832:
857:
can be used to obtain a lower bound for its minimum value. Roof duality may also provide a partial assignment of the variables, indicating some of the values of a minimizer to the polynomial. Several different methods of obtaining lower bounds were developed only to later be shown to be equivalent to
700:
324:
1389:
1188:
1394:
Using the first reduction followed by roof duality, we obtain a lower bound of -3 and no indication on how to assign the three variables. Using the second reduction, we obtain the (tight) lower bound of -2 and the optimal assignment of every variable (which is
841:. Note that minimization of a submodular function is a polynomially solvable problem independent on the presentation form, for e.g. pesudo-Boolean polynomials, opposite to maximization of a submodular function which is NP-hard, Alexander Schrijver (2000).
999:
827:{\displaystyle f({\boldsymbol {x}})+f({\boldsymbol {y}})\geq f({\boldsymbol {x}}\wedge {\boldsymbol {y}})+f({\boldsymbol {x}}\vee {\boldsymbol {y}}),\;\forall {\boldsymbol {x}},{\boldsymbol {y}}\in \mathbf {B} ^{n}\,.}
1629:
550:
130:
2009:
Crowston, R.; Fellows, M.; Gutin, G.; Jones, M.; Rosamond, F.; Thomasse, S.; Yeo, A. (2011). "Simultaneously
Satisfying Linear Equations Over GF(2): MaxLin2 and Max-r-Lin2 Parameterized Above Average".
2048:
1199:
82:
1010:
2129:
1784:
1667:
588:
1525:
1503:
664:
426:
404:
1433:
1859:
880:
1716:
1824:
1804:
1736:
1687:
1461:
608:
446:
362:
1992:
1530:
451:
2095:
1926:
1875:
319:{\displaystyle f({\boldsymbol {x}})=a+\sum _{i}a_{i}x_{i}+\sum _{i<j}a_{ij}x_{i}x_{j}+\sum _{i<j<k}a_{ijk}x_{i}x_{j}x_{k}+\ldots }
2173:
1384:{\displaystyle \displaystyle f({\boldsymbol {x}})=-2x_{1}+x_{2}-x_{3}+4x_{1}x_{2}+4x_{1}x_{3}-2x_{2}x_{3}-2x_{1}x_{2}x_{3}.}
1183:{\displaystyle \displaystyle -x_{1}x_{2}x_{3}=\min _{z\in \mathbf {B} }z(-x_{1}+x_{2}+x_{3})-x_{1}x_{2}-x_{1}x_{3}+x_{1}.}
341:
41:
1953:
1738:
is NP-complete. It is proved in that in polynomial time we can either solve P or reduce the number of variables to
1894:
Hammer, P.L.; Rosenberg, I.; Rudeanu, S. (1963). "On the determination of the minima of pseudo-Boolean functions".
1983:
838:
691:
2057:
32:
2025:
1741:
2062:
1634:
694:
can be viewed as a special class of pseudo-Boolean functions, which is equivalent to the condition
555:
2046:
Ishikawa, H. (2011). "Transformation of general binary MRF minimization to the first order case".
1826:. Then proved that in polynomial time we can either solve P or reduce the number of variables to
1508:
1466:
613:
409:
367:
2083:
2015:
1193:
Different reductions lead to different results. Take for example the following cubic polynomial:
1398:
874:
to obtain an equivalent quadratic problem with additional variables. One possible reduction is
2107:
2075:
1922:
1899:
994:{\displaystyle \displaystyle -x_{1}x_{2}x_{3}=\min _{z\in \mathbf {B} }z(2-x_{1}-x_{2}-x_{3})}
2120:
2150:
2067:
1962:
1870:
1829:
109:
1692:
2029:
2139:"A Combinatorial Algorithm Minimizing Submodular Functions in Strongly Polynomial Time"
1809:
1789:
1721:
1672:
1446:
593:
431:
347:
96:
1967:
2167:
2087:
121:
24:
679:
20:
2099:
334:
2111:
1903:
837:
This is an important class of pseudo-boolean functions, because they can be
2155:
2138:
2079:
1948:
2071:
1624:{\displaystyle f(x)=\sum _{I\subseteq }{\hat {f}}(I)\prod _{i\in I}x_{i}.}
545:{\displaystyle f(x)=\sum _{I\subseteq }{\hat {f}}(I)\prod _{i\in I}x_{i},}
112:
is then a special case, where the values are also restricted to 0 or 1.
