÿ k > 3 ãèïåð-äóãîâàÿ ñîâìåñòíîñòü). Îïðåäåëåíèå (k -ñîâìåñòíîñòü) Çàäà÷à ÓÎ k -ñîâìåñòíà, åñëè ëþáîé ñîâìåñòíûé êîðòåæ çíà÷åíèé ëþáûõ k − 1 ïåðåìåííûõ ìîæåò áûòü ðàñøèðåí ïóòåì çàäàíèÿ çíà÷åíèÿ ëþáîé îäíîé èç îñòàâøèõñÿ ïåðåìåííûõ. Çàäà÷à ÓÎ ñòðîãî k -ñîâìåñòíà òîãäà è òîëüêî òîãäà, êîãäà îíà j -ñîâìåñòíà äëÿ âñåõ j ≤ k . Ñòðîãî n-ñîâìåñòíàÿ ñåòü, ãäå n ÷èñëî ïåðåìåííûõ ÇÓÎ, íàçûâàåòñÿ ãëîáàëüíî ñîâìåñòíîé. Äóãîâàÿ è ïóòåâàÿ ñîâìåñòíîñòü ñîîòâåòñòâóþò 2- è 3-ñîâìåñòíîñòè. Çàìåòèì, ÷òî k -ñîâìåñòíàÿ ÇÓÎ íå îáÿçàòåëüíî ðàçðåøèìà, è íàîáîðîò, ðàçðåøèìîñòü çàäà÷è íå âëå÷åò çà ñîáîé ñîâìåñòíîñòè ëþáîãî óðîâíÿ, à íå òîëüêî 1-ñîâìåñòíîñòü. 35 3.4. Âû÷èñëèòåëüíàÿ ñëîæíîñòü è ïîèñê ëåãêî ðàçðåøèìûõ çàäà÷ óäîâëåòâîðåíèÿ îãðàíè÷åíèé Îäíîé èç íàèáîëåå ôóíäàìåíòàëüíûõ ñëîæíûõ ïðîáëåì ÓÎ è ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ ÿâëÿåòñÿ èçó÷åíèå âû÷èñëèòåëüíîé ñëîæíîñòè çàäà÷ ÓÎ. Â [262] áûëî ïîêàçàíî, ÷òî êëàññ âñåõ çàäà÷ ÓÎ ÿâëÿåòñÿ NP-òðóäíûì, òàê ÷òî âðÿä ëè ñóùåñòâóþò ýôôåêòèâíûå (ïîëèíîìèàëüíûå) àëãîðèòìû îáùåãî íàçíà÷åíèÿ äëÿ ðåøåíèÿ âñåõ âèäîâ çàäà÷ ÓÎ. Öåëüþ ìíîãèõ èññëåäîâàíèé áûëà èäåíòèôèêàöèÿ êëàññîâ ëåãêî ðàçðåøèìûõ çàäà÷ ÓÎ, êîòîðûå ìîãóò áûòü ðåøåíû çà ïîëèíîìèàëüíîå âðåìÿ (ñì. îáçîðû [255], [263], [364]). Ïîäîáíûå êëàññû çàäà÷ îáû÷íî îïèñûâàþòñÿ ëèáî äåðåâüÿìè, äðåâîâèäíûìè ãðàôîâûìè ñòðóêòóðàìè ( [141], [170], [326]) ëèáî îïðåäåëåííîé êîìáèíàöèåé àëãåáðàè÷åñêèõ îïåðàòîðîâ [227]. Äëÿ íåêîòîðûõ êëàññîâ çàäà÷ ÓÎ áûëî ïîêàçàíî, ÷òî èñïîëüçîâàíèå ðàñïðîñòðàíåíèÿ îãðàíè÷åíèé ãàðàíòèðóåò ïîèñê áåç âîçâðàòîâ (backtrack-free search) [136]. Â ðåçóëüòàòå ñðàâíåíèÿ ìíîãèõ ìåòîäîâ ãðàôîâîé äåêîìïîçèöèè ïîêàçàíî, ÷òî íàèëó÷øèìè ÿâëÿþòñÿ ìåòîäû, îñíîâàííûå íà èñïîëüçîâàíèè ïîíÿòèÿ ãèïåðãðàôîâîé äåêîìïîçèöèè (hypertree decomposition), ðàçðàáîòàííîãî â òåîðèè áàç äàííûõ [195]. Ôîðìàëèçàöèÿ ïîíÿòèÿ ÿçûêà îãðàíè÷åíèé (constraint language)11 ìîæåò äàòü èíôîðìàöèþ, ÿâëÿåòñÿ ëè çàäà÷à ÓÎ, âûðàæåííàÿ ñ ïîìîùüþ îïðåäåëåííîãî ÿçûêà îãðàíè÷åíèé, ëåãêî ðàçðåøèìîé [111], [218], [226], [227], [229], [230], [231], [232]. Õîòÿ îáùàÿ çàäà÷à ÓÎ ÿâëÿåòñÿ NP-ïîëíîé, âî ìíîãèõ ïðèëîæåíèÿõ âîçíèêàþùèå çàäà÷è ÓÎ èìåþò ñïåöèàëüíûé âèä, ÷òî ïîçâîëÿåò ðàçðàáîòàòü áîëåå ýôôåêòèâíûå àëãîðèòìû èõ ðåøåíèÿ [90], [112], [260], [310]. Èçâåñòíî, ÷òî áèíàðíûå çàäà÷è ÓÎ ñ äðåâîâèäíîé ñòðóêòóðîé ëåãêî ðàçðåøèìû ([324], ãëàâà 7). Ïîäêëàññ îáùåé çàäà÷è ÓÎ, êîòîðûå ìîãóò ðåøåíû çà ïîëèíîìèàëüíîå âðåìÿ, à òàêæå ðàñïîçíàí çà ïîëèíîìèàëüíîå âðåìÿ, íàçûâàåòñÿ ëåãêî ðàçðåøèìûì ïîäêëàññîì. Ñóùåñòâóþò äâà îñíîâíûõ íàïðàâëåíèÿ ïîèñêà ñïåöèàëüíîé ñòðóêòóðû çàäà÷ ÓÎ, êîòîðûå ìîãóò áûòü ëåãêî ðàçðåøèìûìè: âûäåëåíèå ñòðóêòóðíûõ ñâîéñòâ çàäà÷ ÓÎ (îïèñûâàåìûõ òîïîëîãèåé ãðàôà îãðàíè÷åíèé), ãàðàíòèðóþùèõ ëåãêóþ ðàçðåøèìîñòü íåçàâèñèìî îò òîãî, êàêèå òèïû îãðàíè÷åíèé âêëþ÷åíû â ðàññìîòðåíèå [115], [140], ëèáî âûäåëåíèå òèïîâ îãðàíè÷åíèé, äîñòàòî÷íî îãðàíè÷èòåëüíûõ äëÿ ãàðàíòèðîâàíèÿ ëåãêîé ðàçðåøèìîñòè íåçàâèñèìî îò òîãî, êàê îíè ñî÷åòàþòñÿ [90], [161]. Áëàãîäàðÿ èññëåäîâàíèÿì â ýòîé îáëàñòè áûëè âûäåëåíû ìíîãèå êëàññû ëåãêî ðàçðåøèìûõ çàäà÷ ÓÎ, îñíîâûâàÿñü íà ñòðóêòóðíûõ ñâîéñòâàõ (ñì. [324]). Ïîäîáíûå êëàññû ìîãóò âñòðåòèòüñÿ, â ÷àñòíîñòè, â ñëó÷àå, êîãäà èìååòñÿ íåêîòîðàÿ ñïåöèàëüíàÿ ñòðóêòóðà âçàèìîñâÿçè îãðàíè÷åíèé (èõ ïåðåñå÷åíèé). Íàèáîëåå åñòåñòâåííûì äëÿ èçó÷åíèÿ âçàèìîñâÿçåé îãðàíè÷åíèé ÿâëÿåòñÿ àïïàðàò ãèïåðãðàôîâ. Äëÿ êëàññà çàäà÷ ÓÎ, ó êîòîðûõ àðíîñòü îãðàíè÷åíèé îãðàíè÷åíà íåêîòîðîé ôèêñèðîâàííîé êîíñòàíòîé (òàêèõ, êàê áèíàðíûå çàäà÷è ÓÎ) áûëî ïîêàçàíî, ÷òî 11 ßçûê îãðàíè÷åíèé ÇÓÎ ýòî ìíîæåñòâî îòíîøåíèé â îãðàíè÷åíèÿõ ÇÓÎ [113] 36 (ïðè âûïîëíåíèè íåêîòîðûõ äîïîëíèòåëüíûõ îãðàíè÷åíèé) åäèíñòâåííûì êëàññîì ñòðóêòóð, ãàðàíòèðóþùèì ëåãêóþ ðàçðåøèìîñòü, ÿâëÿåòñÿ êëàññ ñòðóêòóð ñ îãðàíè÷åííîé äðåâîâèäíîé øèðèíîé [198], [209], [23]. Âòîðîå íàïðàâëåíèå èññëåäîâàíèé ñîñòîèò â ðàññìîòðåíèè çàäà÷ ÓÎ ñ áîëåå îãðàíè÷åííûìè òèïàìè îãðàíè÷åíèé. Åñòåñòâåííûì ïîäõîäîì èçó÷åíèÿ òèïîâ îãðàíè÷åíèé ÿâëÿåòñÿ ìàòåìàòè÷åñêàÿ òåîðèÿ îòíîøåíèé è ñîîòâåòñòâóþùèå èì àëãåáðû, ïðèìåíåíèå êîòîðûõ ïîçâîëèëî ïîëó÷èòü õàðàêòåðèçàöèè òîãî, êàêèå âèäû îãðàíè÷åíèé îáåñïå÷èâàþò ëåãêóþ ðàçðåøèìîñòü íåçàâèñèìî îò èõ âçàèìîñâÿçè. Ìíîæåñòâî òèïîâ îãðàíè÷åíèé ñ ýòèì ñâîéñòâîì íàçûâàåòñÿ ëåãêî ðàçðåøèìûì ÿçûêîì îãðàíè÷åíèé. Â îáùåì áûëî ïîêàçàíî, ÷òî ëþáîé ëåãêî ðàçðåøèìûé ÿçûê îãðàíè÷åíèé äîëæåí èìåòü îïðåäåëåííûå àëãåáðàè÷åñêèå ñâîéñòâà, íàçûâàåìûå ïîëèìîðôèçìàìè [225]. Ôóíäàìåíòàëüíîé îòêðûòîé ïðîáëåìîé â ýòîé îáëàñòè ÿâëÿåòñÿ ÷åòêîå îïðåäåëåíèå òèïîâ îãðàíè÷åíèé â çàäà÷àõ ÓÎ, êîòîðûå ìîãóò áûòü ðåøåíû çà ïîëèíîìèàëüíîå âðåìÿ. Ýòà ïðîáëåìà âàæíà ñ òåîðåòè÷åñêîé òî÷êè çðåíèÿ, òàê êàê îíà ïîìîãàåò âûÿñíèòü ãðàíèöó ìåæäó ëåãêîé è òðóäíîé ðàçðåøèìîñòüþ äëÿ øèðîêîãî êðóãà çàäà÷ êîìáèíàòîðíîãî ïîèñêà [125], [161], [225], [252]. Êðîìå òîãî, îíà âàæíà è ñ ïðèêëàäíîé òî÷êè çðåíèÿ, òàê êàê îíà ïîçâîëÿåò ðàçðàáàòûâàòü ÿçûêè ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ, èñïîëüçóþùèå ñóùåñòâîâàíèå ðàçëè÷íûõ ñåìåéñòâ ëåãêî ðàçðåøèìûõ îãðàíè÷åíèé äëÿ ïîèñêà áîëåå ýôôåêòèâíûõ ìåòîäîâ ðåøåíèÿ [260], [310]. Â ðÿäå ïóáëèêàöèé [228], [230], [227], [233] áûëî ïîêàçàíî, ÷òî ñëîæíîñòü ÿçûêîâ îãðàíè÷åíèé íàä êîíå÷íûìè äîìåíàìè ìîæåò áûòü îõàðàêòåðèçîâàíà, èñïîëüçóÿ àëãåáðàè÷åñêèå ñâîéñòâà îòíîøåíèé. Ñîãëàñíî [114], ïåðâûé ýòàï àëãåáðàè÷åñêîãî ïîäõîäà ê ÿçûêàì îãðàíè÷åíèé èñïîëüçóåò èçâåñòíóþ èäåþ, ñîñòîÿùóþ â òîì, ÷òî ê äàííîìó íà÷àëüíîìó ìíîæåñòâó îòíîøåíèé îãðàíè÷åíèé ìîãóò äîáàâëÿòüñÿ äîïîëíèòåëüíûå îãðàíè÷åíèÿ áåç èçìåíåíèÿ ñëîæíîñòè ñîîòâåòñòâóþùåãî êëàññà çàäà÷. Íà ñàìîì äåëå, áûëî ïîêàçàíî, ÷òî ìîæíî äîáàâèòü âñå îãðàíè÷åíèÿ, ïîëó÷åííûå èç èñõîäíûõ îãðàíè÷åíèé, èñïîëüçóÿ íåêîòîðûå ïðîñòûå ïðàâèëà. Áîëüøèå ìíîæåñòâà îòíîøåíèé, ïîëó÷åííûå ïðè èñïîëüçîâàíèè ýòèõ ïðàâèë, íàçûâàþòñÿ ðåëÿöèîííûìè êëîíàìè [146], [306]. Ïåðâûé ýòàï àëãåáðàè÷åñêîãî ïîäõîäà ñîñòîèò â òîì, ÷òî äîñòàòî÷íî àíàëèçèðîâàòü ñëîæíîñòü ëèøü äëÿ òåõ ìíîæåñòâ îòíîøåíèé, êîòîðûå ÿâëÿþòñÿ ðåëÿöèîííûìè êëîíàìè. Âòîðîé ýòàï ñîñòîèò â òîì, ÷òî ðåëÿöèîííûå êëîíû ìîãóò áûòü îïèñàíû ñ ïîìîùüþ èõ ïîëèìîðôèçìîâ, ÿâëÿþùèõñÿ àëãåáðàè÷åñêèìè îïåðàöèÿìè íàä òåì æå îñíîâíûì ìíîæåñòâîì [225],[227]. Ïîìèìî òîãî, ÷òî ïîëèìîðôèçìû äàþò òî÷íûé è óäîáíûé ìåòîä îïèñàíèÿ áîëüøèõ ñåìåéñòâ îòíîøåíèé, îíè îòðàæàþò òàêæå îïðåäåëåííûå àñïåêòû ñòðóêòóðû îòíîøåíèé, êîòîðûå ìîãóò áûòü èñïîëüçîâàíû äëÿ ñîçäàíèÿ ýôôåêòèâíûõ àëãîðèòìîâ. Óêàçàííàÿ ñâÿçü ìåæäó ðåëÿöèîííûìè êëîíàìè è ïîëèìîðôèçìàìè èãðàåò ñåé÷àñ êëþ÷åâóþ ðîëü ïðè îïðåäåëåíèè ìíîãèõ ëåãêî ðàçðåøèìûõ êëàññîâ îãðàíè÷åíèé è ðàçðàáîòêå ýôôåêòèâíûõ àëãîðèòìîâ èõ ðåøåíèÿ ñîîòâåòñòâóþùèõ çàäà÷ ÓÎ [96],[92],[93], [91],[126],[228]. Ïîëíàÿ õàðàêòåðèçàöèÿ âñåâîçìîæíûõ ëåãêî ðàçðåøèìûõ ÿçûêîâ îãðàíè÷åíèé áûëà óñòàíîâëåíà â ñëåäóþùèõ ñëó÷àÿõ: êîíñåðâàòèâíûå ÿçûêè îãðàíè÷åíèé (ò.å. ÿçûêè îãðàíè37 ÷åíèé, ñîäåðæàùèå âñå óíàðíûå îãðàíè÷åíèÿ) [94], è ÿçûêè îãðàíè÷åíèé íàä 2ýëåìåíòíûì äîìåíîì [332] èëè íàä 3-ýëåìåíòíûì äîìåíîì [96]. Êðîìå òîãî, áûëè ðàçðàáîòàíû íîâûå ýôôåêòèâíûå àëãîðèòìû äëÿ ðåøåíèÿ çàäà÷ ÓÎ ñïåöèàëüíîãî âèäà êàê äëÿ êîíå÷íûõ, òàê è äëÿ áåñêîíå÷íûõ äîìåíîâ [75], [92], [94], [112], [121], [126], [251]. Ïîñëåäíèé ýòàï àëãåáðàè÷åñêîãî ïîäõîäà ñâÿçûâàåò ÿçûê îãðàíè÷åíèé è êîíå÷íûå óíèâåðñàëüíûå àëãåáðû (ñì. ðàçäåë 6.4.3 ðàáîòû [114]). ßçûê êîíå÷íûõ óíèâåðñàëüíûõ àëãåáð ïðåäîñòàâëÿåò ìíîæåñòâî î÷åíü óäîáíûõ íîâûõ ñðåäñòâ äëÿ àíàëèçà ñëîæíîñòè îãðàíè÷åíèé, â òîì ÷èñëå ãëóáîêèå ñòðóêòóðíûå ðåçóëüòàòû, ðàçðàáîòàííûå äëÿ êëàññèôèêàöèè ñòðóêòóðû êîíå÷íûõ àëãåáð [210], [274], [358]. Â ñëó÷àå, êîãäà ìíîæåñòâî ðàçðåø¼ííûõ îòíîøåíèé èíâàðèàíòíî îòíîñèòåëüíî íåêîòîðîé ìàëüöåâñêîé îïåðàöèè12 , ïîêàçûâàåòñÿ, ÷òî ñîîòâåòñòâóþùàÿ çàäà÷à ÓÎ ìîæåò áûòü ðåøåíà çà ïîëèíîìèàëüíîå âðåìÿ [2]. Îäíàêî ìíîãèå çàäà÷è ÓÎ íå îáëàäàþò äîñòàòî÷íî îãðàíè÷åííîé ñòðóêòóðîé èëè èñïîëüçóþò íåäîñòàòî÷íî îãðàíè÷åííûé ÿçûê îãðàíè÷åíèé, ÷òîáû ïîïàñòü â êàêîé-ëèáî èç êëàññîâ ëåãêî ðàçðåøèìûõ çàäà÷. Îíè ìîãóò, òåì íå ìåíåå, èìåòü ñâîéñòâà, îáåñïå÷èâàþùèå ýôôåêòèâíîñòü èõ ðåøåíèÿ, íî ýòè ñâîéñòâà îòíîñÿòñÿ êàê ê ñòðóêòóðå, òàê è ê òèïàì îãðàíè÷åíèé. Ýòè ñâîéñòâà èíîãäà íàçûâàþòñÿ ãèáðèäíûìè ïðè÷èíàìè ëåãêîé ðàçðåøèìîñòè [301], îíè ãîðàçäî ìåíåå èçó÷åíû, ÷åì ñâîéñòâà ÿçûêà è îïèñàííûå âûøå ñòðóêòóðíûå ñâîéñòâà. Â [122] îïèñàí íîâûå ãèáðèäíûå ëåãêî ðàçðåøèìûå êëàññû áèíàðíûõ ÇÓÎ, îáîáùàþùèå çàäà÷è ñ äðåâîâèäíîé ñòðóêòóðîé, à òàêæå ãèáðèäíûå ëåãêî ðàçðåøèìûå êëàññû, îñíîâàííûå íà ÿçûêå. Ýòè íîâûå êëàññû îñíîâàíû íà ëîêàëüíûõ ñâîéñòâàõ óïîðÿäî÷åííûõ òðîåê ïåðåìåííûõ. Ïðè ýòîì çàäà÷à îïðåäåëåíèÿ óïîðÿäî÷åíèÿ ïåðåìåííûõ, äëÿ êîòîðîãî âûïîëíÿþòñÿ ýòè ñâîéñòâà, ðåøàåòñÿ çà ïîëèíîìèàëüíîå âðåìÿ. 