475:
383:
851:
problem, each of the original objects has a "color", and it is required that the covering contains exactly one (or at most one) object of each color. Rainbow covering was studied e.g. for covering points by
776:
277:
547:
799:
623:
585:
1060:
If the geometric cover problem admits an r-approximation algorithm, then the conflict-free geometric cover problem admits a similar approximation algorithm in FPT time.
803:(the structures that are covered depend on the combinatorial context). Finally, an optimal solution to the above integer linear program is a covering of minimal cost.
31:
are computational problems that ask whether a certain combinatorial structure 'covers' another, or how large the structure has to be to do that. Covering problems are
723:
696:
183:
835:, the covering problem is defined as the question if for a given marking, there exists a run of the net, such that some larger (or equal) marking can be reached.
743:
669:
649:
393:
288:
1124:
1128:
176:
69:
Covering problems allow the covering primitives to overlap; the process of covering something with non-overlapping primitives is called
1142:
Arkin, Esther M.; Banik, Aritra; Carmi, Paz; Citovsky, Gui; Katz, Matthew J.; Mitchell, Joseph S. B.; Simakov, Marina (2018-12-11).
169:
839:
means here that all components are at least as large as the ones of the given marking and at least one is properly larger.
1090:
748:
79:
995:
224:
205:, one can think of any minimization linear program as a covering problem if the coefficients in the constraint
131:
70:
490:
209:, the objective function, and right-hand side are nonnegative. More precisely, consider the following general
847:
In some covering problems, the covering should satisfy some additional requirements. In particular, in the
1283:
820:
32:
1184:
1054:
782:
590:
552:
119:
36:
853:
816:
210:
1230:
Banik, Aritra; Panolan, Fahad; Raman, Venkatesh; Sahlot, Vibha; Saurabh, Saket (2020-01-01).
701:
674:
206:
138:
59:
8:
1108:"The Stochastic Test Collection Problem: Models, Exact and Heuristic Solution Approaches"
143:
1259:
1212:
1042:
728:
654:
634:
202:
63:
1263:
1251:
1216:
1204:
1165:
1118:
1086:
155:
102:
51:
1105:
1243:
1196:
1155:
150:
95:
44:
24:
1200:
1247:
1078:
470:{\displaystyle x_{i}\in \left\{0,1,2,\ldots \right\}{\text{ for }}i=1,\dots ,n}
1160:
1143:
378:{\displaystyle \sum _{i=1}^{n}a_{ji}x_{i}\geq b_{j}{\text{ for }}j=1,\dots ,m}
1277:
1255:
1231:
1208:
1169:
20:
1107:
1045:
the following for the special case in which the conflict-graph has bounded
812:
126:
40:
1232:"Parameterized Complexity of Geometric Covering Problems Having Conflicts"
937:
107:
55:
1185:"Approximation algorithms for geometric conflict free covering problems"
1057:
tractable (FPT), then the conflict-free geometric cover problem is FPT.
1046:
941:
114:
869:
832:
1026:
is made of disjoint cliques, where each clique represents a color.
1019:
A rainbow set is a conflict-free set in the special case in which
1113:. European Journal of Operational Research, 299 (2022), 945–959}.
823:. Other stochastic related versions of the problem can be found.
880:
1183:
Banik, Aritra; Sahlot, Vibha; Saurabh, Saket (2020-08-01).
1106:
Douek-Pinkovich, Y., Ben-Gal, I., & Raviv, T. (2022).
196:
50:
The most prominent examples of covering problems are the
1229:
1141:
894:
if it contains at most a single interval of each color.
785:
751:
731:
704:
677:
657:
637:
593:
555:
493:
396:
291:
227:
1033:
is the problem of finding a conflict-free subset of
1182:
793:
770:
737:
717:
690:
663:
643:
617:
579:
541:
469:
377:
271:
1275:
811:There are various kinds of covering problems in
771:{\displaystyle A\mathbf {x} \geq \mathbf {b} }
806:
177:
1123:: CS1 maint: multiple names: authors list (
1041:. Banik, Panolan, Raman, Sahlot and Saurabh
483:Such an integer linear program is called a
1127:) CS1 maint: numeric names: authors list (
184:
170:
1159:
947:
913:is contained in at least one interval of
826:
272:{\displaystyle \sum _{i=1}^{n}c_{i}x_{i}}
1077:
924:is the problem of finding a rainbow set
651:types of object and each object of type
542:{\displaystyle a_{ji},b_{j},c_{i}\geq 0}
1144:"Selecting and covering colored points"
1276:
197:General linear programming formulation
842:
725:indicates how many objects of type
13:
1053:If the geometric cover problem is
14:
1295:
787:
764:
756:
778:are satisfied, it is said that
1223:
1176:
1135:
1099:
1071:
968:objects, and a conflict-graph
80:Covering/packing-problem pairs
1:
1064:
1005:, that is, no two objects in
58:, and its special cases, the
54:, which is equivalent to the
16:Type of computational problem
1201:10.1016/j.comgeo.2019.101591
1148:Discrete Applied Mathematics
1009:are connected by an edge in
794:{\displaystyle \mathbf {x} }
618:{\displaystyle j=1,\dots ,m}
580:{\displaystyle i=1,\dots ,n}
7:
876:of points on the real line.