675:
674:
Minimizing (or, equivalently, maximizing) a pseudo-Boolean function is
2020:
105:
2119:
Rother, C.; Kolmogorov, V.; Lempitsky, V.; Szummer, M. (2007).
2049:
IEEE Transactions on
Pattern Analysis and Machine Intelligence
678:. This can easily be seen by formulating, for example, the
333:
of the pseudo-Boolean function is simply the degree of the
120:
Any pseudo-Boolean function can be written uniquely as a
1985:
Generalized Roof
Duality for Pseudo-Boolean Optimization
1919:
Boolean
Methods in Operations Research and Related Areas
1806:
be the degree of the above multi-linear polynomial for
2130:
Conference on
Computer Vision and Pattern Recognition
1832:
1812:
1792:
1744:
1724:
1695:
1675:
1637:
1533:
1511:
1469:
1449:
1401:
1203:
1202:
1014:
1013:
884:
883:
703:
616:
596:
558:
454:
434:
412:
370:
350:
344:), a pseudo-Boolean function is viewed as a function
133:
44:
77:{\displaystyle f:\mathbf {B} ^{n}\to \mathbb {R} ,}
1853:
1818:
1798:
1778:
1730:
1710:
1681:
1661:
1623:
1519:
1497:
1455:
1427:
1383:
1182:
993:
826:
658:
602:
582:
544:
440:
420:
398:
356:
318:
76:
682:problem as maximizing a pseudo-Boolean function.
2122:Optimizing Binary MRFs via Extended Roof Duality
1052:
922:
2100:"Some topics in analysis of Boolean functions"
2004:
2002:
8:
1486:
1470:
1004:There are other possibilities, for example,
853:is a quadratic polynomial, a concept called
653:
629:
387:
371:
342:Fourier analysis of pseudo-Boolean functions
1993:International Conference on Computer Vision
1942:
1940:
1938:
428:. Again in this case we can uniquely write
1917:Hammer, Peter L.; Rudeanu, Sergiu (1968).
788:
2154:
2061:
2019:
1966:
1831:
1811:
1791:
1755:
1743:
1723:
1694:
1674:
1639:
1638:
1636:
1612:
1596:
1572:
1571:
1553:
1532:
1513:
1512:
1510:
1489:
1468:
1448:
1402:
1400:
1371:
1361:
1351:
1335:
1325:
1309:
1299:
1283:
1273:
1257:
1244:
1231:
1210:
1201:
1170:
1157:
1147:
1134:
1124:
1108:
1095:
1082:
1062:
1055:
1042:
1032:
1022:
1012:
981:
968:
955:
932:
925:
912:
902:
892:
882:
870:is greater than 2, one can always employ
820:
814:
809:
800:
792:
777:
769:
752:
744:
727:
710:
702:
615:
595:
560:
559:
557:
533:
517:
493:
492:
474:
453:
433:
414:
413:
411:
390:
369:
349:
304:
294:
284:
268:
246:
233:
223:
210:
194:
181:
171:
161:
140:
132:
67:
66:
57:
52:
43:
1886:
1211:
801:
793:
778:
770:
753:
745:
728:
711:
141:
2137:Schrijver, Alexander (November 2000).
1876:Quadratic pseudo-Boolean optimization
7:
104:is a nonnegative integer called the
1443:Consider a pseudo-Boolean function
1689:the problem P of deciding whether
789:
16:Generalization of binary functions
14:
1982:Kahl, F.; Strandmark, P. (2011).
1947:Boros, E.; Hammer, P. L. (2002).
1669:is integral. Then for an integer
1439:Polynomial Compression Algorithms
858:what is now called roof duality.
