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Graphs, polymorphisms and the complexity of homomorphism problems

Published: 17 May 2008 Publication History

Abstract

We use a connection between polymorphisms and the structure of smooth digraphs to prove the conjecture of Bang-Jensen and Hell from 1990 and, as a consequence, a conjecture of Bang-Jensen, Hell and MacGillivray from 1995. The conjectured characterization of computationally complex coloring problems for smooth digraphs is proved using tools of universal algebra. We cite further graph results obtained using this new approach. The proofs are based in an universal algebraic framework developed for the Constraint Satisfaction Problem and the CSP dichotomy conjecture of Feder and Vardi in particular.

References

[1]
Jørgen Bang-Jensen and Pavol Hell. The effect of two cycles on the complexity of colourings by directed graphs. Discrete Appl. Math., 26(1):1--23, 1990.]]
[2]
Jørgen Bang-Jensen, Pavol Hell, and Gary MacGillivray. Hereditarily hard $H$-colouring problems. Discrete Math., 138(1-3):75--92, 1995. 14th British Combinatorial Conference (Keele, 1993).]]
[3]
Libor Barto, Marcin Kozik, Miklós Maróti, and Todd Niven. Csp dichotomy for special triads. in preparation.]]
[4]
Libor Barto, Marcin Kozik, and Todd Niven. The csp dichotomy holds for digraphs with no sources and no sinks (a positive answer to a conjecture of bang-jensen and hell). submitted.]]
[5]
Andrei Bulatov, Peter Jeavons, and Andrei Krokhin. Classifying the complexity of constraints using finite algebras. SIAM J. Comput., 34(3):720--742 (electronic), 2005.]]
[6]
Andrei A. Bulatov. H-coloring dichotomy revisited. Theoret. Comput. Sci., 349(1):31--39, 2005.]]
[7]
Andrei A. Bulatov, Andrei A. Krokhin, and Peter Jeavons. Constraint satisfaction problems and finite algebras. In Automata, languages and programming (Geneva, 2000), volume 1853 of Lecture Notes in Comput. Sci., pages 272--282. Springer, Berlin, 2000.]]
[8]
Stanley Burris and H. P. Sankappanavar. A course in universal algebra, volume 78 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1981.]]
[9]
Tomás Feder. Classification of homomorphisms to oriented cycles and of k-partite satisfiability. SIAM J. Discrete Math., 14(4):471--480 (electronic), 2001.]]
[10]
Tomás Feder and Moshe Y. Vardi. The computational structure of monotone monadic SNP and constraint satisfaction: a study through Datalog and group theory. SIAM J. Comput., 28(1):57--104 (electronic), 1999.]]
[11]
Wolfgang Gutjahr, Emo Welzl, and Gerhard Woeginger. Polynomial graph-colorings. Discrete Appl. Math., 35(1):29--45, 1992.]]
[12]
P. Hell, J. Nevril, and X. Zhu. Complexity of tree homomorphisms. Discrete Appl. Math., 70(1):23--36, 1996.]]
[13]
P. Hell, J. Nevril, and X. Zhu. Duality and polynomial testing of tree homomorphisms. Trans. Amer. Math. Soc., 348(4):1281--1297, 1996.]]
[14]
P. Hell, J. Nevsetvril, and X. Zhu. Duality of graph homomorphisms. In Combinatorics, Paul Erd\H os is eighty, Vol. 2 (Keszthely, 1993), volume 2 of Bolyai Soc. Math. Stud., pages 271--282. János Bolyai Math. Soc., Budapest, 1996.]]
[15]
Pavol Hell and Jaroslav Ne\vset\vril. On the complexity of $H$-coloring. J. Combin. Theory Ser. B, 48(1):92--110, 1990.]]
[16]
Pavol Hell and Jaroslav Ne\vset\vril. Graphs and homomorphisms, volume 28 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2004.]]
[17]
Pavol Hell, Hui Shan Zhou, and Xuding Zhu. Homomorphisms to oriented cycles. Combinatorica, 13(4):421--433, 1993.]]
[18]
David Hobby and Ralph McKenzie. The structure of finite algebras, volume 76 of Contemporary Mathematics. American Mathematical Society, Providence, RI, 1988.]]
[19]
Peter Jeavons, David Cohen, and Marc Gyssens. Closure properties of constraints. J. ACM, 44(4):527--548, 1997.]]
[20]
Benoit Larose and László Zádori. The complexity of the extendibility problem for finite posets. SIAM J. Discrete Math., 17(1):114--121 (electronic), 2003.]]
[21]
Benoit Larose and László Zádori. Taylor terms, constraint satisfaction and the complexity of polynomial equations over finite algebras. Internat. J. Algebra Comput., 16(3):563--581, 2006.]]
[22]
Gary MacGillivray. On the complexity of colouring by vertex-transitive and arc-transitive digraphs. SIAM J. Discrete Math., 4(3):397--408, 1991.]]
[23]
Miklós Maróti and Ralph McKenzie. Existence theorems for weakly symmetric operations. Algebra Universalis (accepted), 2007.]]
[24]
Ralph N. McKenzie, George F. McNulty, and Walter F. Taylor. Algebras, lattices, varieties. Vol. I. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1987.]]
[25]
Ugo Montanari. Networks of constraints: fundamental properties and applications to picture processing. Information Sci., 7:95--132, 1974.]]
[26]
Thomas J. Schaefer. The complexity of satisfiability problems. In Conference Record of the Tenth Annual ACM Symposium on Theory of Computing (San Diego, Calif., 1978), pages 216--226. ACM, New York, 1978.]]
[27]
Walter Taylor. Varieties obeying homotopy laws. Canad. J. Math., 29(3):498--527, 1977.]]
[28]
Xuding Zhu. A polynomial algorithm for homomorphisms to oriented cycles. J. Algorithms, 19(3):333--345, 1995.]]

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    cover image ACM Conferences
    STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computing
    May 2008
    712 pages
    ISBN:9781605580470
    DOI:10.1145/1374376
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    Published: 17 May 2008

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    Author Tags

    1. computational complexity
    2. constraint satisfaction problem
    3. graph homomorphism
    4. polymorphism
    5. universal algebra

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    May 17 - 20, 2008
    British Columbia, Victoria, Canada

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    Cited By

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    • (2016)Dualities and algebras with a near-unanimity termAlgebra universalis10.1007/s00012-016-0388-x76:1(111-126)Online publication date: 23-Jul-2016
    • (2015)Lower Bounds for the Graph Homomorphism ProblemAutomata, Languages, and Programming10.1007/978-3-662-47672-7_39(481-493)Online publication date: 20-Jun-2015
    • (2014)H-coloring degree-bounded (acyclic) digraphsTheoretical Computer Science10.1016/j.tcs.2014.06.014554:C(40-49)Online publication date: 16-Oct-2014
    • (2013)The complexity of the counting constraint satisfaction problemJournal of the ACM10.1145/252840060:5(1-41)Online publication date: 28-Oct-2013
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    • (2011)The Dichotomy for Conservative Constraint Satisfaction Problems RevisitedProceedings of the 2011 IEEE 26th Annual Symposium on Logic in Computer Science10.1109/LICS.2011.25(301-310)Online publication date: 21-Jun-2011
    • (2011)On Maltsev DigraphsComputer Science – Theory and Applications10.1007/978-3-642-20712-9_14(181-194)Online publication date: 2011
    • (2010)The complexity of global cardinality constraintsLogical Methods in Computer Science10.2168/LMCS-6(4:4)20106:4Online publication date: 27-Oct-2010
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