745:we buy. If the constraints
10:
1300:
1248:10.1007/s00453-019-00600-w
821:Category:Covering problems
807:Kinds of covering problems
671:has an associated cost of
1161:10.1016/j.dam.2018.05.011
952:A more general notion is
868:colored intervals on the
1083:Approximation Algorithms
922:Rainbow covering problem
1031:Conflict-free set cover
132:Maximum independent set
37:integer linear programs
1189:Computational Geometry
1037:that is a covering of
954:conflict-free covering
948:Conflict-free covering
928:that is a covering of
827:Covering in Petri nets
817:computational geometry
795:
772:
739:
719:
692:
665:
645:
619:
581:
543:
471:
379:
312:
273:
248:
211:integer linear program
796:
773:
740:
720:
718:{\displaystyle x_{i}}
693:
691:{\displaystyle c_{i}}
666:
646:
620:
582:
544:
472:
380:
292:
274:
228:
33:minimization problems
783:
749:
729:
702:
675:
655:
635:
591:
553:
491:
394:
289:
225:
127:Minimum vertex cover
60:vertex cover problem
1085:. Springer-Verlag.
956:. In this problem:
940:(by reduction from
897:A set of intervals
108:Maximum set packing
56:hitting set problem
1079:Vazirani, Vijay V.
791:
768:
735:
715:
688:
661:
641:
615:
577:
539:
467:
375:
269:
203:linear programming
201:In the context of
115:Minimum edge cover
64:edge cover problem
1284:Covering problems
909:if each point in
738:{\displaystyle i}
664:{\displaystyle i}
644:{\displaystyle n}
481:
480:
444:
352:
194:
193:
161:
160:
156:Rectangle packing
103:Minimum set cover
91:Covering problems
52:set cover problem
29:covering problems
1291:
1268:
1267:
1227:
1221:
1220:
1180:
1174:
1173:
1163:
1139:
1133:
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1122:
1114:
1112:
1103:
1097:
1096:
1075:
849:rainbow covering
843:Rainbow covering
800:
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797:
792:
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777:
775:
774:
769:
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742:
741:
736:
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713:
697:
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670:
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662:
650:
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642:
624:
622:
621:
616:
586:
584:
583:
578:
548:
546:
545:
540:
532:
531:
519:
518:
506:
505:
485:covering problem
476:
474:
473:
468:
445:
442:
440:
436:
406:
405:
384:
382:
381:
376:
353:
350:
348:
347:
335:
334:
325:
324:
311:
306:
278:
276:
275:
270:
268:
267:
258:
257:
247:
242:
216:
215:
186:
179:
172:
151:Polygon covering
120:Maximum matching
96:Packing problems
87:
86:
76:
75:
45:packing problems
25:computer science
1299:
1298:
1294:
1293:
1292:
1290:
1289:
1288:
1274:
1273:
1272:
1271:
1228:
1224:
1181:
1177:
1140:
1136:
1116:
1115:
1110:
1104:
1100:
1093:
1076:
1072:
1067:
1055:fixed-parameter
1024:
1014:
1003:
996:independent set
973:
960:There is a set
950:
936:The problem is
860:There is a set
845:
829:
809:
786:
784:
781:
780:
763:
755:
750:
747:
746:
730:
727:
726:
709:
705:
703:
700:
699:
682:
678:
676:
673:
672:
656:
653:
652:
636:
633:
632:
592:
589:
588:
554:
551:
550:
527:
523:
514:
510:
498:
494:
492:
489:
488:
443: for
441:
414:
410:
401:
397:
395:
392:
391:
351: for
349:
343:
339:
330:
326:
317:
313:
307:
296:
290:
287:
286:
263:
259:
253:
249:
243:
232:
226:
223:
222:
199:
190:
17:
12:
11:
5:
1297:
1287:
1286:
1270:
1269:
1222:
1175:
1134:
1098:
1091:
1069:
1068:
1066:
1063:
1062:
1061:
1058:
1028:
1027:
1022:
1017:
1012:
1001:
980:
971:
949:
946:
934:
933:
918:
895:
877:
844:
841:
828:
825:
819:and more; see
808:
805:
789:
766:
762:
758:
754:
734:
712:
708:
685:
681:
660:
640:
631:Assume having
614:
611:
608:
605:
602:
599:
596:
576:
573:
570:
567:
564:
561:
558:
538:
535:
530:
526:
522:
517:
513:
509:
504:
501:
497:
479:
478:
466:
463:
460:
457:
454:
451:
448:
439:
435:
432:
429:
426:
423:
420:
417:
413:
409:
404:
400:
389:
386:
385:
374:
371:
368:
365:
362:
359:
356:
346:
342:
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333:
329:
323:
320:
316:
310:
305:
302:
299:
295:
284:
280:
279:
266:
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256:
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246:
241:
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220:
198:
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188:
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174:
166:
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153:
147:
146:
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135:
134:
129:
123:
122:
117:
111:
110:
105:
99:
98:
93:
83:
82:
15:
9:
6:
4:
3:
2:
1296:
1285:
1282:
1281:
1279:
1265:
1261:
1257:
1253:
1249:
1245:
1241:
1237:
1233:
1226:
1218:
1214:
1210:
1206:
1202:
1198:
1194:
1190:
1186:
1179:
1171:
1167:
1162:
1157:
1153:
1149:
1145:
1138:
1130:
1126:
1120:
1109:
1102:
1094:
1092:3-540-65367-8
1088:
1084:
1080:
1074:
1070:
1059:
1056:
1052:
1051:
1050:
1048:
1044:
1040:
1036:
1032:
1025:
1018:
1015:
1008:
1004:
997:
993:
992:conflict-free
989:
985:
981:
978:
974:
967:
963:
959:
958:
957:
955:
945:
943:
939:
931:
927:
923:
919:
916:
912:
908:
904:
900:
896:
893:
889:
885:
882:
878:
875:
871:
867:
863:
859:
858:
857:
855:
850:
840:
838:
834:
824:
822:
818:
814:
804:
802:
801:is a covering
760:
752:
732:
710:
706:
698:. The number
683:
679:
658:
638:
630:
626:
612:
609:
606:
603:
600:
597:
594:
574:
571:
568:
565:
562:
559:
556:
536:
533:
528:
524:
520:
515:
511:
507:
502:
499:
495:
486:
464:
461:
458:
455:
452:
449:
446:
437:
433:
430:
427:
424:
421:
418:
415:
411:
407:
402:
398:
390:
388:
387:
372:
369:
366:
363:
360:
357:
354:
344:
340:
336:
331:
327:
321:
318:
314:
308:
303:
300:
297:
293:
285:
282:
281:
264:
260:
254:
250:
244:
239:
236:
233:
229:
221:
218:
217:
214:
212:
208:
204:
187:
182:
180:
175:
173:
168:
167:
165:
164:
157:
154:
152:
149:
148:
145:
142:
140:
137:
136:
133:
130:
128:
125:
124:
121:
118:
116:
113:
112:
109:
106:
104:
101:
100:
97:
94:
92:
89:
88:
85:
84:
81:
78:
77:
74:
72:
71:decomposition
67:
65:
61:
57:
53:
48:
46:
42:
41:dual problems
38:
34:
30:
26:
22:
21:combinatorics
1239:
1236:Algorithmica
1235:
1225:
1192:
1188:
1178:
1151:
1147:
1137:
1101:
1082:
1073:
1038:
1034:
1030:
1029:
1020:
1010:
1006:
999:
994:if it is an
991:
987:
983:
976:
969:
965:
961:
953:
951:
935:
929:
925:
921:
914:
910:
906:
902:
901:is called a
898:
891:
890:is called a
887:
883:
873:
872:, and a set
865:
861:
848:
846:
836:
830:
813:graph theory
810:
779:
628:
627:
484:
482:
200:
139:Bin covering
90:
68:
49:
35:and usually
28:
18:
1242:(1): 1–19.
892:rainbow set
283:subject to
144:Bin packing
43:are called
1195:: 101591.
1065:References
1047:arboricity
990:is called
942:linear SAT
833:Petri nets
629:Intuition:
1264:254027914
1256:1432-0541
1217:209959954
1209:0925-7721
1170:0166-218X
1154:: 75–86.
982:A subset
870:real line
854:intervals
761:≥
607:…
569:…
534:≥
459:…
434:…
408:∈
367:…
337:≥
294:∑
230:∑
219:minimize
1278:Category
1119:cite web
1081:(2001).
903:covering
549:for all
62:and the
39:, whose
938:NP-hard
1262:
1254:
1215:
1207:
1168:
1089:
881:subset
837:Larger
207:matrix
1260:S2CID
1213:S2CID
1111:(PDF)
1043:prove
1252:ISSN
1205:ISSN
1166:ISSN
1129:link
1125:link
1087:ISBN
920:The
831:For
587:and
23:and
1244:doi
1197:doi
1156:doi
1152:250
998:in
986:of
975:on
964:of
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905:of
886:of
864:of
487:if
19:In
1280::
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1193:89
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1121:}}
1117:{{
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1039:P
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601:1
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566:,
563:1
560:=
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234:i
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171:v
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