1063:
933:
810:
53:
2143:Journal of Combinatorial Theory
1779:{\displaystyle O(k^{2}\log k).}
1896:Studii și cercetări matematice
1848:
1836:
1770:
1748:
1705:
1699:
1656:
1650:
1644:
1589:
1583:
1577:
1566:
1560:
1543:
1537:
1421:
1403:
1215:
1207:
1114:
1072:
987:
942:
782:
766:
757:
741:
732:
724:
715:
707:
623:
617:
577:
571:
565:
510:
504:
498:
487:
481:
464:
458:
448:as a multi-linear polynomial:
145:
137:
63:
1:
1968:10.1016/S0166-218X(01)00341-9
1949:"Pseudo-Boolean Optimization"
1898:(in Romanian) (14): 359–364.
1662:{\displaystyle {\hat {f}}(I)}
1631:Assume that each coefficient
583:{\displaystyle {\hat {f}}(I)}
1954:Discrete Applied Mathematics
1520:{\displaystyle \mathbb {R} }
1498:{\displaystyle \{-1,1\}^{n}}
839:minimized in polynomial time
659:{\displaystyle =\{1,...,n\}}
590:are Fourier coefficients of
421:{\displaystyle \mathbb {R} }
399:{\displaystyle \{-1,1\}^{n}}
340:In many settings (e.g., in
2190:
2174:Mathematical optimization
1428:{\displaystyle {(0,1,1)}}
692:submodular set functions
337:in this representation.
29:pseudo-Boolean function
2156:10.1006/jctb.2000.1989
1855:
1854:{\displaystyle r(k-1)}
1820:
1800:
1780:
1732:
1712:
1683:
1663:
1625:
1521:
1499:
1457:
1429:
1385:
1184:
995:
828:
660:
604:
584:
546:
442:
422:
400:
358:
320:
78:
2072:10.1109/tpami.2010.91
1856:
1821:
1801:
1781:
1733:
1713:
1684:
1664:
1626:
1522:
1500:
1458:
1430:
1386:
1185:
996:
829:
661:
605:
585:
547:
443:
423:
401:
359:
321:
79:
2012:Proc. Of FSTTCS 2011
1830:
1810:
1790:
1742:
1722:
1718:is more or equal to
1711:{\displaystyle f(x)}
1693:
1673:
1635:
1531:
1509:
1467:
1447:
1399:
1200:
1011:
881:
701:
614:
594:
556:
452:
432:
410:
368:
348:
131:
42:
2030:2011arXiv1104.1135C
108:of the function. A
1851:
1816:
1796:
1776:
1728:
1708:
1679:
1659:
1621:
1607:
1570:
1517:
1495:
1463:as a mapping from
1453:
1425:
1381:
1380:
1180:
1179:
1068:
991:
990:
938:
824:
656:
600:
580:
542:
528:
491:
438:
418:
396:
354:
316:
263:
205:
166:
74:
1928:978-3-642-85825-3
1819:{\displaystyle f}
1799:{\displaystyle r}
1731:{\displaystyle k}
1682:{\displaystyle k}
1647:
1592:
1580:
1549:
1456:{\displaystyle f}
1051:
921:
866:If the degree of
603:{\displaystyle f}
568:
513:
501:
470:
441:{\displaystyle f}
357:{\displaystyle f}
242:
190:
157:
2181:
2160:
2158:
2133:
2127:
2115:
2091:
2065:
2056:(6): 1234–1249.
2034:
2033:
2023:
2006:
1997:
1996:
1990:
1979:
1973:
1972:
1970:
1961:(1–3): 155–225.