4. Ñòðóêòóðà è ìåòîäû äåêîìïîçèöèè çàäà÷ ÓÎ Èññëåäîâàíèÿ, êàñàþùèåñÿ ñòðóêòóðû è ñëîæíîñòè çàäà÷ ÓÎ, íà÷àëèñü ñ ðàáîòû (Freuder, 1985) [169], ãäå áûëî ïîêàçàíî, ÷òî ïîèñê íà äåðåâüÿõ ñ ñîâìåñòíûìè äóãàìè ïðîèñõîäèò áåç êàêèõ-ëèáî âîçâðàòîâ. Àíàëîãè÷íûå ðåçóëüòàòû ïðèìåíèòåëüíî ê àöèêëè÷åñêèì ãèïåðãðàôàì áûëè ïîëó÷åíû â îáëàñòè òåîðèè áàç äàííûõ (Beeri et al, 1983) [55]. Ñî âðåìåíè ïóáëèêàöèè ýòèõ ðàáîò äîñòèãíóò çíà÷èòåëüíûé ïðîãðåññ â ÷àñòè ïîëó÷åíèÿ áîëåå îáùèõ ðåçóëüòàòîâ, êàñàþùèõñÿ ñâÿçè ìåæäó ñëîæíîñòüþ ðåøåíèÿ çàäà÷è ÓÎ è ñòðóêòóðîé åå ãðàôà îãðàíè÷åíèé. Â äàííîì ðàçäåëå ðàññìàòðèâàþòñÿ ñïîñîáû, ïîçâîëÿþùèå èñïîëüçîâàòü äëÿ áûñòðîãî ïîèñêà ðåøåíèé ñòðóêòóðó ñàìîé çàäà÷è, ïðåäñòàâëåííóþ â âèäå ãðàôà îãðàíè÷åíèé. Áîëüøèíñòâî îïèñàííûõ çäåñü ïîäõîäîâ ÿâëÿåòñÿ î÷åíü îáùèìè è ïðèìåíèìûìè äëÿ ðåøåíèÿ äðóãèõ äèñêðåòíûõ çàäà÷, êðîìå ÓÎ. 12 Ìàëüöåâñêàÿ ϕ(y, y, x) = x. îïåðàöèÿ ýòî òåðíàðíàÿ îïåðàöèÿ ϕ, óäîâëåòâîðÿþùàÿ ϕ(x, y, y) = 38 4.1. Ãðàôîâûå ìåòîäû Ãðàôîâûå ìåòîäû èñïîëüçóþò ñòðóêòóðó ãðàôà îãðàíè÷åíèé ïðè ðåøåíèè çàäà÷è ÓÎ. Íèæå ðàññìîòðåíû ãðàôîâûå ìåòîäû, èñïîëüçóþùèå òî, ÷òî çàäà÷à ÓÎ ñ äðåâîâèäíûì ãðàôîì îãðàíè÷åíèé ìîæåò áûòü ðåøåíà çà ëèíåéíîå âðåìÿ (ñì. (Miguel & Shen [283]), ðàçäåë 4.4), ïðè÷åì äðåâîâèäíàÿ ñòðóêòóðà íàõîäèòñÿ èëè ñòðîèòñÿ èç ãðàôà îãðàíè÷åíèé äàííîé çàäà÷è ÓÎ. Äëÿ ðàçáèåíèÿ çàäà÷è ÓÎ íà íåçàâèñèìûå ïîäçàäà÷è, ìîæíî ðàññìîòðåòü ñâÿçíûå êîìïîíåíòû ãðàôà îãðàíè÷åíèé. Êàæäûé êîìïîíåíò ñîîòâåòñòâóåò îäíîé ïîäçàäà÷å ÓÎi . Åñëè ïðèñâàèâàíèå Si ÿâëÿåòñÿ ðåøåíèåì ÓÎi , òî ∪i Si ÿâëÿåòñÿ ðåøåíèåì ∪i ÓÎi . Ïðåäïîëîæèì, ÷òî êàæäàÿ ïîäçàäà÷à ÓÎi èìååò c ïåðåìåííûõ èç îáùåãî êîëè÷åñòâà n ïåðåìåííûõ, ãäå c êîíñòàíòà. Â òàêîì ñëó÷àå äîëæíî áûòü n/c ïîäçàäà÷, è äëÿ ðåøåíèÿ êàæäîé èç íèõ òðåáóåòñÿ, ñàìîå áîëüøåå, îáúåì ðàáîòû dc 13 . Ïîýòîìó îáùèé îáúåì ðàáîòû èçìåðÿåòñÿ âåëè÷èíîé O(dc n/c), êîòîðàÿ ëèíåéíî çàâèñèò îò n; áåç òàêîé äåêîìïîçèöèè îáùèé îáúåì ðàáîòû èçìåðÿëñÿ áû âåëè÷èíîé O(dn ), êîòîðàÿ ýêñïîíåíöèàëüíî çàâèñèò îò n. Ïðèâåäåì áîëåå êîíêðåòíûé ïðèìåð èç [12]: äåêîìïîçèöèÿ áóëåâîé çàäà÷è ÓÎ ñ n = 80 íà ÷åòûðå íåçàâèñèìûå ïîäçàäà÷è ñ c = 20 ñîêðàùàåò ïðîäîëæèòåëüíîñòü ïîèñêà ðåøåíèÿ â íàèõóäøåì ñëó÷àå îò âåëè÷èíû, ðàâíîé âðåìåíè ñóùåñòâîâàíèÿ âñåé Âñåëåííîé, äî âåëè÷èíû, ìåíüøåé îäíîé ñåêóíäû. Ïîýòîìó ïîëíîñòüþ íåçàâèñèìûå ïîäçàäà÷è ÿâëÿþòñÿ î÷åíü ïðèâëåêàòåëüíûìè, íî âñòðå÷àþòñÿ ðåäêî. Â áîëüøèíñòâå ñëó÷àåâ ïîäçàäà÷è ëþáîé çàäà÷è ÓÎ ñâÿçàíû äðóã ñ äðóãîì. Ïðîñòåéøèì ñëó÷àåì ÿâëÿåòñÿ òîò, â êîòîðîì ãðàô îãðàíè÷åíèé ÿâëÿåòñÿ äåðåâîì: ëþáûå äâå ïåðåìåííûå ñâÿçàíû íå áîëüøå ÷åì îäíèì ïóòåì. Àëãîðèòì ðåøåíèÿ áëî÷íîé ÇÓÎ èìååò ñëåäóþùèå ýòàïû. • Âûáðàòü â êà÷åñòâå êîðíÿ äåðåâà ëþáóþ ïåðåìåííóþ è óïîðÿäî÷èòü ïåðåìåííûå îò êîðíÿ äî ëèñòüåâ òàêèì îáðàçîì, ÷òîáû ðîäèòåëüñêàÿ âåðøèíà êàæäîé âåðøèíû â äåðåâå ïðåäøåñòâîâàëà ýòîé âåðøèíå â òàêîì óïîðÿäî÷åíèè. Îáîçíà÷èòü ýòè ïåðåìåííûå ïî ïîðÿäêó êàê X1 , . . . , Xn . Òåïåðü êàæäàÿ ïåðåìåííàÿ, êðîìå êîðíÿ, èìååò òîëüêî îäíó ðîäèòåëüñêóþ ïåðåìåííóþ. • Â öèêëå ïî j îò n äî 2 ïðèìåíÿòü ïðîâåðêó ñîâìåñòíîñòè ê äóãàì (Xi , Xj ), ãäå Xi ðîäèòåëüñêàÿ âåðøèíà âåðøèíû Xj , óäàëÿÿ çíà÷åíèÿ èç îáëàñòè îïðåäåëåíèÿ Di ïî ìåðå íåîáõîäèìîñòè. • Â öèêëå ïî j îò 1 äî n ïðèñâàèâàòü Xj ëþáîå çíà÷åíèå, ñîâìåñòíîå ñî çíà÷åíèåì, ïðèñâîåííûì Xi , ãäå Xi ðîäèòåëüñêàÿ âåðøèíà âåðøèíû Xj . Òåïåðü, ïîñëå ñîçäàíèÿ ýôôåêòèâíîãî àëãîðèòìà äëÿ äåðåâüåâ, ñëåäóåò ðàññìîòðåòü âîïðîñ î òîì, ìîæíî ëè êàêèì-òî îáðàçîì ïðèâîäèòü ê äðåâîâèäíûì ñòðóêòóðàì áîëåå îáùèå ãðàôû îãðàíè÷åíèé. Ñóùåñòâóþò äâà îñíîâíûõ ñïîñîáà âûïîëíåíèÿ ýòîé çàäà÷è; îäèí èç íèõ îñíîâàí íà óäàëåíèè âåðøèí, à äðóãîé íà ñëèÿíèè 13 d ðàçìåð äîìåíîâ ïåðåìåííûõ. 39 x1 x2 a) x1 x2 x2 x2 x3 x4 b) x3 x4 Ðèñ. 11. Ýôôåêò ôèêñàöèè ìíîæåñòâà ðàçðûâà öèêëà. âåðøèí äðóã ñ äðóãîì è îáðàçîâàíèè ñóïåð-âåðøèí. Ïåðâûé ïîäõîä ïðåäóñìàòðèâàåò ïðèñâàèâàíèå çíà÷åíèé íåêîòîðûì ïåðåìåííûì òàê, ÷òîáû îñòàâøèåñÿ ïåðåìåííûå îáðàçîâûâàëè äåðåâî. 4.2. Ìåòîä ìíîæåñòâ ðàçðûâà öèêëà Ìåòîä ìíîæåñòâ ðàçðûâà öèêëà (cycle cutset) (Dechter, 1990) [133] îñíîâàí íà òîì, ÷òî ïðèñâîåíèå çíà÷åíèÿ ïåðåìåííîé óñòðàíÿåò åå èç äàëüíåéøåãî ðàññìîòðåíèÿ â ýòîé âåòâè äåðåâà ïîèñêà. Ýòî ñèëüíî ìåíÿåò ñâÿçíîñòü îñòàâøåéñÿ ÷àñòè ãðàôà îãðàíè÷åíèé. Ìíîæåñòâî ðàçðûâà öèêëà ãðàôà îãðàíè÷åíèé ýòî ìíîæåñòâî âåðøèí, ïðè èñêëþ÷åíèè êîòîðûõ (ñ ïîìîùüþ ôèêñàöèè ñîîòâåòñòâóþùèõ ïåðåìåííûõ) îñòàåòñÿ äåðåâî. Ðàññìîòðèì ïðèìåð íà ðèñ. 11 a. Ôèêñèðîâàíèå çíà÷åíèÿ ïåðåìåííîé x2 ¾áëîêèðóåò¿ ïóòü, ïðîõîäÿùèé ÷åðåç ýòó âåðøèíó, òàê ÷òî èìååòñÿ òîëüêî îäèí ïóòü ìåæäó ëþáûìè äâóìÿ ïåðåìåííûìè (âåðøèíàìè). Â ðåçóëüòàòå ïîëó÷àåòñÿ ãðàô îãðàíè÷åíèé (ðèñ. 11 b), â êîòîðîì çàôèêñèðîâàííàÿ ñåé÷àñ ïåðåìåííàÿ x2 ïîâòîðÿåòñÿ äëÿ êàæäîé ñìåæíîé ïåðåìåííîé. Åñëè ìíîæåñòâî ôèêñèðîâàííûõ ïåðåìåííûõ ñîâïàäàåò ñ ìíîæåñòâîì ðàçðûâà öèêëà, òî îñòàâøèéñÿ ãðàô èìååò äðåâîâèäíóþ ñòðóêòóðó è ñîîòâåòñòâóþùàÿ åìó çàäà÷à ìîæåò áûòü ðåøåíà î÷åíü ýôôåêòèâíî. Ýòîò ìåòîä íàõîäèò ëîêàëüíî ñîâìåñòíîå ïðèñâîåíèå äëÿ ïåðåìåííûõ èç ìíîæåñòâà ðàçðûâà öèêëà è ðåøàåò îñòàâøóþñÿ (äðåâîâèäíóþ) çàäà÷ó. Åñëè íå íàéäåíî íè îäíîãî ðåøåíèÿ, òî äîëæíî áûòü íàéäåíî äðóãîå ñîâìåñòíîå ïðèñâîåíèå äëÿ ïåðåìåííûõ èç ìíîæåñòâà ðàçðûâà öèêëà è äåëàåòñÿ ïîïûòêà ðåøåíèÿ çàäà÷è, ñîîòâåòñòâóþùåé îñòàâøåéñÿ ÷àñòè ãðàôà. Íàõîæäåíèå ìèíèìàëüíîãî ìíîæåñòâà ðàçðûâà öèêëà ÿâëÿåòñÿ NP-ïîëíîé çàäà÷åé. Â ñâÿçè ñ ýòèì, öåëåñîîáðàçíî âñòðàèâàòü ìåòîä ìíîæåñòâà ðàçðûâà öèêëà â ìåòîä äåðåâà ïîèñêà, òàêîé, êàê ïîèñê ñ âîçâðàòàìè. Â ñëó÷àå, êîãäà ìíîæåñòâî çàôèêñèðîâàííûõ ïåðåìåííûõ îáðàçóåò ìíîæåñòâî ðàçðûâà öèêëà, îñòàâøàÿñÿ ÷àñòü çàäà÷è ìîæåò áûòü áûñòðî ðåøåíà. Ïî ñðàâíåíèþ ñ ïîèñêîì ñ âîçâðàòàìè íà ñëó÷àéíî ñãåíåðèðîâàííûõ çàäà÷àõ, ìåòîä ïîèñêà ñ âîçâðàòàìè, èñïîëüçóþùèé ìíîæåñòâà ðàçðûâà öèêëà, â ñðåäíåì íà 20% ýôôåêòèâíåå [284]. Ïî ñðàâíåíèþ ñ íàèâíîé âåðñèåé 40 àëãîðèòìà îáðàòíîãî ïåðåõîäà, åãî êîìáèíàöèÿ ñ ìíîæåñòâàìè ðàçðûâà öèêëà â ñðåäíåì íà 25% ýôôåêòèâíåå [284]. Îáùèé àëãîðèòì ðåøåíèÿ óêàçàííûì ñïîñîáîì îïèñàí íèæå [12]. • Âûáðàòü ïîäìíîæåñòâî S èç ìíîæåñòâà ïåðåìåííûõ çàäà÷è ÓÎ, òàêîå, ÷òî ãðàô îãðàíè÷åíèé ïîñëå óäàëåíèÿ S ñòàíîâèòñÿ äåðåâîì. Ïîäìíîæåñòâî S íàçûâàåòñÿ ìíîæåñòâîì ðàçðûâà öèêëà. • Äëÿ êàæäîãî âîçìîæíîãî ïðèñâàèâàíèÿ ïåðåìåííûì â S , êîòîðîå óäîâëåòâîðÿåò âñåì îãðàíè÷åíèÿì â S , âûïîëíèòü ñëåäóþùåå: óäàëèòü èç îáëàñòåé îïðåäåëåíèÿ îñòàâøèõñÿ ïåðåìåííûõ ëþáûå çíà÷åíèÿ, íåñîâìåñòíûå ñ äàííûì ïðèñâàèâàíèåì äëÿ S ; åñëè îñòàâøàÿñÿ çàäà÷à ÓÎ èìååò ðåøåíèå, âåðíóòü ýòî ðåøåíèå âìåñòå ñ ïðèñâàèâàíèåì äëÿ S . Åñëè ìíîæåñòâî ðàçðûâà öèêëà èìååò ðàçìåð c, òî îáùåå âðåìÿ ðàáîòû àëãîðèòìà ñîñòàâëÿåò O(dc · (n − c)d2 ). Â òîì ñëó÷àå, åñëè ãðàô ïî ñâîåé ôîðìå ¾î÷åíü áëèçîê ê äåðåâó¿, òî ìíîæåñòâî ñ áóäåò íåáîëüøèì, à ýêîíîìèÿ âðåìåíè ïî ñðàâíåíèþ ñ ïðÿìûì ïîèñêîì ñ âîçâðàòàìè îêàæåòñÿ îãðîìíîé. Íî â íàèõóäøåì ñëó÷àå êîëè÷åñòâî ýëåìåíòîâ ñ ìîæåò äîñòèãàòü (n − 2). Õîòÿ çàäà÷à ïîèñêà íàèìåíüøåãî ìíîæåñòâà ðàçðûâà öèêëà ÿâëÿåòñÿ N P -òðóäíîé, íî èçâåñòíî íåñêîëüêî ýôôåêòèâíûõ àëãîðèòìîâ ðåøåíèÿ ýòîé çàäà÷è. Â öåëîì äàííûé àëãîðèòìè÷åñêèé ïîäõîä íîñèò íàçâàíèå îïðåäåëåíèÿ óñëîâèé âûáîðà ìíîæåñòâà ðàçðûâà öèêëà (cutset conditioning). 4.3. Ñâÿçíûå êîìïîíåíòû Âûäåëåíèå ñâÿçíûõ êîìïîíåíòîâ ãðàôà (Dechter & Pearl [140]; Freuder [168]) ìîæåò áûòü ïîëåçíî ïðè ðåøåíèè çàäà÷ ÓÎ. Òî÷êà ñî÷ëåíåíèÿ xi â ãðàôå îãðàíè÷åíèé ýòà òàêàÿ âåðøèíà, ÷òî âñå ïóòè ìåæäó îñòàëüíûìè âåðøèíàìè äîëæíû ïðîõîäèòü ÷åðåç íåå. Ðàçäåëèìûé ãðàô ñîäåðæèò òî÷êó ñî÷ëåíåíèÿ, à ñâÿçíûé ãðàô íåò. Ñâÿçíûé êîìïîíåíò ýòî ïîäãðàô áåç òî÷åê ñî÷ëåíåíèÿ. Even [159] ïðåäëîæèë àëãîðèòì ñî ñëîæíîñòüþ O(e) äëÿ íàõîæäåíèÿ âñåõ ñâÿçíûõ êîìïîíåíòîâ è òî÷åê ñî÷ëåíåíèÿ (çäåñü e ÷èñëî îãðàíè÷åíèé). Ïîñëå íàõîæäåíèÿ âñåõ ñâÿçíûõ êîìïîíåíòîâ, íåîáõîäèìî íàéòè âñå âîçìîæíûå ðåøåíèÿ äëÿ êàæäîãî èç íèõ. Ðåøåíèå âñåé çàäà÷è ÓÎ ìîæåò áûòü ïîëó÷åíî ñ ïîìîùüþ ñêëåèâàíèÿ ðåøåíèé äëÿ êîìïîíåíòîâ (ñ ó÷åòîì ñîâìåñòíîñòè çíà÷åíèÿ äëÿ òî÷êè ñî÷ëåíåíèÿ ñìåæíûõ êîìïîíåíòîâ). Dechter & Pearl [140] ïðåäëîæèëè àëãîðèòì ñî ñëîæíîñòüþ O(n · dr ) , èñïîëüçóþùèé îïèñàííûé ïîäõîä, ãäå n ÷èñëî ïåðåìåííûõ, d ðàçìåð ìàêñèìàëüíîãî äîìåíà ïåðåìåííûõ, à r ðàçìåð íàèáîëüøåãî êîìïîíåíòà. 41 Ðèñ. 12. Ïîñòðîåíèå õîðäàëüíîãî ïåðâè÷íîãî ãðàôà. 4.4. Äðåâîâèäíàÿ êëàñòåðèçàöèÿ Äðåâîâèäíàÿ êëàñòåðèçàöèÿ (tree-clustering) (Dechter & Pearl [141]) ïðåîáðàçóåò ïðîèçâîëüíóþ n-àðíóþ çàäà÷ó ÓÎ â àöèêëè÷åñêóþ ôîðìó â ïðåäñòàâëåíèè â âèäå äâîéñòâåííîãî ãðàôà ïóòåì ôîðìèðîâàíèÿ êëàñòåðîâ èç ïåðåìåííûõ, ïðè÷åì ãðàô âçàèìîñâÿçåé èìååò ñòðóêòóðó äåðåâà. n-àðíàÿ çàäà÷à ÓÎ íàçûâàåòñÿ àöèêëè÷åñêîé (ñì. (Miguel & Shen [283])), åñëè åå ïåðâè÷íûé ãðàô îãðàíè÷åíèé îáëàäàåò ñëåäóþùèìè ñâîéñòâàìè: • õîðäàëüíîñòü : êàæäûé öèêë, ñîäåðæàùèé íå ìåíåå 4 âåðøèí, ñîäåðæèò õîðäó (ðåáðî, ñîåäèíÿþùåå 2 íåïîñëåäîâàòåëüíûå âåðøèíû öèêëà). • êîíôîðìíîñòü : êàæäàÿ èç ìàêñèìàëüíûõ êëèê ãðàôà ñîîòâåòñòâóåò îäíîìó îãðàíè÷åíèþ â çàäà÷å ÓÎ. Äëÿ äîñòèæåíèÿ óêàçàííûõ ñâîéñòâ ìîæåò áûòü èñïîëüçîâàí àëãîðèòì òðèàíãóëÿöèè (Tarjan & Yannakakis [361]), èñïîëüçóÿ äâà øàãà: a) Íàéòè óïîðÿäî÷åíèå ñ ìàêñèìàëüíîé ñòåïåíüþ (maximum cardinality) è á) ðåêóðñèâíî äîáàâèòü ðåáðà ìåæäó âåðøèíàìè, ñîåäèíÿþùèìè âåðøèíû, ñ áîëüøèì ïîðÿäêîì ñîãëàñíî óêàçàííîìó óïîðÿäî÷åíèþ. Íà ðèñóíêàõ 12 a è b (èç (Dechter & Pearl [141])) ïîêàçàí ïåðâè÷íûé ãðàô îãðàíè÷åíèé çàäà÷è ÓÎ ñîîòâåòñòâåííî äî è ïîñëå ïðîöåññà òðèàíãóëÿöèè. Ìàêñèìàëüíûìè êëèêàìè ýòîãî ãðàôà ÿâëÿþòñÿ êëàñòåðû, êîòîðûå ñîñòàâëÿþò îãðàíè÷åíèÿ ïðåîáðàçîâàííîé çàäà÷è ÓÎ (ñì. ðèñ. 13 a). Äëÿ ïîñòðîåíèÿ äåðåâà èç ýòîãî ïðåäñòàâëåíèÿ äîñòàòî÷íî óäàëèòü ëèøíèå ðåáðà èç ãðàôà, ò.å. òå ðåáðà, óäàëåíèå êîòîðûõ íå îêàçûâàåò âëèÿíèÿ íà ìíîæåñòâî ðåøåíèé. Â äâîéñòâåííîì ãðàôå äóãà èçáûòî÷íà, åñëè ìíîæåñòâî ïåðåìåííûõ, êîòîðûì îíà ïîìå÷åíà, ÿâëÿåòñÿ îáùèì äëÿ âñåõ äóã íà îòäåëüíîì ïóòè ìåæäó êîíå÷íûìè âåðøèíàìè äóãè. Íà ðèñ. 13 b) ïîêàçàí îäèí ñïîñîá ïîñòðîåíèÿ äåðåâà äëÿ çàäà÷è ÓÎ. Ñëîæíîñòü ýòîé ïðîöåäóðû O(n·r·dr ), ãäå r ðàçìåð íàèáîëüøåé ìàêñèìàëüíîé êëèêè. 