743:
that runs in polynomial time, for which the output to the problem is the number of accepting paths of the Turing machine. Intuitively, such problems count the number of solutions to problems in the complexity class
545:
351:
440:
1025:
1005:
985:
961:
941:
921:
901:
881:
846:
826:
806:
786:
766:
725:
701:
669:
649:
629:
609:
589:
569:
500:
480:
460:
411:
391:
371:
316:
296:
273:
253:
233:
213:
193:
173:
149:
129:
109:
89:
703:
on the inputs to problems that preserves the exact values of the outputs. Such a reduction can be viewed as a polynomial-time counting reduction, by using the
1127:, SIAM Monographs on Discrete Mathematics and Applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, pp. 12–13,
1168:
853:
1140:
1080:
1067:
740:
1210:
20:
278:
These two functions must preserve the correctness of the output. That is, suppose that one transforms an input
24:
512:
1075:, Frontiers in Artificial Intelligence and Applications, vol. 185, IOS Press, pp. 633–654,
71:
A polynomial-time counting reduction is usually used to transform instances of a known-hard problem
735:
A functional problem (specified by its inputs and desired outputs) belongs to the complexity class
680:
507:
60:
32:
1087:
768:
is said to be ♯P-hard if there exists a polynomial-time counting reduction from every problem
1120:
36:
321:
1191:
1150:
964:
503:
943:. If this reduction exists, then there exists a reduction from any other problem in ♯P to
416:
8:
1010:
990:
970:
946:
926:
906:
886:
866:
831:
811:
791:
771:
751:
710:
686:
654:
634:
614:
594:
574:
554:
485:
465:
445:
396:
376:
356:
301:
281:
258:
238:
218:
198:
178:
158:
134:
114:
94:
74:
52:
1182:
1136:
1076:
704:
1177:
1128:
857:
56:
1187:
1146:
849:
745:
152:
1163:
1112:
1204:
1132:
883:
in ♯P to be ♯P-complete is to start with a single known ♯P-complete problem
35:(a transformation from one problem to another) used to define the notion of
1063:
1116:
1059:
1055:
1125:
Complexity classifications of
Boolean constraint satisfaction problems
548:
736:
40:
854:
proving the completeness of the permanent of 0–1 matrices
111:
that is to be proven hard. It consists of two functions
1062:(2009), "Chapter 20. Model Counting", in Biere, Armin;
674:
1166:(1979), "The complexity of computing the permanent",
1013:
993:
973:
949:
929:
909:
889:
869:
834:
814:
794:
774:
754:
713:
689:
657:
637:
617:
597:
577:
557:
515:
488:
468:
448:
419:
399:
379:
359:
324:
304:
284:
261:
241:
221:
201:
181:
161:
137:
117:
97:
77:
1054:
730:
1121:"2.2.2 Parsimonious reductions and ♯P-completeness"
1110:
903:and find a polynomial-time counting reduction from
1019:
999:
979:
955:
935:
915:
895:
875:
840:
820:
800:
780:
760:
719:
695:
663:
643:
623:
603:
583:
563:
539:
494:
474:
454:
434:
413:. It must be the case that the transformed output
405:
385:
365:
345:
310:
290:
267:
247:
227:
207:
187:
167:
143:
123:
103:
83:
860:, is instead used for defining ♯P-completeness.)
1202:
852:. (Sometimes, as in Valiant's original paper
591:to transform the problem into an instance of
442:is the correct output for the original input
16:Problem transformation for counting solutions
462:. That is, if the input-output relations of
1181:
1066:; van Maaren, Hans; Walsh, Toby (eds.),
1162:
547:. Alternatively, expressed in terms of
502:are expressed as functions, then their
45:polynomial many-one counting reductions
1203:
967:a reduction from the other problem to
863:The usual method of proving a problem
611:, solve that instance, and then apply
151:, both of which must be computable in
43:. These reductions may also be called
1106:
1104:
1102:
1100:
1098:
1096:
1050:
1048:
1046:
1044:
1042:
1040:
551:, one possible algorithm for solving
683:is a polynomial-time transformation
675:Relation to other kinds of reduction
13:
1156:
1093:
1037:
91:into instances of another problem
29:polynomial-time counting reduction
14:
1222:
731:Applications in complexity theory
540:{\displaystyle X=g\circ Y\circ f}