1944:
1933:
1932:
1914:
1908:
1907:
1891:
1871:Boolean function
1860:
1858:
1857:
1852:
1825:
1823:
1822:
1817:
1805:
1803:
1802:
1797:
1785:
1783:
1782:
1777:
1760:
1759:
1737:
1735:
1734:
1729:
1717:
1715:
1714:
1709:
1688:
1686:
1685:
1680:
1668:
1666:
1665:
1660:
1649:
1648:
1640:
1630:
1628:
1627:
1622:
1617:
1616:
1606:
1582:
1581:
1573:
1569:
1526:
1524:
1523:
1518:
1516:
1504:
1502:
1501:
1496:
1494:
1493:
1462:
1460:
1459:
1454:
1434:
1432:
1431:
1426:
1424:
1390:
1388:
1387:
1382:
1376:
1375:
1366:
1365:
1356:
1355:
1340:
1339:
1330:
1329:
1314:
1313:
1304:
1303:
1288:
1287:
1278:
1277:
1262:
1261:
1249:
1248:
1236:
1235:
1214:
1189:
1187:
1186:
1181:
1175:
1174:
1162:
1161:
1152:
1151:
1139:
1138:
1129:
1128:
1113:
1112:
1100:
1099:
1087:
1086:
1067:
1066:
1047:
1046:
1037:
1036:
1027:
1026:
1000:
998:
997:
992:
986:
985:
973:
972:
960:
959:
937:
936:
917:
916:
907:
906:
897:
896:
833:
831:
830:
825:
819:
818:
813:
804:
796:
781:
773:
756:
748:
731:
714:
665:
663:
662:
657:
609:
607:
606:
601:
589:
587:
586:
581:
570:
569:
561:
551:
549:
548:
543:
538:
537:
527:
503:
502:
494:
490:
447:
445:
444:
439:
427:
425:
424:
419:
417:
405:
403:
402:
397:
395:
394:
363:
361:
360:
355:
325:
323:
322:
317:
309:
308:
299:
298:
289:
288:
279:
278:
262:
238:
237:
228:
227:
218:
217:
204:
186:
185:
176:
175:
165:
144:
110:Boolean function
103:
93:
83:
81:
80:
75:
70:
62:
61:
56:
2189:
2188:
2184:
2183:
2182:
2180:
2179:
2178:
2164:
2163:
2136:
2125:
2118:
2096:O'Donnell, Ryan
2094:
2063:10.1.1.675.2183
2045:
2042:
2037:
2008:
2007:
2000:
1988:
1981:
1980:
1976:
1946:
1945:
1936:
1929:
1916:
1915:
1911:
1893:
1892:
1888:
1884:
1867:
1828:
1827:
1808:
1807:
1788:
1787:
1751:
1740:
1739:
1720:
1719:
1691:
1690:
1671:
1670:
1633:
1632:
1608:
1529:
1528:
1507:
1506:
1485:
1465:
1464:
1445:
1444:
1441:
1397:
1396:
1367:
1357:
1347:
1331:
1321:
1305:
1295:
1279:
1269:
1253:
1240:
1227:
1198:
1197:
1166:
1153:
1143:
1130:
1120:
1104:
1091:
1078:
1038:
1028:
1018:
1009:
1008:
977:
964:
951:
908:
898:
888:
879:
878:
864:
862:Quadratizations
847:
808:
699:
698:
688:
672:
612:
611:
592:
591:
554:
553:
529:
450:
449:
430:
429:
408:
407:
386:
366:
365:
346:
345:
300:
290:
280:
264:
229:
219:
206:
177:
167:
129:
128:
118:
116:Representations
101:
88:
51:
40:
39:
17:
12:
11:
5:
2187:
2185:
2177:
2176:
2166:
2165:
2162:
2161:
2149:(2): 346–355.
2134:
2116:
2092:
2041:
2038:
2036:
2035:
1998:
1974:
1934:
1927:
1909:
1885:
1883:
1880:
1879:
1878:
1873:
1866:
1863:
1850:
1847:
1844:
1841:
1838:
1835:
1815:
1795:
1775:
1772:
1769:
1766:
1763:
1758:
1754:
1750:
1747:
1727:
1707:
1704:
1701:
1698:
1678:
1658:
1655:
1652:
1646:
1643:
1620:
1615:
1611:
1605:
1602:
1599:
1595:
1591:
1588:
1585:
1579:
1576:
1568:
1565:
1562:
1559:
1556:
1552:
1548:
1545:
1542:
1539:
1536:
1515:
1492:
1488:
1484:
1481:
1478:
1475:
1472:
1452:
1440:
1437:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1392:
1391:
1379:
1374:
1370:
1364:
1360:
1354:
1350:
1346:
1343:
1338:
1334:
1328:
1324:
1320:
1317:
1312:
1308:
1302:
1298:
1294:
1291:
1286:
1282:
1276:
1272:
1268:
1265:
1260:
1256:
1252:
1247:
1243:
1239:
1234:
1230:
1226:
1223:
1220:
1217:
1213:
1209:
1206:
1191:
1190:
1178:
1173:
1169:
1165:
1160:
1156:
1150:
1146:
1142:
1137:
1133:
1127:
1123:
1119:
1116:
1111:
1107:
1103:
1098:
1094:
1090:
1085:
1081:
1077:
1074:
1071:
1065:
1061:
1058:
1054:
1050:
1045:
1041:
1035:
1031:
1025:
1021:
1017:
1002:
1001:
989:
984:
980:
976:
971:
967:
963:
958:
954:
950:
947:
944:
941:
935:
931:
928:
924:
920:
915:
911:
905:
901:
895:
891:
887:
863:
860:
846:
843:
835:
834:
823:
817:
812:
807:
803:
799:
795:
791:
787:
784:
780:
776:
772:
768:
765:
762:
759:
755:
751:
747:
743:
740:
737:
734:
730:
726:
723:
720:
717:
713:
709:
706:
687:
684:
671:
668:
655:
652:
649:
646:
643:
640:
637:
634:
631:
628:
625:
622:
619:
599:
579:
576:
573:
567:
564:
541:
536:
532:
526:
523:
520:
516:
512:
509:
506:
500:
497:
489:
486:
483:
480:
477:
473:
469:
466:
463:
460:
457:
437:
416:
393:
389:
385:
382:
379:
376:
373:
353:
327:
326:
315:
312:
307:
303:
297:
293:
287:
283:
277:
274:
271:
267:
261:
258:
255:
252:
249:
245:
241:
236:
232:
226:
222:
216:
213:
209:
203:
200:
197:
193:
189:
184:
180:
174:
170:
164:
160:
156:
153:
150:
147:
143:
139:
136:
117:
114:
97:Boolean domain
85:
84:
73:
69:
65:
60:
55:
50:
47:
15:
13:
10:
9:
6:
4:
3:
2:
2186:
2175:
2172:
2171:
2169:
2157:
2152:
2148:
2144:
2140:
2135:
2131:
2124:
2123:
2117:
2113:
2109:
2105:
2101:
2097:
2093:
2089:
2085:
2081:
2077:
2073:
2069:
2064:
2059:
2055:
2051:
2050:
2044:
2043:
2039:
2031:
2027:
2022:
2017:
2013:
2005:
2003:
1999:
1994:
1987:
1986:
1978:
1975:
1969:
1964:
1960:
1956:
1955:
1950:
1943:
1941:
1939:
1935:
1930:
1924:
1920:
1913:
1910:
1905:
1901:
1897:
1890:
1887:
1881:
1877:
1874:
1872:
1869:
1868:
1864:
1862:
1845:
1842:
1839:
1833:
1813:
1793:
1773:
1767:
1764:
1761:
1756:
1752:
1745:
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763:
760:
749:
738:
735:
721:
718:
704:
697:
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686:Submodularity
685:
683:
681:
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669:
667:
650:
647:
644:
641:
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635:
632:
626:
620:
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1984:
1977:
1958:
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1921:. Springer.
1918:
1912:
1895:
1889:
1442:
1393:
1192:
1003:
871:
867:
865:
855:roof duality
854:
850:
848:
845:Roof Duality
836:
689:
673:
670:Optimization
339:
330:
328:
124:polynomial:
122:multi-linear
119:
95:
89:
86:
35:of the form
28:
25:optimization
18:
680:maximum cut
21:mathematics
2040:References
872:reductions
364:that maps
335:polynomial
2112:1433-8092
2058:CiteSeerX
2021:1104.1135
1904:0039-4068
1843:−
1765:
1645:^
1601:∈
1594:∏
1578:^
1558:⊆
1551:∑
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1342:−
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1251:−
1222:−
1141:−
1118:−
1076:−
1060:∈
1016:−
975:−
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736:≥
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515:∏
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314:…
244:∑
192:∑
159:∑
64:→
2168:Category
2098:(2008).
2088:17314555
2080:20421673
1865:See also
92:= {0, 1}
33:function
2026:Bibcode
1527:. Then
676:NP-hard
2110:
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1902:
552:where
331:degree
87:where
2145:. B.
2126:(PDF)
2084:S2CID
2016:arXiv
1989:(PDF)
1882:Notes
106:arity
94:is a
31:is a
2108:ISSN
2104:ECCC
2076:PMID
1923:ISBN
1900:ISSN
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690:The
610:and
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