42 Ðèñ. 13. Ïîñòðîåíèå äåðåâà èç äâîéñòâåííîãî ãðàôà. 4.5. Ýëèìèíàöèÿ ïåðåìåííûõ 4.5.1. Ñåãìåíòíàÿ ýëèìèíàöèÿ Ñåãìåíòíàÿ ýëèìèíàöèÿ (Bucket elimination) [137], [142] ïî ñóòè ÿâëÿåòñÿ àíàëîãîì íåñåðèàëüíîãî äèíàìè÷åñêîãî ïðîãðàììèðîâàíèÿ [64], [21], [22], äëÿ ðåøåíèÿ çàäà÷ ÓÎ. Ýòîò àëãîðèòì ìîæåò íàéòè âñå âîçìîæíûå ðåøåíèÿ çàäà÷è. Àëãîðèòì ñåãìåíòíîé ýëèìèíàöèè èìååò ýêñïîíåíöèàëüíûå îöåíêè âðåìåííîé è åìêîñòíîé ñëîæíîñòè ïî èíäóöèðîâàííîé øèðèíå ãðàôà îãðàíè÷åíèé. Áîëüøàÿ àðíîñòü ïðîìåæóòî÷íûõ îãðàíè÷åíèé, êîòîðûå õðàíÿòñÿ â ïàìÿòè â âèäå òàáëèö, ÿâëÿåòñÿ ãëàâíûì ïðåïÿòñòâèåì äëÿ ïðèìåíåíèÿ ìåòîäà ñåãìåíòíîé ýëèìèíàöèè íà ïðàêòèêå. Ïðè äîñòàòî÷íî íåáîëüøîé àðíîñòè îãðàíè÷åíèé àëãîðèòì ñåãìåíòíîé ýëèìèíàöèè äîñòàòî÷íî ýôôåêòèâåí [257]. 4.5.2. Ìåòîä Çåéäåëÿ ïðîöåäóðà èíâàçèè Åùå îäíèì èçâåñòíûì äåêîìïîçèöèîííûì àëãîðèòìîì ÿâëÿåòñÿ àëãîðèòì èíâàçèè Çåéäåëÿ (Seidel, 1981) [337], êîòîðûé íàõîäèò âñå ðåøåíèÿ áèíàðíîé çàäà÷è ÓÎ. Ìåòîä ðàçáèâàåò äàííóþ çàäà÷ó ÓÎ íà ìíîæåñòâî ïîäçàäà÷ ÓÎ, ñâÿçàííûõ ëèíåéíûì ãðàôîì (ïóòåì ÷àñòíûì ñëó÷àåì äåðåâà). Ïðîöåäóðà èíâàçèè14 (Seidel, 1981) [337] ïðåäíàçíà÷åíà äëÿ íàõîæäåíèÿ âñåõ ðåøåíèé çàäà÷è ÓÎ è îñíîâàíà íà ïîñëåäîâàòåëüíîñòè ïîäãðàôîâ G1 , . . . , Gn , íàçûâàåìîé èíâàçèåé ãðàôà îãðàíè÷åíèé G ñ n âåðøèíàìè. ×èñëî âåðøèí â ïîäãðàôå Gi äîëæíî áûòü i, ïðè÷åì Gi äîëæåí áûòü ïîäãðàôîì Gi+1 . Ôðîíò Fi ýòî ìíîæåñòâî âåðøèí ïîäãðàôà Gi , ñìåæíûõ ñ âåðøèíàìè, íå ïðèíàäëåæàùèõ Gi , à äëèíà ôðîíòà fi ýòî ÷èñëî âåðøèí â Fi . Äëèíà ôðîíòà èíâàçèè ýòî ìàêñèìóì äëèí åå îòäåëüíûõ ôðîíòîâ. Ðàññìîòðèì ïðèìåð íà ðèñ. 6a). Èíâàçèÿ ìîæåò áûòü ïîñòðîåíà ñëåäóþùèì îáðàçîì: G1 = {x1 }, G2 = {x1 , x2 }, G3 = {x1 , x2 , x3 }, G4 = {x1 , x2 , x3 , x4 }. Íàéäåì ôðîíòû, ñîîòâåòñòâóþùèå ïîäãðàôàì: F1 = {x1 }, F2 = {x2 }, F3 = {x2 }, F4 = { }. Çàìåòèì, ÷òî Fn âñåãäà ïóñòîå ìíîæåñòâî, òàê êàê Gn ñîâïàäàåò ñ èñõîäíûì ãðàôîì îãðàíè÷åíèé. Äëÿ äàííîé èíâàçèè àëãîðèòì Çåéäåëÿ ñòðîèò ãðàô ðåøåíèé S , ïðåäñòàâëÿþùèé ìíîæåñòâî ðåøåíèé èñõîäíîé çàäà÷è ÓÎ. Âåðøèíû ãðàôà S ñãðóïïèðîâàíû â âèäå ìíîæåñòâ 14 èíâàçèÿ íàøåñòâèå. 43 1 F1 3 4 4 F2 3 F4 F3 2 3 s4 4 3 1 s0 4 5 5 4 2 4 Ðèñ. 14. Ãðàô ðåøåíèé äëÿ àëãîðèòìà Çåéäåëÿ. S0 , . . . , Sn , òàêèõ, ÷òî Si (i 6= 0, i 6= n) ÿâëÿåòñÿ ìíîæåñòâîì êîðòåæåé ïðèñâîåííûõ çíà÷åíèé äëÿ ôðîíòà Fi . Sn è S0 ñîäåðæàò ïî îäíîé âåðøèíå, sn è s0 , ñîîòâåòñòâåííî (ýòî íà÷àëî è êîíåö ðåøåíèÿ). Íàïðàâëåííîå ðåáðî èç êîðòåæà èç Si ê êîðòåæó èç Si−1 ïîìå÷àåòñÿ ïðèñâîåíèåì çíà÷åíèÿ ïåðåìåííîé xi (ãäå xi ïåðåìåííàÿ â Gi , êîòîðàÿ íå ïðèíàäëåæèò Gi−1 ). Ýòî ïðèñâîåíèå äîëæíî óäîâëåòâîðÿòü âñåì îãðàíè÷åíèÿì, ñîäåðæàùèì xi è âñå ïåðåìåííûå èç ôðîíòà Fi−1 äëÿ äàííûõ ïðèñâîåíèé, ïðåäñòàâëåííûõ êîíêðåòíîìó êîðòåæó èç Si−1 . Ïóòü èç sn â s0 ñ ôèêñèðîâàíèåì xi íà çíà÷åíèÿõ, ñîîòâåòñòâóþùèõ êàæäîìó ïðîéäåííîìó ðåáðó, ñîîòâåòñòâóåò îäíîìó ðåøåíèþ. Ãðàô ðåøåíèé çàäà÷è ñ ðèñ. 6 a) ïîêàçàí íà ðèñ. 14. Ïðè ïîñòðîåíèè Si êàæäûé êîðòåæ ïðåäûäóùåãî ìíîæåñòâà êîðòåæåé Si−1 ðàññìàòðèâàåòñÿ ïî î÷åðåäè äëÿ âñåõ âîçìîæíûõ ïðèñâîåíèé xi . Êîðòåæ äîáàâëÿåòñÿ ê Si äëÿ êàæäîãî ïðèñâîåíèÿ ïåðåìåííîé xi , êîòîðîå óäîâëåòâîðÿåò âñåì îãðàíè÷åíèÿì ñ êîíêðåòíûì ïðèñâîåíèåì êîðòåæà â Si−1 . Äîáàâëÿåòñÿ ðåáðî, ñîåäèíÿþùåå êàæäûé âíîâü ïîñòðîåííûé êîðòåæ èç Si ñ êîðòåæåì èç Si−1 , è ïîìåòèì åãî âåëè÷èíîé xi . Âðåìåííàÿ ñëîæíîñòü ýòîãî àëãîðèòìà O(e · df +1 ), ãäå e ÷èñëî îãðàíè÷åíèé, d ìàêñèìàëüíûé ðàçìåð äîìåíà, à f äëèíà ôðîíòà èíâàçèè. Êàê è äëÿ äðóãèõ äåêîìïîçèöèîííûõ ìåòîäîâ, ýôôåêòèâíîñòü àëãîðèòìà èíâàçèè çàâèñèò îò ìàêñèìàëüíîãî ðàçìåðà ïîäçàäà÷ ÓÎ. Òàêèì îáðàçîì, ýôôåêòèâíîñòü ýòîãî àëãîðèòìà çàâèñèò îò íàõîæäåíèÿ õîðîøåé èíâàçèè, ò.å. èíâàçèè ñ íåáîëüøîé äëèíîé ôðîíòà. Ê ñîæàëåíèþ, âîçìîæíîñòü ñóùåñòâîâàíèÿ õîðîøèõ èíâàçèé äëÿ ëþáûõ ãðàôîâ íå ãàðàíòèðóåòñÿ ïîëíûé ãðàô ñ n âåðøèíàìè èìååò èíâàçèè òîëüêî ñ äëèíîé ôðîíòà n − 1. Äëÿ íåêîòîðûõ çàäà÷, òàêèõ êàê çàäà÷à î n ôåðçÿõ, ìàêñèìàëüíûé ðàçìåð ïîäçàäà÷ ÓÎ áóäåò ðàâåí n. 4.6. Äðåâîâèäíàÿ äåêîìïîçèöèÿ Ïîíÿòèå øèðèíû äåðåâà áûëî ââåäåíî â òåîðèè ãðàôîâ Ðîáåðòñîí è Ñåéìóð (Robertson & Seymour, 1986) [320]. Dechter & Pearl, 1987, 1989) [140], [141], îñíîâûâàÿñü íà ðàáîòå Ôðåéäåðà (Freuder, 1985) [169], ïðèìåíÿëè òî æå ïîíÿòèå (íàçâàííîå èìè èíäóöèðîâàííîé øèðèíîé ) ê çàäà÷àì ÓÎ è ðàçðàáîòàëè ïîäõîä, èñ44 ïîëüçóþùèé äðåâîâèäíóþ äåêîìïîçèöèþ (tree decomposition) ãðàôà îãðàíè÷åíèé è ñîçäàíèè ìíîæåñòâà ñâÿçàííûõ ïîäçàäà÷. Êàæäàÿ ïîäçàäà÷à ðåøàåòñÿ íåçàâèñèìî, à çàòåì èòîãîâûå ðåøåíèÿ êîìáèíèðóþòñÿ. Êàê è áîëüøèíñòâî àëãîðèòìîâ, äåéñòâóþùèõ ïî ïðèíöèïó ¾ðàçäåëÿé è âëàñòâóé¿, ýòîò àëãîðèòì ðàáîòàåò óñïåøíî, åñëè ïîäçàäà÷è íå ñëèøêîì âåëèêè. Ëþáàÿ äðåâîâèäíàÿ äåêîìïîçèöèÿ äîëæíà óäîâëåòâîðÿòü òðåì ïðèâåäåííûì íèæå òðåáîâàíèÿì. • Êàæäàÿ ïåðåìåííàÿ èç ïåðâîíà÷àëüíîé çàäà÷è ïîÿâëÿåòñÿ ïî ìåíüøåé ìåðå â îäíîé èç ïîäçàäà÷. • Åñëè äâå ïåðåìåííûå ïåðâîíà÷àëüíîé çàäà÷è ñâÿçàíû îãðàíè÷åíèåì, òî äîëæíû ïîÿâèòüñÿ âìåñòå (íàðÿäó ñ ýòèì îãðàíè÷åíèåì) ïî ìåíüøåé ìåðå â îäíîé èç ïîäçàäà÷. • Åñëè êàêàÿ-òî ïåðåìåííàÿ ïîÿâëÿåòñÿ â äâóõ ïîäçàäà÷àõ â äåðåâå, òî äîëæíà ïîÿâëÿòüñÿ â êàæäîé ïîäçàäà÷å âäîëü ïóòè, ñîåäèíÿþùåãî ýòè ïîäçàäà÷è. Ïåðâûå äâà óñëîâèÿ ãàðàíòèðóþò, ÷òî â äåêîìïîçèöèè áóäóò ïðåäñòàâëåíû âñå ïåðåìåííûå è îãðàíè÷åíèÿ. Òðåòüå óñëîâèå íà ïåðâûé âçãëÿä êàæåòñÿ äîâîëüíî ôîðìàëüíûì, õîòÿ îíî ïðîñòî îòðàæàåò òî îãðàíè÷åíèå, ÷òî ëþáàÿ êîíêðåòíàÿ ïåðåìåííàÿ äîëæíà èìåòü îäíî è òî æå çíà÷åíèå â êàæäîé ïîäçàäà÷å, â êîòîðîé ïîÿâëÿåòñÿ; ñîáëþäåíèå ýòîãî îãðàíè÷åíèÿ ãàðàíòèðóþò ñâÿçè ìåæäó ïîäçàäà÷àìè â äåðåâå. Êàæäàÿ ïîäçàäà÷à ðåøàåòñÿ íåçàâèñèìî; åñëè êàêàÿ-ëèáî èç íèõ íå èìååò ðåøåíèÿ, òî èçâåñòíî, ÷òî âñÿ çàäà÷à òàêæå íå èìååò ðåøåíèÿ. Åñëè óäàåòñÿ ðåøèòü âñå ïîäçàäà÷è, òî ìîæåò áûòü ïðåäïðèíÿòà ïîïûòêà ñîñòàâèòü ãëîáàëüíîå ðåøåíèå ñëåäóþùèì îáðàçîì. Ïðåæäå âñåãî, êàæäàÿ ïîäçàäà÷à ðàññìàòðèâàåòñÿ êàê ¾ñóïåð-ïåðåìåííàÿ¿, îáëàñòüþ îïðåäåëåíèÿ êîòîðîé ÿâëÿåòñÿ ìíîæåñòâî âñåõ ðåøåíèé ýòîé ïîäçàäà÷è. Çàòåì ðåøàåòñÿ çàäà÷à ñ îãðàíè÷åíèÿìè, ñâÿçûâàþùèìè ïîäçàäà÷è; äëÿ ýòîãî èñïîëüçóåòñÿ ýôôåêòèâíûé àëãîðèòì äëÿ äåðåâüåâ, ïðèâåäåííûé âûøå. Îãðàíè÷åíèÿ, ñâÿçûâàþùèå ïîäçàäà÷è, óêàçûâàþò íà òî, ÷òî ðåøåíèÿ ïîäçàäà÷ äîëæíû áûòü ñîãëàñîâàííûìè ïî èõ îáùèì ïåðåìåííûì. Ëþáîé êîíêðåòíûé ãðàô îãðàíè÷åíèé äîïóñêàåò áîëüøîå êîëè÷åñòâî äðåâîâèäíûõ äåêîìïîçèöèé; ïðè âûáîðå äåêîìïîçèöèè íóæíî ñòðåìèòüñÿ ê òîìó, ÷òîáû ïîäçàäà÷è áûëè êàê ìîæíî ìåíüøå. Øèðèíà äåðåâà äðåâîâèäíîé äåêîìïîçèöèè ãðàôà íà åäèíèöó ìåíüøå ðàçìåðà íàèáîëüøåé ïîäçàäà÷è; øèðèíà äåðåâà ñàìîãî ãðàôà îïðåäåëÿåòñÿ êàê ìèíèìàëüíàÿ øèðèíà äåðåâà ñðåäè âñåõ åãî äðåâîâèäíûõ äåêîìïîçèöèé. Åñëè ãðàô èìååò øèðèíó äåðåâà w è äàíà ñîîòâåòñòâóþùàÿ äðåâîâèäíàÿ äåêîìïîçèöèÿ, òî ñîîòâåòñòâóþùàÿ çàäà÷à ìîæåò áûòü ðåøåíà çà âðåìÿ O(ndw+1 ). Ýòî îçíà÷àåò, ÷òî çàäà÷è ÓÎ ñ ãðàôàìè îãðàíè÷åíèé, õàðàêòåðèçóþùèìèñÿ êîíå÷íîé øèðèíîé äåðåâà, ìîãóò áûòü ðåøåíû çà ïîëèíîìèàëüíîå âðåìÿ. Ê ñîæàëåíèþ, çàäà÷à ïîèñêà äåêîìïîçèöèè ñ ìèíèìàëüíîé øèðèíîé äåðåâà ÿâëÿåòñÿ N P -òðóäíîé, íî ñóùåñòâóþò ýâðèñòè÷åñêèå ìåòîäû, êîòîðûå õîðîøî ðàáîòàþò íà ïðàêòèêå. 45 4.7. Äðóãèå ìåòîäû äåêîìïîçèöèè Îïèðàÿñü íà ðàáîòû (Robertson & Seymour, 1986) [320], (Dechter & Pearl, 1987, 1989) [140], [141], è íà ðåçóëüòàòû èç îáëàñòè òåîðèè áàç äàííûõ, Ãîòòëîá è äð. ( Gottlob et al., 1999) [195], [196] ðàçðàáîòàëè ïîíÿòèå øèðèíû ãèïåðäåðåâà, êîòîðîå îñíîâàíî íà ìåòîäå õàðàêòåðèçàöèè çàäà÷è ÓÎ ñ ïîìîùüþ ãèïåðãðàôà. Îíè íå òîëüêî ïîêàçàëè, ÷òî ëþáóþ çàäà÷ó ÓÎ ñ øèðèíîé ãèïåðäåðåâà w ìîæíî ðåøèòü çà âðåìÿ O(nw+1 log n), íî è îáîñíîâàëè óòâåðæäåíèå, ÷òî êðèòåðèé øèðèíû ãèïåðäåðåâà ïðåâîñõîäèò âñå ðàíåå ïðåäëîæåííûå êðèòåðèè ¾øèðèíû¿ â òîì ñìûñëå, ÷òî â íåêîòîðûõ ñëó÷àÿõ øèðèíà ãèïåðäåðåâà ÿâëÿåòñÿ êîíå÷íîé, òîãäà êàê øèðèíà, îïðåäåëÿåìàÿ äðóãèìè êðèòåðèÿìè, íåîãðàíè÷åííîé. Â (Pearson & Jeavons, 1997) [301] ïðèâåäåí îáçîð ëåãêî ðàçðåøèìûõ êëàññîâ çàäà÷ ÓÎ è îïèñàíû êàê ìåòîäû ñòðóêòóðíîé äåêîìïîçèöèè, òàê è ìåòîäû, îñíîâàííûå íà ñâîéñòâàõ îáëàñòåé îïðåäåëåíèÿ èëè ñâîéñòâàõ ñàìèõ îãðàíè÷åíèé. Â ðàáîòå [115] ïðåäëîæåí îáùèé êëàññ ìåòîäîâ ñòðóêòóðíîé äåêîìïîçèöèè äëÿ çàäà÷ ÓÎ, íàçâàííûé îñòîðîæíîé äåêîìïîçèöèåé (guarded decomposition). Ïîêàçàíî, ÷òî ìíîãèå ñóùåñòâóþùèå ìåòîäû äåêîìïîçèöèè ìîãóò õàðàêòåðèçîâàòüñÿ â òåðìèíàõ íàõîæäåíèÿ îñòîðîæíîé äåêîìïîçèöèè, óäîâëåòâîðÿþùåé íåêîòîðûì ñïåöèàëüíûì äîïîëíèòåëüíûì óñëîâèÿì. 5. Ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ 5.1. Îñíîâíûå ïîäõîäû è ñîñòîÿíèå äåë Ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ (constraint programming) ïðîãðàììíàÿ òåõíîëîãèÿ äëÿ äåêëàðàòèâíîãî îïèñàíèÿ è ýôôåêòèâíîãî ðåøåíèÿ áîëüøèõ êîìáèíàòîðíûõ çàäà÷ â îáëàñòÿõ ïëàíèðîâàíèÿ è êàëåíäàðíîãî ïëàíèðîâàíèÿ. Öåëü ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ (ñì. (Colmerauer, 1990) [120], (Jaar & Lassez, 1987) [220], (Van Hentenryck, 1989) [368]) ñîñòîèò â ðàçðàáîòêå ÿçûêîâ ïðîãðàììèðîâàíèÿ äëÿ çàäàíèÿ îãðàíè÷åíèé è ïðîöåäóð ïîèñêà çàäà÷ ÓÎ. Ïðîöåäóðà ïîèñêà â ÓÎ íåÿâíî îïèñûâàåò îáõîä äåðåâà ïîèñêà, íå îïèñûâàÿ ïðè ýòîì, êàê îáõîäèòü ýòî äåðåâî, ýòî óæå çàäà÷à ñòðàòåãèé ïîèñêà. Îáû÷íî äåðåâî ïîèñêà îáõîäèòñÿ ñ ïîìîùüþ ïîèñêà â ãëóáèíó. Òåì íå ìåíåå, èìåþòñÿ è äðóãèå ñòðàòåãèè ïîèñêà (ñì. (Harvey & Ginsberg, 1995) [206], (Korf, 1996) [249], (Meseguer, 1997) [279], (Walsh, 1997) [381]), ïðåäñòàâëÿþùèå áîëüøîé èíòåðåñ. Â ÷àñòíîñòè, áûëî ïîêàçàíî, ÷òî ïîèñê ñ îãðàíè÷åííîé íåñîâìåñòíîñòüþ (limited discrepancy search (LDS)) è åãî âàðèàöèè äîñòèãàþò íà íåêîòîðûõ ïðèêëàäíûõ çàäà÷àõ ñóùåñòâåííî ëó÷øèõ ðåçóëüòàòîâ ïî ñðàâíåíèþ ñ ïîèñêîì â ãëóáèíó. Áëàãîäàðÿ óñïåøíîìó ðåøåíèþ ìíîãèõ ïðèêëàäíûõ çàäà÷ (ñì. (Harvey & Ginsberg, 1995) [206], (Laburthe & Caseau, 1998) [256], (Perron, 1999) [304]), ñòðàòåãèè ïîèñêà ñòàëè íåîòúåìëåìîé ÷àñòüþ ÿçûêîâ ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ (ñì. (Laburthe & Caseau, 1998) [256], (Perron, 1999) [304], (Schulte, 1997) [336], (Van Hentenryck et al., 2000) [375]). Â ÷àñòíîñòè, ïðîãðàììíàÿ ñèñòåìà Oz âïåðâûå ââåëà ñïåöèôèêàöèþ îáùèõ 46 ñòðàòåãèé ïîèñêà, êîòîðûå ìîãóò áûòü çàäàíû íåçàâèñèìî îò ïðîöåäóðû ïîèñêà (Schulte, 1997) [335]. Â ðåçóëüòàòå, ñòðàòåãèÿ ïîèñêà ìîæåò áûòü ïðèìåíåíà ê ìíîãèì ïðîöåäóðàì ïîèñêà è ïðîöåäóðà ïîèñêà ìîæåò áûòü îñíàùåíà íåñêîëüêèìè ñòðàòåãèÿìè ïîèñêà. Ïîèñêîâûé ÿçûê SALSA15 (Laburthe & Caseau, 1998) [256] òàêæå ñîäåðæèò ðÿä çàäàííûõ îáùèõ ñòðàòåãèé ïîèñêà. Â ðàáîòå (Perron, 1999) [304] ïîêàçàíî, ÷òî ìíîãî ñòðàòåãèé ïîèñêà ìîæåò áûòü çàäàíî ñ ïîìîùüþ âñåãî äâóõ ôóíêöèé: ôóíêöèè îöåíêè, êîòîðàÿ ñòàâèò îöåíêó â ñîîòâåòñòâèå êàæäîé âåðøèíå äåðåâà ïîèñêà è ôóíêöèè ïîäâåøèâàíèÿ, êîòîðàÿ íåÿâíî çàäàåò, êàêàÿ âåðøèíà áóäåò ñëåäóþùåé. Îñíîâíûå èäåè, ëåæàùèå â îñíîâå ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ, ïðîñòû [97]: äåêëàðàòèâíîå ïðåäñòàâëåíèå îãðàíè÷åíèé çàäà÷è, ñîâìåùåííîå ñ îáùèìè ìåòîäàìè ðåøåíèÿ òèïà õðîíîëîãè÷åñêîãî ïîèñêà ñ âîçâðàòàìè èëè ëîêàëüíîãî ïîèñêà. Ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ èìååò ìíîæåñòâî ñèëüíûõ ñòîðîí: ìîùíûå ÿçûêè ìîäåëèðîâàíèÿ, ïîçâîëÿþùèå ïðåäñòàâèòü ñëîæíûå è äèíàìè÷åñêèå ïðèêëàäíûå çàäà÷è; áûñòðûå ìåòîäû âûâîäà îáùåãî íàçíà÷åíèÿ òèïà âûíóæäåíèÿ äóãîâîé ñîâìåñòíîñòè, äëÿ óñå÷åíèÿ ÷àñòè ïðîñòðàíñòâà ïîèñêà; áûñòðûå ìåòîäû âûâîäà ñïåöèàëüíîãî íàçíà÷åíèÿ, ñâÿçàííûå ñ ãëîáàëüíûìè îãðàíè÷åíèÿìè; ãèáðèäíûå ìåòîäû, ñî÷åòàþùèå ïðåèìóùåñòâà ìåòîäîâ ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ è ïîäõîäîâ èññëåäîâàíèÿ îïåðàöèé; ìåòîäû ëîêàëüíîãî ïîèñêà, ïîçâîëÿþùèå áûñòðî íàõîäèòü ðåøåíèÿ, áëèçêèå ê îïòèìàëüíûì; áîëüøîé äèàïàçîí ðàñøèðåíèé òèïà ìÿãêèõ îãðàíè÷åíèé è ðàñïðåäåëåííîãî ðåøåíèÿ îãðàíè÷åíèé, ñ ïîìîùüþ êîòîðûõ âîçìîæíî áîëåå àäåêâàòíîå ìîäåëèðîâàíèå ïðàêòè÷åñêèõ çàäà÷. 5.2. Ëîãè÷åñêîå ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ Çà ïîñëåäíèå 10-15 ëåò ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ [266] ïðîøëî ïóòü îò íàó÷íîé èäåè äî ìîùíîé ïàðàäèãìû ïðîãðàììèðîâàíèÿ, êîòîðàÿ âñå áîëüøå èñïîëüçóåòñÿ äëÿ ìîäåëèðîâàíèÿ è ðåøåíèÿ ìíîãèõ òðóäíûõ ïðàêòè÷åñêè âàæíûõ çàäà÷. Ïîä ïðîãðàììèðîâàíèåì â îãðàíè÷åíèÿõ (constraint programming) ïîíèìàåòñÿ ïðîãðàììèðîâàíèå àëãîðèòìîâ ðåøåíèÿ çàäà÷ ÓÎ [81]. Ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ ñ÷èòàåòñÿ â íàñòîÿùåå âðåìÿ îäíèì èç ñòðàòåãè÷åñêèõ íàïðàâëåíèé èíôîðìàòèêè [372]. Ëîãè÷åñêîå ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ (ËÏÎ) (Constraint Logic Programming (CLP)) [97] áûëî ïðåäëîæåíî â ðàáîòàõ Jaar & Lassez [220], Marriott & Stuckey [266], [37] êàê ðàñøèðåíèå ëîãè÷åñêîãî ïðîãðàììèðîâàíèÿ. Ëîãè÷åñêîå ïðîãðàììèðîâàíèå ýòî ïàðàäèãìà ïðîãðàììèðîâàíèÿ, îñíîâàííàÿ íà ëîãèêå [116]. Åãî êîíñòðóêòû áóëåâû èìïëèêàöèè (ò.å. q ⇒ p èëè p : −q ), êîìïîçèöèé, èñïîëüçóþùèõ áóëåâû îïåðàöèè êîíúþíêöèè è äèçúþíêöèè. Ëîãè÷åñêîå ïðîãðàììèðîâàíèå ìîæåò ðàññìàòðèâàòüñÿ êàê ïðîöåäóðíûé ÿçûê, â êîòîðîì ïðîöåäóðû ýòî áóëåâû ôóíêöèè, ðåçóëüòàò ðàáîòû ïðîãðàììû âñåãäà èñòèíà èëè ëîæü. Ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ ÿâëÿåòñÿ äåêëàðàòèâíîé ïàðàäèãìîé, ïîç15 SALSA: A Language for Search Algorithms. 47 âîëÿþùåé âûðàçèòü ìíîãèå ïðèêëàäíûå çàäà÷è ñ ïîìîùüþ îãðàíè÷åíèé, ò.å. îòíîøåíèé ìåæäó ïåðåìåííûìè çàäà÷è. Ïîäîáíûé ïóòü çàäàíèÿ çàäà÷ ìîæåò ïðîèçâîäèòüñÿ äëÿ ìíîãèõ òðàäèöèîííûõ ñðåä ïðîãðàììèðîâàíèÿ, íî ëó÷øå âñåãî ïîäõîäÿò òå ñðåäû, êîòîðûå èñïîëüçóþò äåêëàðàòèâíóþ ïàðàäèãìó ïðîãðàììèðîâàíèÿ, ïîäîáíûå ïàðàäèãìå ëîãè÷åñêîãî ïðîãðàììèðîâàíèÿ [261], êîòîðàÿ òàêæå îñíîâàíà íà èñïîëüçîâàíèè îòíîøåíèé. Ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ (Marriott & Stuckey, 1998) [266] ñîñòîèò èç äâóõ íåîáõîäèìûõ êîìïîíåíòîâ: ðåøàòåëÿ îãðàíè÷åíèé è ìàøèíû ïîèñêà. Õîòÿ ðàíåå è áûëè ðàçðàáîòàíû íåêîòîðûå ÿçûêè äëÿ ïîääåðæàíèÿ îãðàíè÷åíèé ñïåöèàëüíîãî âèäà (ThingLab [78], Bertrand [259], è ALICE [258]), íî íå áûëî íèêàêîé îáùåé ëèíãâèñòè÷åñêîé ïîääåðæêè äëÿ ðåøåíèÿ çàäà÷ ÓÎ è ðàñïðîñòðàíåíèÿ îãðàíè÷åíèé äî ðàçðàáîòêè ñõåìû ËÏÎ. Ýòà ñõåìà ìîæåò ðàññìàòðèâàòüñÿ êàê îáîëî÷êà ëîãè÷åñêîãî ïðîãðàììèðîâàíèÿ, ïîääåðæèâàåìàÿ ñèñòåìîé îãðàíè÷åíèé, ñ ïîìîùüþ êîòîðîé ìîæåò çàïèñûâàòü è ðåøàòü ëþáûå îãðàíè÷åíèÿ. Ïî ýòîé ïðè÷èíå ñõåìà ËÏÎ ìîæåò îáîçíà÷àòüñÿ êàê ËÏÎ(X), ãäå X îïðåäåëÿåò êëàññ îãðàíè÷åíèé, ñ êîòîðûìè ìû õîòèì ðàáîòàòü. Ïîíÿòíî, ÷òî îïèñàíèå çàäà÷è â âèäå ñèñòåìû îãðàíè÷åíèé ÿâëÿåòñÿ äåêëàðàòèâíûì, ïðè÷åì îïèñûâàåòñÿ, ¾÷òî¿ äîëæíî âûïîëíÿòüñÿ, íî íå óêàçûâàåòñÿ, ¾êàê¿ ýòî ñäåëàòü. Ðàçðàáîòàíî ìíîãî ìåòîäîâ ðåøåíèÿ, ïîçâîëÿþùèõ (òî÷íî èëè ïðèáëèæåííî) íàõîäèòü îäíî èëè íåñêîëüêî ðåøåíèé çàäà÷è ÓÎ. Íàèáîëåå èçâåñòíûì ÿâëÿåòñÿ îïèñàííûé âûøå ïîèñê ñ âîçâðàòàìè, êîòîðûé ìîæåò â îáùåì ñëó÷àå îáîéòè âñå äåðåâî âîçìîæíûõ çíà÷åíèé äëÿ íàõîæäåíèÿ ðåøåíèÿ, èëè óñòàíîâëåíèÿ ôàêòà îòñóòñòâèÿ ðåøåíèé. Òîò ôàêò, ÷òî ñèñòåìû óäîâëåòâîðåíèÿ îãðàíè÷åíèé è ëîãè÷åñêîå ïðîãðàììèðîâàíèå [261] äåêëàðàòèâíûå ïàðàäèãìû, èñïîëüçóþùèå îòíîøåíèÿ è ïîèñê ñ âîçâðàòàìè, ñäåëàë èõ èíòåãðàöèþ ëåãêîé è åñòåñòâåííîé. ×ðåçâû÷àéíî âàæíûì ÿâèëñÿ ôàêò îáúåäèíåíèÿ ËÏÎ ñ îáëàñòüþ èñêóññòâåííîãî èíòåëëåêòà, íàçûâàåìîãî óäîâëåòâîðåíèå îãðàíè÷åíèé. ËÏÎ ðàñøèðåíèå ëîãè÷åñêîãî ïðîãðàììèðîâàíèÿ, â êîòîðîì óíèôèêàöèÿ ìîæåò áûòü çàìåíåíà èëè äîïîëíåíà äðóãèìè âèäàìè îãðàíè÷åíèé, îïðåäåëåííûõ íà äîìåíàõ (îáëàñòÿõ îïðåäåëåíèÿ) èñïîëüçóåìûõ ïåðåìåííûõ. ËÏÎ è óäîâëåòâîðåíèå îãðàíè÷åíèé èìåþò îáùèå öåëè, à òàêæå èñïîëüçóþò êîìïàêòíîå äåêëàðàòèâíîå ôîðìóëèðîâàíèå êîìáèíàòîðíûõ çàäà÷ è ýôôåêòèâíûå ñòðàòåãèè ïîèñêà. Òàê, ðàñïðîñòðàíåíèå îãðàíè÷åíèé èñïîëüçóåòñÿ êàê äëÿ ðåøåíèÿ îãðàíè÷åíèé ñ êîíå÷íûìè äîìåíàìè, òàê è â ïðîãðàììèðîâàíèè â îãðàíè÷åíèÿõ. Îáùåïðèíÿòûé ïîäõîä ê ðåøåíèþ çàäà÷ ÓÎ ñî÷åòàåò äåðåâî ïîèñêà ñ âîçâðàòàìè ñ ðàñïðîñòðàíåíèåì îãðàíè÷åíèé. Ýòî ðåàëèçîâàíî â ñèñòåìàõ ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ ñ êîíå÷íûìè äîìåíàìè, òàêèõ êàê SICStus Prolog [99], ILOG Solver [216], è Gecode [183], óñïåøíî èñïîëüçîâàííûõ äëÿ ðåøåíèÿ ìíîãèõ ïðàêòè÷åñêèõ çàäà÷. Êîãäà ÇÓÎ âñòðîåíà â ëîãè÷åñêîå ïðîãðàììèðîâàíèå, òî îãðàíè÷åíèå ìîæåò áûòü îïðåäåëåíî â ïðîãðàììå êàê ìíîæåñòâî ôàêòîâ èëè äàæå ìíîæåñòâî ïðàâèë. Îñíîâíàÿ èäåÿ ËÏÎ ñîñòîèò â ðàñøèðåíèè ëîãè÷åñêîãî ïðîãðàììèðîâàíèÿ 48 òàê, ÷òî ïîäñòàíîâêè çàìåíÿþòñÿ îãðàíè÷åíèÿìè, óíèôèêàöèÿ çàìåíÿåòñÿ ðåøåíèåì çàäà÷è ÓÎ, à âñå ñåìàíòè÷åñêèå ñâîéñòâà ëîãè÷åñêîãî ïðîãðàììèðîâàíèÿ (â îñíîâíîì, ñóùåñòâîâàíèå äåêëàðàòèâíûõ, äåíîòàöèîíàëüíûõ è ïðîöåäóðàëüíûõ ñåìàíòèê è ýêâèâàëåíòíîñòü ýòèõ òðåõ îáúåêòîâ) âñå åùå âûïîëíÿþòñÿ. Jaar & Lassez (1987) [220] ïîêàçàëè, ÷òî òåîðåòè÷åñêèå îñíîâû ÿçûêîâ ëîãè÷åñêîãî ïðîãðàììèðîâàíèÿ îñòàþòñÿ ñïðàâåäëèâûìè äëÿ ÿçûêîâ ËÏÎ. Íåêîòîðûå ÿçûêè ËÏÎ äî ñèõ ïîð èñïîëüçóþòñÿ ñïåöèàëèñòàìè ïî ëîãè÷åñêîìó ïðîãðàììèðîâàíèþ: PROLOG III è IV, CLP(R)16 , CHIP17 CLP(BNR)18 . 5.3. Ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ ñ êîíå÷íûìè äîìåíàìè Ìíîãèå ÿçûêè ËÏÎ áûëè ñîçäàíû äëÿ ðàáîòû ñ îãðàíè÷åíèÿìè ñ êîíå÷íûìè äîìåíàìè. ËÏÎ ñ êîíå÷íûìè äîìåíàìè âïåðâûå áûëî ðåàëèçîâàíî â êîíöå 1980õ ãîäîâ Van Hentenryck [368] â ÿçûêå CHIP [149]. Äðóãèìè ïðèìåðàìè ìîãóò ñëóæèòü ÿçûêè ECLiPSe19 [158], [378] è CLP(FD)20 [109]. Îäíèì èç ãëàâíûõ ïðåèìóùåñòâ èñïîëüçîâàíèÿ ñèñòåìû îãðàíè÷åíèé â ðàìêàõ ÿçûêà ËÏÎ ÿâëÿåòñÿ êîíòðîëü çà ïðîöåññîì ïîèñêà. Èñïîëüçîâàíèå ìåòîäîâ ÷àñòè÷íîãî ïîèñêà (partial search) ñóùåñòâåííî óëó÷øèëî ìåòîäû ïîèñêà. Ìåòîäû ÷àñòè÷íîãî ïîèñêà îãðàíè÷èâàþò ÷èñëî àëüòåðíàòèâ, èçó÷àåìûõ â ðàçëè÷íûõ âåðøèíàõ äåðåâà ïîèñêà. Òàêèì îáðàçîì, îáû÷íûé ïîèñê â ãëóáèíó ñ âîçâðàòàìè, èñïîëüçóåìûé â ËÏÎ, êîòîðûé èíîãäà ïðîâåðÿåò ëèøü íåáîëüøóþ ÷àñòü äåðåâà ïîèñêà, ìîæåò áûòü çàìåíåí íà äðóãèå ïîèñêîâûå ñõåìû (òàêèå êàê LDS [206] èëè ïîèñê [57]), êîòîðûå ìîãóò ðàññìàòðèâàòü ìíîãî ðàçëè÷íûõ ïóòåé â äåðåâå ïîèñêà. Âûøå, â ðàçäåëå 2.2 îïèñàíû ãëîáàëüíûå îãðàíè÷åíèÿ, ïîçâîëÿþùèå ñíèçèòü ïåðåáîð ïðè ðåøåíèè çàäà÷ ÓÎ. ßçûêè, ïîäîáíûå CHIP [149], èìåþò ñëîæíóþ ìîðôîëîãèþ ãëîáàëüíûõ îãðàíè÷åíèé [56]. Äðóãèå ÿçûêè, òàêèå êàê SICStus Prolog [99], ECLiPSe [158], [378], IF Prolog http://www.ifcomputer.de/Products/Prolog/home_en.html, à òàêæå Oz [207], [353], [204], [292], âêëþ÷àþò íåêîòîðûå èç ãëîáàëüíûõ îãðàíè÷åíèé, èñïîëüçóåìûõ â CHIP. Ðàáîòû ïî èñïîëüçîâàíèþ ðàñïðîñòðàíåíèÿ îãðàíè÷åíèé äëÿ ãëîáàëüíûõ îãðàíè÷åíèé (â ÷àñòíîñòè, îãðàíè÷åíèÿ all-dierent) ïîêàçûâàþò ïðàêòè÷åñêóþ öåííîñòü èñïîëüçîâàíèÿ îãðàíè÷åíèé ýòîãî âèäà [313], [338]. Ãëîáàëüíûå îãðàíè÷åíèÿ áûëè èñïîëüçîâàíû òàêæå äëÿ ÷àñòè÷íûõ çàäà÷ ÓÎ, â êîòîðûõ íåêîòîðûå îãðàíè÷åíèÿ ìîãóò íàðóøàòüñÿ ïðè íàõîæäåíèè ðåøåíèÿ [43]. Çàäà÷è ñ äèíàìè÷åñêèìè îãðàíè÷åíèÿìè ýòî çàäà÷è ÓÎ, èçìåíÿþùèåñÿ âî âðåìåíè, ñ ïîìîùüþ äîáàâëåíèÿ èëè èñêëþ÷åíèÿ íåêîòîðûõ îãðàíè÷åíèé, ëèáî êîìáèíàöèè ýòèõ äâóõ îïåðàöèé. Â òî âðåìÿ êàê äîáàâëåíèå îãðàíè÷åíèé ÷àñòî èñïîëüçóåòñÿ íà êàæäîì øàãå âû÷èñëåíèé, èñêëþ÷åíèå îãðàíè÷åíèé ðàññìàòðèâàåòñÿ ðåäêî, õîòÿ ýòî áûëî áû âåñüìà ïîëåçíî äëÿ èíòåðàêòèâíûõ èëè äèíàìè16 CLP(R): Constraint Logic Programming (Reals). Constraint Handling in Prolog. 