856:, a weaker notion of reduction,
741:non-deterministic Turing machine
21:computational complexity theory
429:
423:
340:
334:
49:weakly parsimonious reductions
1:
1030:
66:
1183:10.1016/0304-3975(79)90044-6
1169:Theoretical Computer Science
651:into the correct answer for
7:
828:itself belongs to ♯P, then
631:to transform the output of
10:
1227:
1069:Handbook of Satisfiability
39:for the complexity class
987:with the reduction from
59:and they generalize the
51:; they are analogous to
1133:10.1137/1.9780898718546
748:. A functional problem
235:transforms outputs for
61:parsimonious reductions
1211:Reduction (complexity)
1021:
1001:
981:
957:
937:
917:
897:
877:
842:
822:
802:
782:
762:
721:
697:
681:parsimonious reduction
665:
645:
625:
605:
585:
565:
541:
496:
476:
456:
436:
407:
387:
373:, and then one solves
367:
347:
346:{\displaystyle y=f(x)}
312:
292:
269:
249:
229:
209:
189:
175:transforms inputs for
169:
145:
125:
105:
85:
1058:; Sabharwal, Ashish;
1022:
1002:
982:
958:
938:
918:
898:
878:
843:
823:
803:
783:
763:
722:
698:
679:As a special case, a
666:
646:
626:
606:
586:
566:
542:
497:
477:
457:
437:
408:
393:to produce an output
388:
368:
348:
313:
293:
270:
250:
230:
210:
190:
170:
146:
126:
106:
86:
1086:. See in particular
1011:
991:
971:
947:
927:
907:
887:
867:
832:
812:
792:
772:
752:
711:
687:
655:
635:
615:
595:
575:
555:
513:
504:function composition
486:
466:
446:
435:{\displaystyle g(z)}
417:
397:
377:
357:
322:
302:
282:
259:
239:
219:
199:
179:
159:
135:
115:
95:
75:
808:. If, in addition,
215:, and the function
53:many-one reductions
1017:
997:
977:
953:
933:
913:
893:
873:
838:
818:
798:
778:
758:
739:if there exists a
717:
693:
661:
641:
621:
601:
581:
571:would be to apply
561:
537:
492:
472:
452:
432:
403:
383:
363:
343:
308:
288:
265:
245:
225:
205:
185:
165:
141:
121:
101:
81:
1111:Creignou, Nadia;
1020:{\displaystyle Y}
1000:{\displaystyle X}
980:{\displaystyle X}
956:{\displaystyle Y}
936:{\displaystyle Y}
916:{\displaystyle X}
896:{\displaystyle X}
876:{\displaystyle Y}
841:{\displaystyle Y}
821:{\displaystyle Y}
801:{\displaystyle Y}
781:{\displaystyle X}
761:{\displaystyle Y}
720:{\displaystyle g}
705:identity function
696:{\displaystyle f}
664:{\displaystyle X}
644:{\displaystyle Y}
624:{\displaystyle g}
604:{\displaystyle Y}
584:{\displaystyle f}
564:{\displaystyle X}
495:{\displaystyle Y}
475:{\displaystyle X}
455:{\displaystyle x}
406:{\displaystyle z}
386:{\displaystyle y}
366:{\displaystyle Y}
311:{\displaystyle X}
291:{\displaystyle x}
268:{\displaystyle X}
255:into outputs for
248:{\displaystyle Y}
228:{\displaystyle g}
208:{\displaystyle Y}
188:{\displaystyle X}
168:{\displaystyle f}
144:{\displaystyle g}
124:{\displaystyle f}
104:{\displaystyle Y}
84:{\displaystyle X}
57:decision problems
25:counting problems
1218:
1195:
1194:
1185:
1160:
1154:
1153:
1108:
1091:
1085:
1074:
1052:
1026:
1024:
1023:
1018:
1006:
1004:
1003:
998:
986:
984:
983:
978:
962:
960:
959:
954:
942:
940:
939:
934:
922:
920:
919:
914:
902:
900:
899:
894:
882:
880:
879:
874:
858:Turing reduction
847:
845:
844:
839:
827:
825:
824:
819:
807:
805:
804:
799:
787:
785:
784:
779:
767:
765:
764:
759:
726:
724:
723:
718:
707:as the function
702:
700:
699:
694:
670:
668:
667:
662:
650:
648:
647:
642:
630:
628:
627:
622:
610:
608:
607:
602:
590:
588:
587:
582:
570:
568:
567:
562:
546:
544:
543:
538:
501:
499:
498:
493:
481:
479:
478:
473:
461:
459:
458:
453:
441:
439:
438:
433:
412:
410:
409:
404:
392:
390:
389:
384:
372:
370:
369:
364:
352:
350:
349:
344:
317:
315:
314:
309:
297:
295:
294:
289:
274:
272:
271:
266:
254:
252:
251:
246:
234:
232:
231:
226:
214:
212:
211:
206:
195:into inputs for
194:
192:
191:
186:
174:
172:
171:
166:
150:
148:
147:
142:
130:
128:
127:
122:
110:
108:
107:
102:
90:
88:
87:
82:
1226:
1225:
1221:
1220:
1219:
1217:
1216:
1215:
1201:
1200:
1199:
1198:
1161:
1157:
1143:
1113:Khanna, Sanjeev
1109:
1094:
1083:
1072:
1056:Gomes, Carla P.