18 CLP(BNR): Constraint Logic Programming (Booleans, Naturals and Reals). 19 ECLiPSe: Constraint Logic Programming System. 20 CLP(FD): Constraint Logic Programming over Finite Domains. 17 CHIP: 49 ÷åñêèõ ñèñòåì, â òîì ÷èñëå ïðè íåîáõîäèìîñòè ïåðåñ÷åòà ðåøåíèÿ ïðè èçìåíåíèè óñëîâèé çàäà÷è ÓÎ, à òàêæå ïîèñêà óñòîé÷èâûõ ðåøåíèé [380]. Ïî ýòîé ïðè÷èíå, â ðÿäå ðàáîò ââåäåíî ïîíÿòèå èñêëþ÷åíèÿ (èëè îòìåíû) îãðàíè÷åíèé, è ïðåäëîæåíû ýôôåêòèâíûå àëãîðèòìû äëÿ ýòîãî â ðàìêàõ êëàññè÷åñêèõ ÿçûêîâ ËÏÎ. Íàïðèìåð, ïðîãðàììíàÿ ñõåìà CLP(FD) áûëà ðàñøèðåíà êàê â ïëàíå ñèíòàêñèñà, ñåìàíòèêè è âîçìîæíîñòè îòìåíû îãðàíè÷åíèé â [184]. 5.4. Íåëèíåéíûå îãðàíè÷åíèÿ â ÿçûêàõ ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ. Âîçìîæíîñòü èñïîëüçîâàòü è ðåøàòü íåëèíåéíûå îãðàíè÷åíèÿ èìååò âàæíîå çíà÷åíèå âî ìíîãèõ îáëàñòÿõ ïðèëîæåíèé (ýêîíîìèêà, ìåõàíèêà, õèìè÷åñêîå ïðîèçâîäñòâî, òåõíè÷åñêèå ïðèëîæåíèÿ è äð.). Îáû÷íî ðåøåíèå çàäà÷ òàêîãî òèïà äîñòàòî÷íî ñëîæíî è òðåáóåò ïðèâëå÷åíèÿ áîëüøèõ âû÷èñëèòåëüíûõ ðåñóðñîâ. Ïîìèìî êëàññè÷åñêèõ ìåòîäîâ, â íàñòîÿùåå âðåìÿ çäåñü èñïîëüçóþòñÿ ìåòîäû èíòåðâàëüíîãî àíàëèçà [295]. Ïî ýòîé æå ïðè÷èíå, èíòåðâàëüíûå îãðàíè÷åíèÿ [62] èñïîëüçóþòñÿ â òàêèõ ÿçûêàõ ËÏÎ, êàê ECLiPSe [158], [378],[385] è CLP(BNR), à ÿçûêè Helios [370] è Numerica [371] áûëè ðàçðàáîòàíû äëÿ ðåøåíèÿ ñèñòåì íåëèíåéíûõ îãðàíè÷åíèé, èñïîëüçóÿ àëãîðèòìû èíòåðâàëüíîãî àíàëèçà [295]. 5.5. Ìîäåëèðîâàíèå. ×ðåçâû÷àéíî âàæíî èìåòü õîðîøèé ÿçûê ìîäåëèðîâàíèÿ äëÿ çàïèñè çàäà÷ ÓÎ. Â íàñòîÿùåå âðåìÿ áîëüøèíñòâî ñèñòåì ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ ÿâëÿþòñÿ ëèáî ðàñøèðåíèåì äåêëàðàòèâíîãî ÿçûêà ïðîãðàììèðîâàíèÿ (òèïà Prolog), ëèáî áèáëèîòåêàìè, èñïîëüçóåìûìè ñîâìåñòíî ñ îáû÷íûìè ÿçûêàìè ïðîãðàììèðîâàíèÿ (òàêèìè êàê C/C++). Ñåé÷àñ èìååòñÿ ìíîãî èññëåäîâàíèé ïî ðàçðàáîòêå ÿçûêîâ ìîäåëèðîâàíèÿ îãðàíè÷åíèé, êîòîðûå äåëàþò ôàçó ìîäåëèðîâàíèÿ áîëåå ëåãêîé è åñòåñòâåííîé (ïðèìåðû: Helios [370], Numerica [371] è OPL [374]). Íåêîòîðûå èç ýòèõ ÿçûêîâ èñïîëüçóþò ñðåäñòâà âèçóàëèçàöèè äëÿ îïèñàíèÿ è ãåíåðèðîâàíèÿ ïðîãðàìì â îãðàíè÷åíèÿõ. Íàïðèìåð, ñèñòåìà FORWARD êàëåíäàðíîãî ïëàíèðîâàíèÿ íåôòåïåðåãîííîãî çàâîäà [191] èñïîëüçóåò ñðåäñòâî ãðàôè÷åñêîé ñïåöèôèêàöèè äëÿ CHIP [344], ïðè ýòîì ìîäåëü íåôòåïåðåãîííîãî çàâîäà ââîäèòñÿ ñ ïîìîùüþ ãðàôè÷åñêîãî èíòåðôåéñà. CML21 ÿçûê ìîäåëèðîâàíèÿ îãðàíè÷åíèé, êîòîðûé áûë ðàçðàáîòàí äëÿ îáëåã÷åíèÿ ïîñòàíîâêè è ìîäèôèêàöèè êîìáèíàòîðíûõ çàäà÷ [32]: CML-ñïåöèôèêàöèè çàäà÷è ïîçâîëÿþò ôîðìóëèðîâàòü îãðàíè÷åíèÿ, îòðàæàþùèå âûðàæåíèÿ åñòåñòâåííîãî ÿçûêà, êîòîðûå çàòåì ïåðåâîäÿòñÿ â ýôôåêòèâíûå CHIP-ïðîãðàììû. Äðóãèì ÿçûêîì ìîäåëèðîâàíèÿ ÿâëÿåòñÿ VISUAL SOLVER [373], êîòîðûé ïîçâîëÿåò çàäàòü êàê ïîñòàíîâêó çàäà÷è, òàê è ýâðèñòèêè, èñïîëüçóåìûå äëÿ ïîèñêà ðåøåíèé. Äðóãèå ïîäõîäû ê ìîäåëèðîâàíèþ ðåàëüíûõ çàäà÷ â âèäå çàäà÷è ÓÎ èñïîëüçóþò èíòåðàêòèâíûå ìåòîäû: ðåøàòåëü çàäà÷è ÓÎ îáùàåòñÿ ñ ïîëüçîâàòåëåì, ïðåä21 CML: Constraint Modelling Language. 50 ëàãàþùèì ÷àñòè÷íûå ðåøåíèÿ, è ïîëó÷àåò äàëüíåéøóþ èíôîðìàöèþ î çàäà÷å îò ÷åëîâåêà íà îñíîâàíèè îöåíêè ðåøàòåëåì ýòèõ ÷àñòè÷íûõ ðåøåíèé [173]. Â ÷èñëå äðóãèõ ìåòîäîâ ñòîèò óïîìÿíóòü ìåòîäû ìàøèííîãî îáó÷åíèÿ: ïðè èñïîëüçîâàíèè ìÿãêèõ îãðàíè÷åíèé ïîëüçîâàòåëü îöåíèâàåò ðåøåíèÿ, à ñèñòåìà èçó÷àåò çàäà÷ó ÓÎ [327], ÷òî âåñüìà óïðîùàåò ðåøåíèå çàäà÷è ñ ìÿãêèìè îãðàíè÷åíèÿìè. Äðóãîé èíòåðåñíîé ñõåìîé ìîäåëèðîâàíèÿ ÿâëÿåòñÿ ñõåìà, îñíîâàííàÿ íà àáäóêöèîííîì ËÏÎ (Abductive Constraint Logic Programming (ACLP) (ñì. http://www.cs.ucy.ac.cy/aclp/), ïîñòðîåííàÿ íà îñíîâå ECLiPSe [158]. 5.6. Îòëàäêà. Îäíîé èç ðàñïðîñòðàíåííûõ ïðîáëåì, âîçíèêàþùèõ ó ïîëüçîâàòåëåé ñèñòåì ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ ÿâëÿåòñÿ èíîãäà íåïðåäñêàçóåìîå ïîâåäåíèå ìîäåëè ñ îãðàíè÷åíèÿìè: äàæå íåáîëüøèå èçìåíåíèÿ ïðîãðàììû èëè äàííûõ ìîæåò ïðèâîäèòü ê ñóùåñòâåííîìó èçìåíåíèþ ïðîèçâîäèòåëüíîñòè ïðîãðàììû. Â òî âðåìÿ, êàê ïðîñòûå ïðèëîæåíèÿ ìîãóò áûòü ñîçäàíû äîñòàòî÷íî áûñòðî, îçíàêîìëåíèå ñî âñåìè âîçìîæíîñòÿìè ñèñòåìû â îãðàíè÷åíèÿõ ìîæåò çàíÿòü ìíîãî âðåìåíè. Îäíîé èç ïîïûòîê ðåøåíèÿ ýòèõ ïðîáëåì ÿâëÿåòñÿ âèçóàëèçàöèÿ äåðåâà ïîèñêà [100]. Ïðîãðàììíûå ñðåäñòâà òèïà Oz Explorer [335], ïîèñêîâîãî äåðåâà CHIP [345] (îáçîð ðàííèõ ïðèëîæåíèé CHIP ïðèâåäåí â ðàáîòå [150]), ñèñòåìà, íåäàâíî ðàçðàáîòàííàÿ â ðàìêàõ ñèñòåìû CIAO [208], êîòîðàÿ îáåñïå÷èâàåò 3D-âèçóàëèçàöèþ ñ ïîìîùüþ VRML [352], ïîçâîëÿþò íàìíîãî ãëóáæå ïîíÿòü ïðîöåññ ïîèñêà, ïîñêîëüêó îíè ïðåäîñòàâëÿþò äîñòàòî÷íóþ èíôîðìàöèþ äëÿ àíàëèçà ðàñïðîñòðàíåíèÿ îãðàíè÷åíèé è ñîñòîÿíèÿ äîìåíîâ ïåðåìåííûõ â êàæäîé âåðøèíå äåðåâà. 5.7. Ìÿãêèå îãðàíè÷åíèÿ. Â ðàáîòå [109] ÿçûê ËÏÎ CLP(FD), ðàáîòàþùèé ñ êîíå÷íûìè äîìåíàìè, áûë îáîáùåí äëÿ ðàáîòû ñ îãðàíè÷åíèÿìè èç ïîëóêîëüöà, â ðåçóëüòàòå ïîëó÷åí ÿçûê CLP(FD,S) 22 [185], S êàêîå-ëèáî ïîëóêîëüöî (semiring), âûáèðàåìîå ïîëüçîâàòåëåì. Âûáèðàÿ êîíêðåòíîå ïîëóêîëüöî, ïîëüçîâàòåëü ðåøàåò èñïîëüçîâàòü êîíêðåòíûé êëàññ ìÿãêèõ îãðàíè÷åíèé: íå÷åòêèõ, îïòèìèçèðóåìûõ, âåðîÿòíîñòíûõ, ëèáî êëàññè÷åñêèõ æåñòêèõ îãðàíè÷åíèé. Äëÿ îãðàíè÷åíèé ñ îöåíêàìè ïîêà íåò ïîëíûõ ÿçûêîâ ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ äëÿ ðàáîòû ñ íèìè, íî èìååòñÿ ìíîãî ìåòîäîâ íàõîæäåíèÿ õîðîøèõ íèæíèõ îöåíîê äëÿ èõ îïòèìàëüíûõ ðåøåíèé [98], [143]. 5.8. ßçûêè çàïðîñîâ ñ îãðàíè÷åíèÿìè (Constraint query languages). Ïîñêîëüêó îãðàíè÷åíèÿ èñïîëüçóþò îòíîøåíèÿ (êàê è áàçû äàííûõ (ÁÄ)), ìíîãèå èññëåäîâàòåëè èçó÷àëè âçàèìîñâÿçü ìåæäó ËÏÎ è ÁÄ. Â ýòîé îáëàñòè èññëåäîâàíèé èìåþòñÿ äâà ãëàâíûõ íàïðàâëåíèÿ. Ñ îäíîé ñòîðîíû, ðåçóëüòàòû 22 CLP(FD,S): Semiring-based Constraint Logic Programming Language over Finite Domains. 51 èç òåîðèè ÁÄ (íàïðèìåð, ïî äåêîìïîçèöèè è âûâîäó (ïîëó÷åíèþ) èíôîðìàöèè) áûëè îáîáùåíû è ïðèìåíåíû äëÿ çàäà÷ ÓÎ, ÷òî ïîçâîëèëî ðàçðàáîòàòü ïîëåçíûå èíñòðóìåíòû äëÿ ïðîâåðêè íåêîòîðûõ êëàññîâ çàäà÷ ÓÎ íà ëåãêóþ ðàçðåøèìîñòü è ýôôåêòèâíîãî ðåøåíèÿ ýòèõ çàäà÷ [201], [110]. Ñ äðóãîé ñòîðîíû, êëàññè÷åñêèå ÿçûêè îáðàáîòêè çàïðîñîâ â ÁÄ áûëè ðàñøèðåíû äëÿ ðàáîòû ñ îãðàíè÷åíèÿìè, êîòîðûå èñïîëüçóþòñÿ íå òîëüêî äëÿ óñòàíîâëåíèÿ öåëîñòíîñòè èíôîðìàöèè, íî òàêæå êàê ñðåäñòâî äëÿ ïðåäñòàâëåíèÿ ìíîæåñòâà êîðòåæåé ÁÄ êîìïàêòíûì îáðàçîì. Äåéñòâèòåëüíî, êîðòåæ ðåëÿöèîííîé ÁÄ ìîæåò ðàññìàòðèâàòüñÿ êàê îñîáûé âèä îãðàíè÷åíèÿ, à èìåííî, êîíúþíêöèè îãðàíè÷åíèé-ðàâåíñòâ ìåæäó àòðèáóòàìè êîðòåæà è çíà÷åíèé èç äàííûõ äîìåíîâ. Ââåäåíèå íîâûõ ëîãè÷åñêèõ ôîðìóë äëÿ âûðàæåíèÿ âçàèìîîòíîøåíèé (ò.å. îãðàíè÷åíèé) ìåæäó àòðèáóòàìè ÁÄ ïðèâîäèò ê îïðåäåëåíèþ ÁÄ ñ îãðàíè÷åíèÿìè (Constraint Databases) êàê íîâóþ èíòåðåñíóþ îáëàñòü èññëåäîâàíèé [83], [237]. Îãðàíè÷åíèÿ ìîãóò áûòü äîáàâëåíû ê ðåëÿöèîííûì ÁÄ íà ðàçíûõ óðîâíÿõ. Íà óðîâíå äàííûõ îãðàíè÷åíèÿ ìîãóò âûðàæàòü îáîáùåííûå êîðòåæè, ñïîñîáíûå çàäàâàòü áåñêîíå÷íûå ìíîæåñòâà êîðòåæåé. Îãðàíè÷åíèÿ ïðåäñòàâëÿþò ñîáîé ìîùíûé ìåõàíèçì äëÿ ìîäåëèðîâàíèÿ ïðîñòðàíñòâåííûõ è òåìïîðàëüíûõ ïîíÿòèé, âîçíèêàþùèõ â òåîðèè ÁÄ, ãäå èìååòñÿ íåîáõîäèìîñòü â êîìïàêòíîì ïðåäñòàâëåíèè êîíòèíóàëüíîé èíôîðìàöèè [60], [199]. Äëÿ ðàáîòû ñ îãðàíè÷åíèÿìè áûëè ðàñøèðåíû ðåëÿöèîííîå èñ÷èñëåíèå è ðåëÿöèîííàÿ àëãåáðà êëàññè÷åñêèõ ÁÄ [192], [236], [237], ÷òî ïðèâåëî ê íîâûì áîëåå âûðàçèòåëüíûì ÿçûêàì îáðàáîòêè çàïðîñîâ. Ïîäîáíûé ÿçûê áûë ïîëó÷åí ïóòåì äîáàâëåíèÿ ëèíåéíûõ îãðàíè÷åíèé ê Datalog â [317]. Äðóãèìè ïðèìåðàìè ìîãóò ñëóæèòü áåçîïàñíûé ñòðàòèôèöèðîâàííûé Datalog, ðàññìîòðåííûé â [316], à òàêæå èíòåãðàöèÿ ëèíåéíûõ îãðàíè÷åíèé ñ ÁÄ, îïòèìèçàöèÿ çàïðîñîâ äëÿ êîòîðûõ áûëà èññëåäîâàíà â [84]. Îïòèìèçàöèÿ çàïðîñîâ áûëà òàêæå ðàññìîòðåíà â [362] â êîíòåêñòå äåäóêòèâíûõ çàïðîñîâ â ÁÄ ñ îãðàíè÷åíèÿìè. 5.9. Êîíêóðèðóþùèå è ðàñïðåäåëåííûå ñèñòåìû. Èñïîëüçîâàíèå îãðàíè÷åíèé ïðè ìîäåëèðîâàíèè ïîâåäåíèÿ êîíêóðèðóþùèõ ñèñòåì íà÷àëîñü ñ ðàçâèòèåì ïðîãðàììèðîâàíèÿ â êîíêóðèðóþùèõ îãðàíè÷åíèÿõ (concurrent constraint programming), â êîòîðîé êîíêóðèðóþùèå àãåíòû äåëÿò ìåæäó ñîáîé õðàíèëèùå îãðàíè÷åíèé, ïðè÷åì îíè ìîãóò ëèáî ñîîáùàòü î íàëè÷èè äîïîëíèòåëüíûõ îãðàíè÷åíèé (tell operation), îáìåíèâàòüñÿ ñîîáùåíèÿìè ñ äðóãèìè àãåíòàìè, èëè ñïðàøèâàòü, ÷òî èìååòñÿ ëè â íàëè÷èè (ask operation), ñèíõðîíèçèðîâàòüñÿ ñ äðóãèìè àãåíòàìè. ßçûêè ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ, òàêèå êàê Oz [207], [353], [204], [292], AKL23 [205], [224], [290], CHR24 [175], CIAO [208] èñïîëüçîâàëè è âíåäðèëè ýòó ìîäåëü, îáåñïå÷èâ òåì ñàìûì ïðîãðàììíóþ îñíîâó äëÿ êîíêóðèðóþùèõ ñèñòåì. Òåìïîðàëüíîå êîíêóðåíòíîå ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ (Temporal con23 AKL: 24 CHR: Andorra Kernel Language. Constraint Handling Rules. 52 current constraint programming) è íåäåòåðìèíèñòñêîå òåìïîðàëüíîå êîíêóðåíòíîå ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ (non-deterministic temporal concurrent constraint programming (NTCC)) ÿâëÿþòñÿ âàðèàíòàìè ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ, ðàáîòàþùèìè ñî âðåìåíåì. Ðàñøèðåíèÿìè ýòîé ìîäåëè ÿâëÿþòñÿ ðàñïðåäåëåííûå ñèñòåìû [386] è ñèñòåìû ðåàëüíîãî âðåìåíè [129]. 5.10. Ñîâðåìåííûå ïàêåòû ïðîãðàììíîãî îáåñïå÷åíèÿ CHIP [149] áûë ïåðâûì ÿçûêîì ËÏÎ, èñïîëüçóþùèì ðàñïðîñòðàíåíèå îãðàíè÷åíèé [368]. Äðóãèìè ïðèìåðàìè ñèñòåì ËÏÎ ìîãóò ñëóæèòü áèáëèîòåêè ïîääåðæêè îãðàíè÷åíèé ILOG [216] è COSYTEC [123], à òàêæå ÿçûêè ëîãè÷åñêîãî ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ Prolog III [119], [120], Prolog IV [307], CLP(R) [222] , ECLiPSe [158], [378], CIAO [208], CLP(FD) [109]. Èñïîëüçóÿ ýòè ñèñòåìû, ìíîãèå ñëîæíûå ïðèêëàäíûå çàäà÷è áûëè óñïåøíî ðåøåíû ñ ïîìîùüþ òåõíîëîãèé ÓÎ. Â ÷èñëå ïðèìåðîâ óïîìÿíåì ïðîâåðêó ýëåêòðîííûõ ñõåì, êàëåíäàðíîå ïëàíèðîâàíèå, ðàñïðåäåëåíèå ðåñóðñîâ, ñîñòàâëåíèå ðàñïèñàíèé, óïðàâëÿþùèå ñèñòåìû, ãðàôè÷åñêèå èíòåðôåéñû, à òàêæå ìíîæåñòâî êîìáèíàòîðíûõ çàäà÷ [368], [377]. Ñèíòàêñèñ äëÿ âûðàæåíèÿ îãðàíè÷åíèé íàä êîíå÷íûìè äîìåíàìè çàâèñèò îò ÿçûêà ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ. Íèæå ïðèâåäåíà ïðîãðàììà íà ÿçûêå Prolog, ðåøàþùàÿ êëàññè÷åñêèé êðîññâîðä èç ãëàâû 1 SEND+MORE=MONEY: sendmore(Digits) :Digits = [S,E,N,D,M,O,R,Y], Digits :: [0..9], S #\= 0, M #\= 0, alldifferent(Digits), % % % % % % % Ñîçäàíèå ïåðåìåííûõ Äîìåíû ïåðåìåííûõ Îãðàíè÷åíèå: S äîëæíî îòëè÷àòüñÿ îò 0 Îãðàíè÷åíèå: M äîëæíî îòëè÷àòüñÿ îò 0 âñå ïåðåìåííûå äîëæíû ïðèíèìàòü ðàçëè÷íûå çíà÷åíèÿ Äðóãèå îãðàíè÷åíèÿ 1000*S + 100*E + 10*N + D + 1000*M + 100*O + 10*R + E #= 10000*M + 1000*O + 100*N + 10*E + Y, labeling(Digits). % Íà÷àòü ïîèñê Èíòåðïðåòàòîð ñîçäàåò ïåðåìåííóþ äëÿ êàæäîé áóêâû êðîññâîðäà. Ñèìâîë :: èñïîëüçóåòñÿ äëÿ çàäàíèÿ äîìåíîâ ýòèõ ïåðåìåííûõ, òàê ÷òî îíè ïðèíèìàþò ìíîæåñòâî çíà÷åíèé {0, 1, 2, 3, . . . , 9}. Îãðàíè÷åíèÿ S# = 0 è M # = 0 îçíà÷àþò, ÷òî ýòè ïåðåìåííûå íå ìîãóò ïðèíèìàòü çíà÷åíèå 0. Ïðè îöåíêå èíòåðïðåòàòîðîì ýòèõ îãðàíè÷åíèé îí óìåíüøàåò äîìåíû ýòèõ äâóõ ïåðåìåííûõ, óäàëÿÿ çíà÷åíèå 0 èç íèõ. Çàòåì ðàññìàòðèâàåòñÿ è çàïîìèíàåòñÿ îãðàíè÷åíèå alldierent(Digits). Ïîñëåäíåå àëãåáðàè÷åñêîå îãðàíè÷åíèå âûðàæàåò òîò ôàêò, ÷òî öèôðû, ïðèñâîåííûå áóêâàì, äîëæíû áûòü òàêèìè, ÷òîáû ¾SEND+MORE=MONEY¿ âûïîëíÿëîñü ïðè çàìåíå êàæäîé áóêâû íà ñîîòâåòñòâóþùóþ åé öèôðó. 53 Ïåðå÷èñëèì ïîïóëÿðíûå ÿçûêè ËÏÎ: • B-Prolog (íà áàçå Prolog, êîììåð÷åñêèé) http://www.probp.com/ • CHIP V5 (íà áàçå Prolog, âêëþ÷àåò áèáëèîòåêè C++ è C, êîììåð÷åñêèé) http://www.cosytec.com/production_scheduling/chip/optimization_product_chip.htm • Ciao Prolog (íà áàçå Prolog, ñâîáîäíî ðàñïðîñòðàíÿåìîå ÏÎ: GPL/LGPL) http://clip.dia.fi.upm.es/Software/Ciao/ • ECLiPSe (íà áàçå Prolog, open source) http://eclipse-clp.org/ • SICStus (íà áàçå Prolog, êîììåð÷åñêèé) http://www.sics.se/isl/sicstuswww/site/index.html • GNU Prolog (ñâîáîäíî ðàñïðîñòðàíÿåìîå ÏÎ) http://www.gprolog.org/ • Oz http://www.dmoz.org/Computers/Programming/Languages/Oz/ • YAP Prolog http://www.dcc.fc.up.pt/~vsc/Yap/documentation.html • SWI Prolog ñâîáîäíàÿ ñèñòåìà Prolog, ñîäåðæàùàÿ íåñêîëüêî áèáëèîòåê äëÿ ðåøåíèÿ îãðàíè÷åíèé http://www.swi-prolog.org/ • Claire http://www.claire-language.com/cgi-bin/trac.cgi • Curry (íà áàçå Haskell, ñâîáîäíî ðàñïðîñòðàíÿåìîå) http://www.curry-language.org/ • HAL: [147], [145] Ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ ðåàëèçóåòñÿ â èìïåðàòèâíîì ïðîãðàììèðîâàíèè ñ ïîìîùüþ ñîçäàíèÿ îòäåëüíûõ áèáëèîòåê. Òàêèìè ïîïóëÿðíûìè áèáëèîòåêàìè äëÿ ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ ÿâëÿþòñÿ: Choco (áèáëèîòåêà Java, ñâîáîäíî ðàñïðîñòðàíÿåìîå ÏÎ) http://www.emn.fr/x-info/choco-solver/doku.php Comet (ÿçûê C-ñòèëÿ äëÿ ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ, ëîêàëüíûé ïîèñê, îñíîâàííûé íà îãðàíè÷åíèÿõ, è ìàòåìàòè÷åñêîå ïðîãðàììèðîâàíèå, ñâîáîäíî ðàñïðîñòðàíÿåìûå èñïîëíèìûå ôàéëû äëÿ àêàäåìè÷åñêèõ öåëåé) http://dynadec.com/ Disolver (áèáëèîòåêà C++, êîììåð÷åñêèé) Gecode (C++ library, ñâîáîäíîå ÏÎ: X11 style) http://www.gecode.org/ ILOG CP Optimizer (C++, Java, áèáëèîòåêè .NET libraries, êîììåð÷åñêèé) http://www.ilog.com/products/cpoptimizer/ ILOG CP (áèáëèîòåêà C++, êîììåð÷åñêèé) http://www.ilog.com/products/cp/ JaCoP(áèáëèîòåêà Java, open source) http://jacop.osolpro.com/ JOpt (áèáëèîòåêà Java, ñâîáîäíî ðàñïðîñòðàíÿåìîå ÏÎ) http://jopt.sourceforge.net/ Koalog Constraint Solver (áèáëèîòåêà Java, êîììåð÷åñêèé) http://linux.softpedia.com/get/Programming/Libraries/ 54 Koalog-Constraint-Solver-8073.shtml Minion (C++ program, GPL) http://minion.sourceforge.net/ Python-constraint (áèáëèîòåêà Python, GPL) http://labix.org/python-constraint Cream (áèáëèîòåêà Java, ñâîáîäíî ðàñïðîñòðàíÿåìîå ÏÎ: LGPL) http://bach.istc.kobe-u.ac.jp/cream/ Emma (áèáëèîòåêà Python, êîììåð÷åñêèé) http://www.eveutilities.com/products/emma ßçûêè ñ ïîääåðæêîé ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ Common Lisp via Screamer (ñâîáîäíî ðàñïðîñòðàíÿåìîå ÏÎ, êîòîðîå ïîä- äåðæèâàåò ïîèñê ñ âîçâðàòàìè è âîçìîæíîñòè CLP(R), CHiP ) http://www.cl-user.net/asp/d86x/sdataQkvrOGEiQe5uDQ3jQRpX8yBX8yBXnMq= /sdataQu3F$sSHnB== Bertrand ÿçûê äëÿ ïîñòðîåíèÿ ñèñòåì ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ [259]. 5.11. Ñðàâíåíèå ñ äðóãèìè ÿçûêàìè ìîäåëèðîâàíèÿ Àëãåáðàè÷åñêèå ÿçûêè ìîäåëèðîâàíèÿ, ïîÿâèâøèåñÿ â 1970-õ ãîäàõ, èñïîëüçóåìûå äëÿ ðåøåíèÿ çàäà÷ ìàòåìàòè÷åñêîãî ïðîãðàììèðîâàíèÿ (íàïðèìåð, GAMS (Brooke et al., 1988) [85] è MGG (Simons, 1987) [351]), áûëè ñóùåñòâåííûì ïðîäâèæåíèåì âïåðåä. Âî-ïåðâûõ, îíè áûëè ðàçðàáîòàíû äëÿ óïðîùåíèÿ ðîëè ïîëüçîâàòåëÿ â ðåøåíèè çàäà÷ ìàòåìàòè÷åñêîãî ïðîãðàììèðîâàíèÿ, îáåñïå÷èâàÿ áëèçêèé ê ìàòåìàòè÷åñêîé çàïèñè ñèíòàêñèñ. Âî-âòîðûõ, îíè èìåþò äåêëàðàòèâíóþ ïðèðîäó, îïèñûâàÿ çàäà÷ó, à íå òî, êàê ðåøåíèÿ ìîãóò áûòü íàéäåíû. Ýòî ñíèìàåò íàãðóçêó ñ ïîëüçîâàòåëÿ, è ïîçâîëÿåò ïðîñòî ïðèìåíÿòü ðàçëè÷íûå ðåøàòåëè äëÿ ðåøåíèÿ îäíîé è òîé æå çàäà÷è. Äðóãèå ïîëåçíûå ÿçûêè ìîäåëèðîâàíèÿ áûëè ñîçäàíû â äðóãèõ îáëàñòÿõ, íàïðèìåð, ACE (Fuchs & Schwitter, 1996) [178], ÿçûê ðàññóæäåíèé î áàçàõ çíàíèé, ïîäîáíûé àíãëèéñêîìó. Óñïåõ ÿçûêîâ ìîäåëèðîâàíèÿ ïîêàçàë, ÷òî ïîäîáíûé ïîäõîä ìîã áû áûòü ïëîäîòâîðíûì äëÿ ðåøåíèÿ çàäà÷ ÓÎ. Ðàííèå ïîïûòêè â ýòîì íàïðàâëåíèè, ÿçûê ALICE (Lauriere, 1978) [258] èìååò ìíîãî àñïåêòîâ, ïîäîáíûõ àëãåáðàè÷åñêèì ÿçûêàì ìîäåëèðîâàíèÿ òîãî âðåìåíè, â òîì ÷èñëå äåêëàðàòèâíóþ ïðèðîäó. ßçûêè ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ ïîñòåïåííî ðàñøèðÿëè äèàïàçîí òèïîâ ðåøàþùèõ ïåðåìåííûõ. Íàïðèìåð, Eclipse (Gervet, 1994) [186] ïîääåðæèâàåò ðåøàþùèå ïåðåìåííûå, ýëåìåíòû äîìåíîâ êîòîðûõ ÿâëÿþòñÿ ìíîæåñòâàìè; àíàëîãè÷íî F ïîääåðæèâàåò ôóíêöèè, ESRA ïîääåðæèâàåò îòíîøåíèÿ è ôóíêöèè, à NP-Spec ïîääåðæèâàåò ìíîæåñòâà, ïåðåñòàíîâêè, ðàçáèåíèÿ è öåëûå ôóíêöèè. Óñèëåíèå àáñòðàêöèè ïîçâîëÿåò ïîëüçîâàòåëþ èçáåæàòü äîïîëíèòåëüíûõ óñèëèé ïðè ìîäåëèðîâàíèè, ïåðåäàâàÿ ýòî âñå êîìïèëÿòîðó. ESSENCE ñäåëàë áîëüøîé ñêà÷îê â ýòîì íàïðàâëåíèè, îáåñïå÷èâàÿ áîëåå øèðîêèé äèàïàçîí òèïîâ, ÷åì ýòî áûëî â ïðåäøåñòâóþùèõ ÿçûêàõ è, ÷òî óíèêàëüíî, êîíñòðóêòîðû òèïîâ, êîòîðûå ìîãóò áûòü êàê óãîäíî âëîæåííûìè. Èìåííî ýòè ôàêòîðû äåëàþò ESSENCE ñêîðåå ÿçûêîì ñïåöèôèêàöèé, à íå ïðîñòî ÿçûêîì ìîäåëèðîâàíèÿ. 55 Ïðè ðåøåíèè ñëîæíûõ êîìáèíàòîðíûõ çàäà÷ îáû÷íî èñïîëüçóåòñÿ äâóõøàãîâàÿ ïðîöåäóðà. Íà ïåðâîì øàãå ðàçðàáàòûâàåòñÿ êîíöåïòóàëüíàÿ ìîäåëü, îïèñûâàþùàÿ, â ÷åì ñîñòîèò çàäà÷à, áåç îïèñàíèÿ, êàê ôàêòè÷åñêè åå ðåøàòü. Âòîðîé øàã ñîñòîèò â îòîáðàæåíèè êîíöåïòóàëüíîé ìîäåëè íà ïðîåêòíóþ ìîäåëü, êîòîðàÿ óæå ìîæåò áûòü íåïîñðåäñòâåííî ðåøåíà. Â èäåàëå, îäíà è òà æå êîíöåïòóàëüíàÿ ìîäåëü ìîæåò áûòü ïðåîáðàçîâàíà â ðàçëè÷íûå ïðîåêòíûå ìîäåëè, ïîçâîëÿÿ òåì ñàìûì ñïåöèàëèñòàì ïî ìîäåëèðîâàíèþ ëåãêî ¾ïîäêëþ÷àòü è ðàáîòàòü¿ (plug and play) ñ ðàçëè÷íûìè ðåøàòåëÿìè. Íîâûé ÿçûê ìîäåëèðîâàíèÿ Zinc, ðàçðàáîòàííûé â ðàìêàõ ýòîé ìåòîäîëîãèè, îïèñàí â [268]. Zinc (Marriott et al., 2006), [267]; (de la Banda et al., 2006) [144] ýòî íîâûé ÿçûê ñïåöèôèêàöèé, èñïîëüçóþùèé, êàê è ESSENCE, âëîæåííûå êîíñòðóêòîðû òèïîâ, íî, êðîìå òîãî, â îòëè÷èå îò ESSENCE, îí èìååò âîçìîæíîñòü îïðåäåëÿòü ïðåäèêàòû. Êðîìå ESSENCE èçâåñòíû ëèøü òðè ÿçûêà ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ ñ ïîäîáíûìè âîçìîæíîñòÿìè: ESRA, F è LOCALIZER (Michel & Van Hentenryck, 2000) [281]. Äðóãîé ïîäõîä ê ñïåöèôèêàöèè çàäà÷è, êîòîðûé èñïîëüçóåò áîëåå ìîùíûé è îáùèé ÿçûê ñïåöèôèêàöèé, òàêîé êàê Z, áûë èññëåäîâàí â ðàáîòå (Renker & Ahriz, 2004) [315], ñîçäàâøèõ èíñòðóìåíòàëüíûå ñðåäñòâà ñõåìû Z äëÿ ïîääåðæêè îáùèõ ãëîáàëüíûõ îãðàíè÷åíèé è èñïîëüçîâàâøèõ ýòî äëÿ ïîñòðîåíèÿ îáøèðíîãî êàòàëîãà ñïåöèôèêàöèé [318]. Íåäîñòàòîê ýòîãî ïîäõîäà ñîñòîèò â òîì, ÷òî Z ÿâëÿåòñÿ ÷åðåñ÷óð îáùèì äëÿ çàäà÷ ÓÎ, òàê êàê ïîçâîëÿåò ñïåöèôèêàöèè çàäà÷, êîòîðûå åñòåñòâåííûì îáðàçîì ê ÇÓÎ íå ñâîäÿòñÿ. Ê íåäîñòàòêàì ñïåöèôèêàöèé ÿçûêà Z îòíîñèòñÿ ãîðàçäî ìåíüøàÿ èõ åñòåñòâåííîñòü ïî ñðàâíåíèþ ñî ñïåöèôèêàöèÿìè ESSENCE. ßçûê Alloy (Jackson, 2006) [219] ïîçâîëÿåò èçáåæàòü íåêîòîðûõ íåäîñòàòêîâ ÿçûêà Z ïóòåì îãðàíè÷èâàíèÿ åãî äî ëîãèêè ïåðâîãî ïîðÿäêà. Alloy äàåò åñòåñòâåííûé è âûðàçèòåëüíûé ñïîñîá ñïåöèôèêàöèè çàäà÷ â òåðìèíàõ îòíîøåíèé è àòîìîâ è îòîáðàæàåò ýòè ñïåöèôèêàöèè â ýôôåêòèâíûå ìîäåëè âûïîëíèìîñòè SAT. 6. Ïðèëîæåíèÿ ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ è ÓÎ Ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ â íàñòîÿùåå âðåìÿ èìååò øèðîêèé êðóã ïðèëîæåíèé. Òàêèå êîìïàíèè, êàê American Express, BMW, Coors, Danone, eBay, France Telecom, General Electric, HP, JB Hunt, LL Bean, Mitsubishi Chemical, Nippon Steel, Orange, Porsche, QAD, Royal Bank of Scotland, Shell, Travelocity, US Postal Service, Visa,Wal-Mart, Xerox, Yves Rocher, Zurich Insurance èñïîëüçóþò ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ äëÿ îïòèìèçàöèè ñâîèõ áèçíåñ-ïðîöåññîâ [325]. Ðàññìîòðèì ãëàâíûå îáëàñòè ïðèëîæåíèé, â êîòîðûõ áûëà èñïîëüçîâàíà òåõíîëîãèÿ ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ è îêàçàëàñü êîíêóðåíòíîñïîñîáíîé êàê ïî ãèáêîñòè ìîäåëèðîâàíèÿ, òàê è ïî ýôôåêòèâíîñòè ðåøåíèÿ. Â ÷èñëå îáçîðíûõ ðàáîò ïî ïðèëîæåíèÿì ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ ìîæíî óïîìÿíóòü [221], [348], [349], [377]. 