1053:
1038:
1033:
1012:
1009:
1008:
992:
989:
988:
972:
969:
968:
948:
945:
944:
928:
925:
924:
908:
905:
904:
888:
885:
884:
868:
865:
864:
833:
830:
829:
813:
810:
809:
793:
790:
789:
773:
770:
769:
753:
750:
749:
733:
712:
709:
708:
688:
685:
684:
677:
656:
653:
652:
636:
633:
632:
616:
613:
612:
596:
593:
592:
576:
573:
572:
556:
553:
552:
514:
511:
510:
487:
484:
483:
467:
464:
463:
447:
444:
443:
418:
415:
414:
398:
395:
394:
378:
375:
374:
358:
355:
354:
323:
320:
319:
303:
300:
299:
283:
280:
279:
260:
257:
256:
240:
237:
236:
220:
217:
216:
200:
197:
196:
180:
177:
176:
160:
157:
156:
155:. The function
153:polynomial time
136:
133:
132:
116:
113:
112:
96:
93:
92:
76:
73:
72:
69:
17:
12:
11:
5:
1224:
1214:
1213:
1197:
1196:
1176:(2): 189–201,
1164:Valiant, L. G.
1155:
1141:
1092:
1081:
1035:
1034:
1032:
1029:
1016:
996:
976:
963:, obtained by
952:
932:
912:
892:
872:
848:is said to be
837:
817:
797:
777:
757:
732:
729:
716:
692:
676:
673:
660:
640:
620:
600:
580:
560:
536:
533:
530:
527:
524:
521:
518:
506:must obey the
491:
471:
451:
431:
428:
425:
422:
402:
382:
362:
342:
339:
336:
333:
330:
327:
307:
287:
264:
244:
224:
204:
184:
164:
140:
120:
100:
80:
68:
65:
15:
9:
6:
4:
3:
2:
1223:
1212:
1209:
1208:
1206:
1193:
1189:
1184:
1179:
1175:
1171:
1170:
1165:
1159:
1152:
1148:
1144:
1142:0-89871-479-6
1138:
1134:
1130:
1126:
1122:
1118:
1114:
1107:
1105:
1103:
1101:
1099:
1097:
1089:
1084:
1082:9781586039295
1078:
1071:
1070:
1065:
1064:Heule, Marijn
1061:
1057:
1051:
1049:
1047:
1045:
1043:
1041:
1036:
1028:
1014:
994:
974:
966:
950:
930:
910:
890:
870:
861:
859:
855:
851:
835:
815:
795:
775:
755:
747:
742:
738:
728:
714:
706:
690:
682:
672:
658:
638:
618:
598:
578:
558:
550:
534:
531:
528:
525:
522:
519:
516:
509:
505:
489:
469:
449:
426:
420:
400:
380:
360:
337:
331:
328:
325:
305:
285:
276:
262:
242:
222:
202:
182:
162:
154:
138:
118:
98:
78:
64:
62:
58:
54:
50:
46:
42:
38:
34:
31:is a type of
30:
26:
22:
1173:
1167:
1158:
1124:
1117:Sudan, Madhu
1068:
1060:Selman, Bart
862:
734:
678:
353:for problem
318:to an input
298:for problem
277:
70:
48:
44:
37:completeness
28:
18:
1088:pp. 634–635
850:♯P-complete
1031:References
549:algorithms
67:Definition
965:composing
788:in ♯P to
532:∘
526:∘
33:reduction
1205:Category
1119:(2001),
508:identity
1192:0526203
1151:1827376
19:In the
1190:
1149:
1139:
1079:
1073:(PDF)
1137:ISBN
1077:ISBN
482:and
131:and
55:for
27:, a
1178:doi
1129:doi
1007:to
923:to
47:or
23:of
1207::
1188:MR
1186:,
1172:,
1147:MR
1145:,
1135:,
1123:,
1115:;
1095:^
1039:^
1027:.
746:NP
737:♯P
727:.
671:.
275:.
63:.
41:♯P
1180::
1174:8
1131::
1090:.
1015:Y
995:X
975:X
951:Y
931:Y
911:X
891:X
871:Y
836:Y
816:Y
796:Y
776:X
756:Y
715:g
691:f
659:X
639:Y
619:g
599:Y
579:f
559:X
535:f
529:Y
523:g
520:=
517:X
490:Y
470:X
450:x
430:)
427:z
424:(
421:g
401:z
381:y
361:Y
341:)
338:x
335:(
332:f
329:=
326:y
306:X
286:x
263:X
243:Y
223:g
203:Y
183:X
163:f
139:g
119:f
99:Y
79:X
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