56 6.1. Çàäà÷è î íàçíà÷åíèÿõ. Çàäà÷è î íàçíà÷åíèÿõ áûëè îäíèì èç ïåðâûõ òèïîâ ïðèêëàäíûõ çàäà÷, êîòîðûå áûëè ðåøåíû ñ ïîìîùü òåõíîëîãèè ËÏÎ. Ýòè çàäà÷è îáû÷íî ó÷èòûâàþò äâà òèïà ðåñóðñîâ è îãðàíè÷åíèé, ñâÿçûâàþùèõ èõ, è ïûòàþòñÿ ïðèïèñàòü îäèí ðåñóðñ ïåðâîãî òèïà îäíîìó ðåñóðñó âòîðîãî òèïà òàê, ÷òîáû âñå îãðàíè÷åíèÿ óäîâëåòâîðÿëèñü. Îäíèì èç ïðèìåðîì ýòîé çàäà÷è î íàçíà÷åíèÿõ ìîæåò ñëóæèòü çàäà÷à ðàçìåùåíèÿ ñòîÿíîê ñàìîëåòîâ â àýðîïîðòó, ãäå ñàìîëåòû (ðåñóðñû ïåðâîãî òèïà) äîëæíû ïàðêîâàòüñÿ íà äîñòóïíûõ ñòîÿíêàõ (âòîðîé òèï ðåñóðñîâ) âî âðåìÿ èõ ïðåáûâàíèÿ â àýðîïîðòó [107], [108], [303], [348]. 6.2. Íàçíà÷åíèå ïåðñîíàëà Çàäà÷è íàçíà÷åíèÿ ïåðñîíàëà ÿâëÿþòñÿ ÷àñòíûì ñëó÷àåì çàäà÷ î íàçíà÷åíèÿõ, â êîòîðûõ îäíèì òèïîì ðåñóðñîâ ÿâëÿþòñÿ ëþäè, ÷òî äåëàåò ýòè çàäà÷è äîñòàòî÷íî ñïåöèàëüíûìè äëÿ òîãî, ÷òîáû ðàññìàòðèâàòü èõ îòäåëüíî. Â ñàìîì äåëå, èçìåíÿþùèåñÿ ïðàâèëà ðàáîòû è âñåâîçìîæíûå ïðåäïèñàíèÿ ïîðîæäàþò ñëîæíûå îãðàíè÷åíèÿ, çàâèñÿùèå îò âðåìåíè. Ïîìèìî ýòîãî, ïîëüçîâàòåëüñêèå ïðåäïî÷òåíèÿ ÷àñòî âåäóò ê ïåðåîãðàíè÷åííûì çàäà÷àì, íå èìåþùèì ðåøåíèé, óäîâëåòâîðÿþùèõ âñåì îãðàíè÷åíèÿì. Äðóãèì âàæíûì àñïåêòîì ÿâëÿåòñÿ íåîáõîäèìîñòü ñîáëþäåíèÿ áàëàíñà îáúåìà ðàáîòû ìåæäó ðàçíûìè ëþäüìè, ÷òî ïðèâîäèò ê òðóäíûì çàäà÷àì îïòèìèçàöèè. Ðàçëè÷íûå ïðèëîæåíèÿ ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ â îáëàñòè íàçíà÷åíèÿ ïåðñîíàëà îïèñàíû â ðàáîòàõ [102], [103] (ðàçðàáîòêà ñìåííûõ ãðàôèêîâ ìåäñåñòåð), [118] (ãðàôèêè ðàáîòû íà òåëåâèäåíèè Ôðàíöèè), [155], [246], [298]. 6.3. Óïðàâëåíèå ñåòÿìè Ñèñòåìà LOCARIM [347] áûëà ðàçðàáîòàíà êîìïàíèåé COSYTEC äëÿ France Telecom: íà÷èíàÿ ñ àðõèòåêòóðíîãî ïðîåêòà çäàíèÿ, îíà ïðåäëàãàåò ïðîêëàäêó êàáåëÿ òåëåêîììóíèêàöèîííîé ñåòè çäàíèÿ. Ñèñòåìà PLANETS (PLanning Activities on NETworkS) [124], ðàçðàáîòàííàÿ Óíèâåðñèòåòîì Êàòàëîíèÿ (University of Catalonia) â Áàðñåëîíå äëÿ èñïàíñêîé ýëåêòðè÷åñêîé êîìïàíèè, êàê èíñòðóìåíò ïåðåñòðîéêè ñòðóêòóðû ýëåêòðè÷åñêîé ñåòè, êîòîðûé ïîçâîëÿåò ñòðîèòü êàëåíäàðíûé ïëàí òåõíè÷åñêîãî îáñëóæèâàíèÿ äåÿòåëüíîñòè ñ ïîìîùüþ âûäåëåíèÿ ó÷àñòêîâ ñåòè áåç íàðóøåíèÿ îáñëóæèâàíèÿ ïîòðåáèòåëåé. Â ðàáîòå [36] îïèñàíî èñïîëüçîâàíèå ÿçûêà CHIP â óïðàâëåíèè âîäíîé ñèñòåìîé. Ïëàíèðîâàíèå òåëåêîììóíèêàöèîííîé ñåòè ìîáèëüíîé ñâÿçè áûëî îñóùåñòâëåíî ñ ïîìîùüþ ñèñòåìû POPULAR [177], çàïðîãðàììèðîâàííîé íà ECLiPSe. 6.4. Êàëåíäàðíîå ïëàíèðîâàíèå Íàâåðíîå, íàèáîëåå óñïåøíîé ïðèêëàäíîé îáëàñòüþ äëÿ ëîãè÷åñêîãî ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ ÿâëÿåòñÿ êàëåíäàðíîå ïëàíèðîâàíèå [44], [101]. Äëÿ 57 çàäàííîãî ìíîæåñòâà ðåñóðñîâ ñ äàííûìè ìîùíîñòÿìè, ìíîæåñòâî âèäîâ äåÿòåëüíîñòè ñ äàííîé ïðîäîëæèòåëüíîñòüþ è ïîòðåáíîñòÿìè â ðåñóðñàõ, ìíîæåñòâî òåìïîðàëüíûõ îãðàíè÷åíèé ìåæäó âèäàìè äåÿòåëüíîñòè, çàäà÷à ¾÷èñòîãî¿ êàëåíäàðíîãî ïëàíèðîâàíèÿ ñîñòîèò â ðåøåíèè, êîãäà âûïîëíèòü êàæäóþ ðàáîòó òàê, ÷òîáû âûïîëíÿëèñü òåìïîðàëüíûå îãðàíè÷åíèÿ è îãðàíè÷åíèÿ ïî ðåñóðñàì. Ñîâðåìåííûå ðåçóëüòàòû èññëåäîâàíèé ïî ñëîæíîñòè êëàññîâ çàäà÷ ÓÎ ïîêàçûâàþò, ÷òî òàêèå çàäà÷è êàëåíäàðíîãî ïëàíèðîâàíèÿ íà ñàìîì äåëå ëåãêî ðàçðåøèìû è ïîýòîìó èìåþòñÿ ýôôåêòèâíûå àëãîðèòìû èõ ðåøåíèÿ [310]. Òèïè÷íûì ïðèìåðîì ïðèëîæåíèÿ êàëåíäàðíîãî ïëàíèðîâàíèÿ, îñíîâàííîãî íà ÓÎ, ÿâëÿåòñÿ ATLAS [346], îñóùåñòâëÿþùèé êàëåíäàðíîå ïëàíèðîâàíèå ïðîèçâîäñòâà ãåðáèöèäîâ íà çàâîäå Monsanto â Àíòâåðïåíå. Ñèñòåìà PLANE [59] èñïîëüçîâàíà êîìïàíèåé Dassault Aviation äëÿ ïëàíèðîâàíèÿ ïðîèçâîäñòâà âîåííîãî ñàìîëåòà Mirage 2000 è ñàìîëåòà Falcon. Ñèñòåìà MOSES [28], [350] áûëà ðàçðàáîòàíà êîìïàíèåé COSYTEC äëÿ ïðîèçâîäèòåëÿ ïèòàíèÿ äëÿ æèâîòíûõ â Âåëèêîáðèòàíèè. Ñèñòåìà FORWARDC [191] ïðåäñòàâëÿåò ñîáîé ñèñòåìó ïîääåðæêè ïðèíÿòèÿ ðåøåíèé (ÑÏÏÐ) [13], ðåàëèçîâàííóþ íà CHIP, è èñïîëüçóåòñÿ íà òðåõ íåôòåïåðåãîííûõ çàâîäàõ â Åâðîïå äëÿ ðåøåíèÿ çàäà÷ êàëåíäàðíîãî ïëàíèðîâàíèÿ, âîçíèêàþùèõ ïðè ïðèâîçå ñûðîé íåôòè, åå îáðàáîòêå, ñìåøèâàíèè è äîñòàâêå. Xerox èñïîëüçîâàë ñèñòåìó ÓÎ äëÿ êàëåíäàðíîãî ïëàíèðîâàíèÿ ðàçëè÷íûõ ðàáîò íà êîïèðîâàëüíûõ ìàøèíàõ [174]. Óïðåæäàþùèå (preemptive) çàäà÷è êàëåíäàðíîãî ïëàíèðîâàíèÿ ýòî òå çàäà÷è, â êîòîðûõ ðàáîòû ìîãóò ïðåðûâàòüñÿ âî âðåìåíè. È äëÿ òàêèõ çàäà÷ ìîãóò óñïåøíî ïðèìåíÿòüñÿ òåõíîëîãèè ëîãè÷åñêîãî ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ: òàê, CLAIRE Scheduler ýòî áèáëèîòåêà ïðîãðàìì â îãðàíè÷åíèÿõ, ïîçâîëÿþùàÿ ðàáîòàòü ñ ïîäîáíûìè çàäà÷àìè, â òîì ÷èñëå ïðè íàëè÷èè ïðåäïî÷òåíèé [296], [297]. Ïîäîáíûå ôóíêöèè âûïîëíÿåò è ILOG Scheduler [217]. Åñëè ñðàâíèâàòü çàäà÷è êàëåíäàðíîãî ïëàíèðîâàíèÿ ïî ìåòîäàì ðåøåíèÿ, òî îíè ðåøàþòñÿ ñ ïîìîùüþ èìïåðàòèâíûõ ÿçûêîâ (íàïðèìåð, C/C++), íåñïåöèàëèçèðîâàííûõ ÿçûêîâ ËÏÎ (íàïðèìåð, CHIP), è ñïåöèàëüíûõ ñðåäñòâ ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ. Ðàáîòà [41] ïîêàçàëà, ÷òî ïîäõîä ËÏÎ ïðåâîñõîäèò äðóãèå ïî ìíîãèì ïàðàìåòðàì: âðåìÿ ðàçðàáîòêè, âåðøèíû, ïðîñìîòðåííûå â äåðåâå ïîèñêà, ïî ÷èñëó ïîñòðîåííûõ äîïóñòèìûõ ðåøåíèé, äàæå ïî ýôôåêòèâíîñòè. Ñîãëàñíî ýòîìó èññëåäîâàíèþ, ãëàâíîé ïðè÷èíîé óñïåõà ÿçûêîâ ËÏÎ ïðè ðåøåíèè ýòèõ çàäà÷ ñîñòîèò â èñïîëüçîâàíèè ðàñïðîñòðàíåíèÿ îãðàíè÷åíèé, êîòîðîå ïîìîãàåò óìåíüøèòü êàê âðåìÿ ðàçðàáîòêè, òàê è âðåìÿ âûïîëíåíèÿ. Âûÿñíèëîñü, ÷òî äëÿ íåêîòîðûõ êëàññîâ çàäà÷ êàëåíäàðíîãî ïëàíèðîâàíèÿ íå ñóùåñòâóåò ìåòîäà, ëó÷øåãî âñåõ îñòàëüíûõ ìåòîäîâ, à ëó÷øå ïðèìåíÿòü ñî÷åòàíèå íåñêîëüêèõ ìåòîäîâ. Â ñâÿçè ñ ýòèì áûëè ïðåäëîæåíû ãèáðèäíûå àëãîðèòìû, èñïîëüçóþùèå êàê ÿçûêè ËÏÎ, òàê è äðóãèå ìåòîäû. Òàê, ñî÷åòàíèå ËÏÎ è öåëî÷èñëåííîãî ïðîãðàììèðîâàíèÿ áûëî èñïîëüçîâàíî äëÿ ðåøåíèÿ çàäà÷ êàëåíäàðíîãî ïëàíèðîâàíèÿ ñ ìíîãèìè ïîäúåìíèêàìè [321]. Îòìåòèì ñëåäóþùèå ñèñòåìû ïëàíèðîâàíèÿ. 58 TAP-AI [52] àêòèâíàÿ ñèñòåìà ïëàíèðîâàíèÿ, ïðåäíàçíà÷åííàÿ äëÿ íàçíà÷åíèÿ ýêèïàæåé ïî àâèàëèíèÿì SAS. Îíà ïðåäíàçíà÷åíà äëÿ åæåäíåâíîãî óïðàâëåíèÿ äåÿòåëüíîñòüþ (îïåðàöèÿìè). OPTISERVICE [118] ïðîãðàììíûé ïàêåò äëÿ íàçíà÷åíèÿ ïåðñîíàëà äëÿ âñåõ çàãðàíè÷íûõ òåëå- è ðàäèîñòàíöèé ñåòè RFO. Îí íàçíà÷àåò êîìàíäû êâàëèôèöèðîâàííûõ æóðíàëèñòîâ è òåõíè÷åñêèõ ðàáîòíèêîâ ïî ïåðåäà÷àì, ó÷èòûâàÿ îãðàíè÷åíèÿ ïî âðåìåíè è óñëîâèÿì îïëàòû. Ñèñòåìà MOSAR (http://www.cosytec.com/constraint_programming/cases_studies/administration.htm), ðàçðàáîòàííàÿ êîìïàíèÿìè Cisi è COSYTEC äëÿ ìèíèñòåðñòâà þñòèöèè Ôðàíöèè, íàçíà÷àåò îõðàííèêîâ òþðåì ïî 200 òþðüìàì Ôðàíöèè ïî ÷åðåäóþùèìñÿ ñìåíàì. 6.5. Òðàíñïîðòíûå ïðîáëåìû. Ìíîãèå òðàíñïîðòíûå ïðîáëåìû áûëè ðåøåíû ñ èñïîëüçîâàíèåì òåõíîëîãèè îãðàíè÷åíèé. Ýòè çàäà÷è ÷àñòî î÷åíü ñëîæíû â ñâÿçè ñ èõ áîëüøèìè ðàçìåðàìè, ÷èñëó è ìíîæåñòâó âèäîâ îãðàíè÷åíèé, à òàêæå èç-çà íàëè÷èÿ ñëîæíûõ îãðàíè÷åíèé, íàêëàäûâàåìûõ íà âîçìîæíûå ìàðøðóòû. Êðîìå òîãî, ýòè çàäà÷è ÷àñòî òðåáóþò ðåøåíèÿ ïîäçàäà÷è íàçíà÷åíèÿ ïåðñîíàëà, ïðè÷åì ïðè âûïîëíåíèè ðÿäà ñëîæíûõ óñëîâèé. Ðàññìîòðèì ñîîòâåòñòâóþùèå ïðèëîæåíèÿ. Ñèñòåìà COBRA [350] ðàçðàáàòûâàåò äèàãðàììû ðàáî÷èõ ïëàíîâ ìàøèíèñòîâ ïîåçäîâ êîìïàíèè North Western Trains â Âåëèêîáðèòàíèè. Ïðîåêò DAYSY Esprit (ïàêåò SAS-Pilot) [52] îñóùåñòâëÿåò ïåðåíàçíà÷åíèå ëåòíûõ ýêèïàæåé ïî ïîëåòàì. Ýòà æå çàäà÷à ðåøàåòñÿ ñ ïîìîùüþ äðóãîé ñèñòåìû [164], ñî÷åòàþùåé ìåòîäû ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ è èññëåäîâàíèÿ îïåðàöèé. Ñèñòåìà, îïèñàííàÿ â [269], èñïîëüçóåò áèáëèîòåêè îãðàíè÷åíèé ILOG äëÿ ðàçðàáîòêè è îïòèìèçàöèè ïîåçäíûõ îïåðàòèâíûõ ïëàíîâ äëÿ ãðóçîâîé æåëåçíîäîðîæíîé êîìïàíèè. Äëÿ çàäà÷ ïëàíèðîâàíèÿ òðàíñïîðòà (êàê è äëÿ çàäà÷ êàëåíäàðíîãî ïëàíèðîâàíèÿ), â ðÿäå ñëó÷àåâ íàèëó÷øèì ìåòîäîì îêàçàëàñü êîìáèíàöèÿ íåñêîëüêèõ ìåòîäîâ. Íàïðèìåð, ÓÎ è ëîêàëüíûé ïîèñê áûëè ñîâìåñòíî èñïîëüçîâàíû ïðè ðåøåíèè çàäà÷ ìàðøðóòèçàöèè òðàíñïîðòíûõ ñðåäñòâ â [340], à â ðàáîòå [305] òàêèå æå çàäà÷è ðåøàëèñü ñ ïîìîùüþ êîìáèíàöèè ìåòîäîâ ÓÎ è ìåòîäîâ èññëåäîâàíèÿ îïåðàöèé. Ìàðøðóòèçàöèÿ òðàíñïîðòíûõ ñðåäñòâ (Vehicle Routing) ýòî çàäà÷à ïîñòðîåíèÿ ìàðøðóòîâ äëÿ òðàíñïîðòíûõ ñðåäñòâ äëÿ ïåðåâîçîê ïðîäóêöèè ïîëüçîâàòåëÿì ñ ìèíèìàëüíîé ñòîèìîñòüþ. 6.6. Ñèñòåìû óïðàâëåíèÿ ýëåêòðîìåõàíè÷åñêèìè ñèñòåìàìè Òåõíîëîãèÿ ÓÎ èñïîëüçóåòñÿ äëÿ ðàçðàáîòêè ÏÎ äëÿ óïðàâëåíèÿ ýëåêòðîìåõàíè÷åñêèìè ñèñòåìàìè ñ êîíå÷íûì ÷èñëîì âõîäîâ è âûõîäîâ è âíóòðåííèõ ñîñòîÿíèé. 59 Ñèñòåìû óïðàâëåíèÿ òðåáóþòñÿ äëÿ òàêèõ ïðèëîæåíèé, êàê ëèôòû, êîïèðîâàëüíûå ìàøèíû, ñáîðî÷íûå ëèíèè è ýëåêòðîñòàíöèè. Â êà÷åñòâå ïðèìåðîâ òàêîé ïðîãðàììíîé ñèñòåìû ìîæíî ïðèâåñòè ñèñòåìû âåðèôèêàöèè: SVE (system verication environment) [163], à òàêæå ñèñòåìà âåðèôèêàöèè ïàðàëëåëüíûõ ñèñòåì [276], èñïîëüçóþùàÿ ñåòè Ïåòðè è ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ (ÿçûê ïðîãðàììèðîâàíèÿ 2LP). 6.7. Ýëåêòðîííûå òàáëèöû ñ îãðàíè÷åíèÿìè Ýëåêòðîííàÿ òàáëèöà â íàñòîÿùåå âðåìÿ îäíî èç íàèáîëåå ðàñïðîñòðàí¼ííûõ ïðîãðàììíûõ ñðåäñòâ ïîääåðæêè ïðèíÿòèÿ ðåøåíèé. Îäíàêî, îáû÷íûå ýëåêòðîííûå òàáëèöû èìåþò ñëåäóþùèå äâà îãðàíè÷åíèÿ: îíè âû÷èñëÿþò ëèøü â ôèêñèðîâàííîì íàïðàâëåíèè, (îò âõîäíûõ ÿ÷ååê ê ÿ÷åéêàì-ðåçóëüòàòàì), ïðè ýòîì ðåçóëüòàòû ìîãóò áûòü îïðåäåëåíû ïîñëå çàäàíèÿ çíà÷åíèé âõîäíûõ ÿ÷ååê. Ñ äðóãîé ñòîðîíû, îãðàíè÷åíèÿ ïîçâîëÿþò ïåðåìåííûì âîçäåéñòâîâàòü äðóã íà äðóãà âî âñåõ íàïðàâëåíèÿõ (êàê â îãðàíè÷åíèè X ≤ Y ), è ðàáîòàòü åñòåñòâåííûì îáðàçîì ñ ÷àñòè÷íîé èíôîðìàöèåé Èäåÿ ïðèìåíåíèÿ òåõíîëîãèè ËÏÎ äëÿ ïðåîäîëåíèÿ ýòèõ îãðàíè÷åíèé ñóùåñòâóþùèõ ýëåêòðîííûõ òàáëèö áûëà èñïîëüçîâàíà ïðè ñîçäàíèè íåñêîëüêèõ ïðîãðàììíûõ ñèñòåì. Ñèñòåìà êðàòêîñðî÷íîãî ïëàíèðîâàíèÿ (The Short Term Planning (STP)) äëÿ êîìïàíèè Renault [105] ðåøàåò çàäà÷ó òðàíñïîðòèðîâêè àâòîìîáèëåé çàêàç÷èêàì ñ ó÷åòîì ìíîæåñòâà îãðàíè÷åíèé. Áûëè ðàçðàáîòàíû ýëåêòðîííûå òàáëèöû íà îñíîâå îãðàíè÷åíèé äëÿ ïîìîùè ïîëüçîâàòåëþ â ðåøåíèè ýòîé çàäà÷è. Âî âðåìÿ ïðîöåññà ïëàíèðîâàíèÿ, âåëè÷èíû â ýëåêòðîííûõ òàáëèöàõ íå òî÷íûå ÷èñëà, à èíòåðâàëû [min, max]. Îãðàíè÷åíèÿ ïðèïèñàíû ê ðàçëè÷íûì ÿ÷åéêàì òàáëèöû è èñïîëüçóþòñÿ äëÿ ðàñïðîñòðàíåíèÿ ïîñëåäñòâèé èçìåíåíèé, íàëîæåííûõ ïîëüçîâàòåëåì, êîòîðûé ìîæåò ëèáî ââåñòè êîíêðåòíîå çíà÷åíèå â ÿ÷åéêó, èëè ïðÿìî îãðàíè÷èòü äèàïàçîí âîçìîæíûõ çíà÷åíèé ïóòåì ñóæåíèÿ èíòåðâàëà. Äðóãèì ïåðñïåêòèâíûì ïðèëîæåíèåì ýëåêòðîííûõ òàáëèö ñ îãðàíè÷åíèÿìè ÿâëÿåòñÿ ôèíàíñîâîå ïëàíèðîâàíèå [215]: ýëåêòðîííûå òàáëèöû ïîçâîëÿþò ñïåöèàëèñòó ïî ïëàíèðîâàíèþ ôèíàíñîâ èñïîëüçîâàòü ïîòåíöèàëüíûå èíâåñòèöèè. Ôèíàíñîâûå ïðèëîæåíèÿ ÷àñòî ñîäåðæàò íåëèíåéíûå îãðàíè÷åíèÿ, êîòîðûå ìîãóò ðåøàòüñÿ ñ ïîìîùüþ ìåòîäîâ èíòåðâàëüíîãî àíàëèçà. Ïëàíèðîâàíèå áþäæåòîâ ðàéîíîâ Ìîñêâû è ïîñëåäóþùèé êîíòðîëü çà ýòèìè áþäæåòàìè áûëè îñóùåñòâëåíû ñ ïîìîùüþ ýëåêòðîííûõ òàáëèö íà îñíîâå îãðàíè÷åíèé, ãäå ìåòîä ðåøåíèÿ îãðàíè÷åíèé áûë âñòðîåí â ÿçûê ECLiPSe [342]. 6.8. Èíòåðàêòèâíîå ðåøåíèå çàäà÷ Òåõíîëîãèÿ ËÏÎ, â îñîáåííîñòè ðàñïðîñòðàíåíèå îãðàíè÷åíèé, ìîæåò áûòü èñïîëüçîâàíà ïðè îðãàíèçàöèè ó÷àñòèÿ ïîëüçîâàòåëÿ â ðàçðàáîòêå ðåøåíèé. Ôàêòè÷åñêè íà êàæäîì øàãå ïîñëåäñòâèÿ êîíêðåòíîãî âûáîðà ÿâíî äåìîíñòðèðóþòñÿ ïîëüçîâàòåëþ ñ ïîìîùüþ ìåõàíèçìà ðàñïðîñòðàíåíèÿ îãðàíè÷åíèé. Ïîëüçîâàòåëü â ýòîì ñëó÷àå ëèáî îòêàçûâàåòñÿ îò ýòîãî ðåøåíèÿ, ëèáî èñïîëüçóåò ðàñïðîñòðà60 íåííóþ èíôîðìàöèþ äëÿ âûðàáîòêè ñëåäóþùåãî âûáîðà èëè ðàçðåøàåò ñèñòåìå ñäåëàòü ýòîò âûáîð àâòîìàòè÷åñêè. Íàïðèìåð, ñèñòåìû ËÏÎ òàêîãî òèïà áûëè èñïîëüçîâàíû äëÿ ðåøåíèÿ çàäà÷ ñîñòàâëåíèÿ ðàñïèñàíèé â Banque Bruxelles Lambert [153] è ìåäèöèíñêîãî ôàêóëüòåòà Óíèâåðñèòåòà èì. Ãóìáîëüòà (Humbolt University) [193]. Èñïîëüçîâàíèå ñèñòåì ËÏÎ îïèñàíî òàêæå â ðàáîòàõ [26], [63], [182], [187]. 6.9. Ãðàôè÷åñêèå èíòåðôåéñû ïîëüçâàòåëÿ Îäíîé èç îáøèðíûõ è ñòàðåéøèõ îáëàñòåé ïðèëîæåíèÿ äëÿ òåõíîëîãèè ÓÎ ÿâëÿåòñÿ ðàçðàáîòêà ãðàôè÷åñêèõ èíòåðôåéñîâ. Ðîëü ÓÎ ñîñòîèò çäåñü â ñîõðàíåíèè ãðàôè÷åñêèõ îáúåêòîâ â ïðàâèëüíîì îòíîøåíèè äðóã ê äðóãó ïîñëå îáíîâëåíèÿ ãðàôè÷åñêîãî îêíà (îáû÷íî ñ ïîìîùüþ ìûøêè). Çäåñü öåëü îòíîñèòåëüíî áûñòðîå íàõîæäåíèå äîïóñòèìîãî ðåøåíèÿ. Êîììåð÷åñêàÿ áèáëèîòåêà ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ äëÿ ïîñòðîåíèÿ èíòåðàêòèâíûõ ãðàôè÷åñêèõ èíòåðôåéñîâ ïîëüçîâàòåëÿ ïðåäëîæåíà â [80]. Ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ áûëî èñïîëüçîâàíî ïðè ñîçäàíèè àëãîðèòìîâ ïîñòðîåíèÿ ãðàôîâ [360] è ïðè ñîçäàíèè ñðåäñòâ âèçóàëèçàöèè è àíèìàöèè [359]. Ãðàôè÷åñêèé èíòåðôåéñ, èñïîëüçóþùèé òåõíîëîãèþ ÓÎ, áûë èñïîëüçîâàí äëÿ ðåøåíèÿ çàäà÷è ñáîðêè â ïðîôåññèîíàëüíûõ ñèñòåìàõ CAD/CAM [339]. Ðàçðàáîòàí ÿçûê EaCL [365] äëÿ ñîäåéñòâèÿ ïîëüçîâàòåëÿì â ïîñòðîåíèè çàäà÷ ÓÎ. 6.10. Ïåðåîãðàíè÷åííûå çàäà÷è Ìíîãèå ïðèêëàäíûå çàäà÷è ÿâëÿþòñÿ ïåðåîãðàíè÷åííûìè: ïîëüçîâàòåëü ââîäèò íàñòîëüêî ìíîãî îãðàíè÷åíèé, ÷òî ñòàíîâèòñÿ íåâîçìîæíûì óäîâëåòâîðèòü èõ âñå îäíîâðåìåííî. Â ýòèõ ñëó÷àÿõ íóæíî âíà÷àëå ïîêàçàòü, ÷òî ðåøåíèé íåò (âûÿñíåíèå ýòîãî ìîæåò çàíÿòü ìíîãî âðåìåíè), à çàòåì ðåøèòü, êàêèå èç îãðàíè÷åíèé îñëàáèòü, ÷òîáû çàäà÷à ñòàëà ðàçðåøèìîé. Äëÿ ðåøåíèÿ ýòîé âòîðîé çàäà÷è ïðåäëàãàåòñÿ ïîïðîñèòü êîíå÷íîãî ïîëüçîâàòåëÿ íàéòè íåêîòîðûå ðåøåíèÿ, òàê, êàê ýòî äåëàåòñÿ â èíòåðàêòèâíîì ðåøåíèè çàäà÷. Â ðåçóëüòàòå ïîëó÷àåòñÿ èåðàðõèÿ îãðàíè÷åíèé (ãäå âûøåñòîÿùèå îãðàíè÷åíèÿ ÿâëÿþòñÿ áîëåå âàæíûìè) [79], èëè â áîëåå îáùåì ñëó÷àå èñïîëüçóþòñÿ ìÿãêèå îãðàíè÷åíèÿ [172], [154], [328], [334], [68]. Òèïè÷íûìè ïåðåîãðàíè÷åííûìè çàäà÷àìè ÿâëÿþòñÿ çàäà÷è ñîñòàâëåíèÿ ðàñïèñàíèé, â êîòîðûõ îãðàíè÷åíèÿ èç ðàçíûõ èñòî÷íèêîâ (íàïðèìåð, äîñòóïíîñòü àóäèòîðèé è ïðåäïî÷òåíèÿ ïðåïîäàâàòåëåé). Ñîâðåìåííûå ñèñòåìû ñîñòàâëåíèÿ ðàñïèñàíèé èñïîëüçóþò êàê æåñòêèå, òàê è ìÿãêèå îãðàíè÷åíèÿ äëÿ ìîäåëèðîâàíèÿ è ðåøåíèÿ çàäà÷ ñîñòàâëåíèÿ ðàñïèñàíèé, êàê â University of Siegen, Germany [53], à äðóãàÿ ñèñòåìà ñîñòàâëåíèÿ ðàñïèñàíèé ïî èíôîðìàòèêå â Òåõíè÷åñêîì óíèâåðñèòåòå Ôëîðèäû èñïîëüçóåò äâà ýòàïà ÓÎ (ïåðâûé ýòàï íàçíà÷èòü ôàêóëüòåò êóðñàì, à âòîðîé ïðèïèñàòü âðåìåííûå èíòåðâàëû êóðñàì ëåêöèé) [72]. Åùå îäíà ñèñòåìà, ðåøàþùàÿ ïåðåîãðàíè÷åííûå çàäà÷è ñ ïîìîùüþ èåðàðõè÷åñêîé ñèñòåìîé ÓÎ, èñïîëüçóåòñÿ â íàñòîÿùåå âðåìÿ â áîëüíèöå Neuwied äëÿ ðåøåíèÿ çàäà÷è êàëåíäàðíîãî ïëàíèðîâàíèÿ ðàáîòû ìåäñåñòåð [280]. 61 6.11. Äðóãèå îáëàñòè ïðèëîæåíèé Ñëîæíàÿ çàäà÷à ïðîãíîçà ñòðóêòóðû áåëêà (îäíà èç íàèáîëåå âàæíûõ çàäà÷ âû÷èñëèòåëüíîé áèîëîãèè) ðåøàåòñÿ ñ ïîìîùüþ òåõíîëîãèè ËÏÎ [40]. Ýòà çàäà÷à ñîñòîèò â íàõîæäåíèè ñòðóêòóðû áåëêà ñ ìèíèìàëüíîé ýíåðãèåé. Îêàçàëîñü, ÷òî èñïîëüçîâàíèå îãðàíè÷åíèé ïîìîãàåò ñíèçèòü îáúåì ïîèñêà äëÿ ðåøåíèÿ ýòîé NPïîëíîé çàäà÷è. Òåõíîëîãèÿ îãðàíè÷åíèé áûëà òàêæå èñïîëüçîâàíà Sony â íåñêîëüêèõ ïðèëîæåíèÿõ, ñâÿçàííûõ ñ ìóçûêîé [299]. 7. Çàêëþ÷åíèå. ÓÎ è ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ èíòåðåñíûå è ìíîãîîáåùàþùèå òåõíîëîãèè èñêóññòâåííîãî èíòåëëåêòà, ïîçâîëÿþùèå â ñî÷åòàíèè ñ ìåòîäàìè èññëåäîâàíèÿ îïåðàöèé ðåøàòü ñëîæíûå êîìáèíàòîðíûå çàäà÷è. Àâòîð íàäååòñÿ, ÷òî äàííûé îáçîð, ïîñâÿùåííûé ÷ðåçâû÷àéíî èíòåðåñíîìó è ïåðñïåêòèâíîìó íàïðàâëåíèþ èñêóññòâåííîãî èíòåëëåêòà, áóäåò èíòåðåñåí è ïîëåçåí ñïåöèàëèñòàì â îáëàñòè èñêóññòâåííîãî èíòåëëåêòà, èññëåäîâàíèÿ îïåðàöèé, äèñêðåòíîé ìàòåìàòèêè, ñèñòåì ïîääåðæêè ïðèíÿòèÿ ðåøåíèé. Ñïèñîê öèòèðóåìûõ èñòî÷íèêîâ 1. 2. 3. 4. 5. 6. 7. 8. 9. Áðàòêî È. Àëãîðèòìû èñêóññòâåííîãî èíòåëëåêòà íà ÿçûêå Prolog, 3-å èçäàíèå / Ïåð. ñ àíãë. Ì.: Èçäàòåëüñêèé äîì ¾Âèëüÿìñ¿, 2004. 640 ñ. Áóëàòîâ À.À. Ïîëèíîìèàëüíîñòü ìàëüöåâñêèõ çàäà÷ CSP // Àëãåáðà è ëîãèêà. 2006. Ò. 45. Ñ. 655686. Ãýðè Ì., Äæîíñîí Ä. Âû÷èñëèòåëüíûå ìàøèíû è òðóäíîðåøàåìûå çàäà÷è. Ì.: Ìèð, 1982. 416 ñ. Åìåëè÷åâ Â. À., Ìåëüíèêîâ Î. È., Ñàðâàíîâ Â. È., Òûøêåâè÷ Ð. È. Ëåêöèè ïî òåîðèè ãðàôîâ. Èçä. 2-å, èñïð. Ì.: Êíèæíûé äîì ¾ËÈÁÐÎÊÎÌ¿, 2009. 392 ñ. NeMo+: Îáúåêòíî-îðèåíòèðîâàííàÿ ñèñòåìà ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ íà îñíîâå íåäîîïðåäåëåííûõ ìîäåëåé / Ã.Á. Çàãîðóëüêî [è äð.] // Òð. øåñòîé íàö. êîíô. ïî èñêóññòâåííîìó èíòåëëåêòó (ÊÈÈ'98). Ïóùèíî. 1998. Ñ. 524530. Îáñòàíîâêà äëÿ ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ íà îñíîâå íåäîîïðåäåëåííûõ ìîäåëåé NeMo+ (ÿçûê, àðõèòåêòóðà, èíòåðôåéñ) / Ã.Á. Çàãîðóëüêî [è äð.] // Íàó÷íîòåõí. îò÷åò N 7 / Ðîññèéñêèé ÍÈÈ èñêóññòâåííîãî èíòåëëåêòà, Èíñòèòóò ñèñòåì èíôîðìàòèêè èì. À. Ï. Åðøîâà ÑÎ ÐÀÍ. Ìîñêâà, Íîâîñèáèðñê, 1998. 107 ñ. Ïðîãðàììèðîâàíèå â îãðàíè÷åíèÿõ è íåäîîïðåäåëåííûå ìîäåëè À.Ñ. Íàðèíüÿíè [è äð.] // Èíôîðìàöèîííûå òåõíîëîãèè. 1998. N 7. Ñ. 1322. Òåõíîëîãèÿ ðåøåíèÿ çàäà÷ â îáúåêòíî-îðèåíòèðîâàííîé ñðåäå ÍåÌî+ / À.À. Êóçíåöîâ [è äð.] // Ïÿòàÿ íàöèîíàëüíàÿ êîíôåðåíöèÿ ñ ìåæäóíàðîäíûì ó÷àñòèåì ¾Èñêóññòâåííûé èíòåëëåêò 96¿. Ñáîðíèê íàó÷íûõ òðóäîâ â òðåõ òîìàõ. Òîì III. Êàçàíü, 1996. Ñ. 408414. Íàðèíüÿíè À.Ñ., Íàïðååíêî Â.Ã., Ñìèðíîâ Å.Ï. Êîìïüþòåðíàÿ ìîäåëü ýêîíîìèêè ðåãèîíà íà îñíîâå èíòåëëåêòóàëüíîé òåõíîëîãèè íîâîãî ïîêîëåíèÿ. Êîíöåïöèÿ, 62 òåõíîëîãèÿ è èíñòðóìåíòû ôîðìèðîâàíèÿ óïðàâëåí÷åñêèõ ðåøåíèé â ñîöèàëüíîýêîíîìè÷åñêîé è òåõíîëîãè÷åñêîé ñôåðàõ ñóáúåêòîâ Ðîññèéñêîé Ôåäåðàöèè. Èçäàíèå ðàáî÷åé ãðóïïû ïî ñòðàòåãèè ðåãèîíàëüíîãî ðàçâèòèÿ è áåçîïàñíîñòè Ãîñóäàðñòâåííîãî Ñîâåòà Ðîññèéñêîé Ôåäåðàöèè. Ìîñêâà, 2005. 10. Ïåòðîâ Å.Ñ. Îïûò èíòåãðàöèè ëîãè÷åñêîãî ïðîãðàììèðîâàíèÿ è ïðîãðàììèðîâàíèÿ â îãðàíè÷åíèÿõ // Ïðîãðàììèðîâàíèå. 1998. N 3. Ñ.4049. 11. Ïåòðîâ Å.Ñ., ßõíî Ò.Ì. Íåäîîïðåäåëåííûå ìîäåëè è ëîãè÷åñêîå ïðîãðàììèðîâàíèå: ðåàëèçàöèÿ îãðàíè÷åíèé // Ïðîãðàììèðîâàíèå. 2001, 2. Ñ.60-67. 12. Ðàññåë Ñ., Íîðâèã Ï. Èñêóññòâåííûé èíòåëëåêò: ñîâðåìåííûé ïîäõîä. 2å èçä. / Ïåð. ñ àíãë. Ì.: Âèëüÿìñ, 2006. 1408 ñ. èìååòñÿ êîïèÿ: http://rriai.org.ru/zadachi-udovletvoreniya-ogranicheniy.html (äàòà îáðàùåíèÿ 07.08.2010). 13. Ñàðàåâ À.Ä., Ùåðáèíà Î.À. Ñèñòåìíûé àíàëèç è ñîâðåìåííûå èíôîðìàöèîííûå òåõíîëîãèè // Òðóäû Êðûìñêîé àêàäåìèè íàóê. Ñèìôåðîïîëü: ÑÎÍÀÒ. 2006. Ñ. 47-59. 14. Ñåìåíîâ À.Ë. 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