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The Complexity of Boolean Surjective General-Valued CSPs

Published:21 November 2018Publication History
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Abstract

Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with a (Q ∪ {∞ })-valued objective function given as a sum of fixed-arity functions. In Boolean surjective VCSPs, variables take on labels from D = {0,1}, and an optimal assignment is required to use both labels from D. Examples include the classical global Min-Cut problem in graphs and the Minimum Distance problem studied in coding theory.

We establish a dichotomy theorem and thus give a complete complexity classification of Boolean surjective VCSPs with respect to exact solvability. Our work generalises the dichotomy for {0, ∞}-valued constraint languages (corresponding to surjective decision CSPs) obtained by Creignou and Hébrard. For the maximisation problem of Q≥0-valued surjective VCSPs, we also establish a dichotomy theorem with respect to approximability.

Unlike in the case of Boolean surjective (decision) CSPs, there appears a novel tractable class of languages that is trivial in the non-surjective setting. This newly discovered tractable class has an interesting mathematical structure related to downsets and upsets. Our main contribution is identifying this class and proving that it lies on the borderline of tractability. A crucial part of our proof is a polynomial-time algorithm for enumerating all near-optimal solutions to a generalised Min-Cut problem, which might be of independent interest.

References

  1. Walter Bach and Hang Zhou. 2011. Approximation for Maximum Surjective Constraint Satisfaction Problems. Technical Report. Retrieved from http://arxiv.org/abs/1110.2953.Google ScholarGoogle Scholar
  2. Libor Barto and Marcin Kozik. 2014. Constraint satisfaction problems solvable by local consistency methods. J. ACM 61, 1 (2014). Retrieved from Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. Libor Barto, Andrei Krokhin, and Ross Willard. 2017. Polymorphisms, and how to use them. See Reference Krokhin and Živný {33}, 1--44.Google ScholarGoogle Scholar
  4. Manuel Bodirsky, Jan Kára, and Barnaby Martin. 2012. The complexity of surjective homomorphism problems—A survey. Discrete Appl. Math. 160, 12 (2012), 1680--1690. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Andrei Bulatov. 2006. A dichotomy theorem for constraint satisfaction problems on a 3-element set. J. ACM 53, 1 (2006), 66--120. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. Andrei Bulatov. 2017. A dichotomy theorem for nonuniform CSP. In Proceedings of the 58th Annual IEEE Symposium on Foundations of Computer Science (FOCS’17). IEEE, 319--330.Google ScholarGoogle ScholarCross RefCross Ref
  7. Andrei Bulatov, Andrei Krokhin, and Peter Jeavons. 2005. Classifying the complexity of constraints using finite algebras. SIAM J. Comput. 34, 3 (2005), 720--742. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Andrei A. Bulatov, Víctor Dalmau, Martin Grohe, and Dániel Marx. 2012. Enumerating homomorphisms. J. Comput. Syst. Sci. 78, 2 (2012), 638--650. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Hubie Chen. 2014. An algebraic hardness criterion for surjective constraint satisfaction. Algebra Universalis 72, 4 (2014), 393--401.Google ScholarGoogle ScholarCross RefCross Ref
  10. David A. Cohen. 2004. Tractable decision for a constraint language implies tractable search. Constraints 9 (2004), 219--229. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. David A. Cohen, Martin C. Cooper, Páidí Creed, Peter Jeavons, and Stanislav Živný. 2013. An algebraic theory of complexity for discrete optimisation. SIAM J. Comput. 42, 5 (2013), 915--1939.Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. David A. Cohen, Martin C. Cooper, Peter G. Jeavons, and Andrei A. Krokhin. 2006. The complexity of soft constraint satisfaction. Artific. Intell. 170, 11 (2006), 983--1016. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. Nadia Creignou. 1995. A dichotomy theorem for maximum generalized satisfiability problems. J. Comput. Syst. Sci. 51, 3 (1995), 511--522. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Nadia Creignou and Jean-Jacques Hébrard. 1997. On generating all solutions of generalized satisfiability problems. Info. Théor. Appl. 31, 6 (1997), 499--511.Google ScholarGoogle Scholar
  15. P. Crescenzi. 1997. A short guide to approximation preserving reductions. In Proceedings of the 12th Annual IEEE Conference on Computational Complexity (CCC’97). IEEE Computer Society, Washington, DC, 262. Retrieved from http://dl.acm.org/citation.cfm?id=791230.792302. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Rina Dechter and Alon Itai. 1992. Finding all solutions if you can find one. In Proceedings of the AAAI 1992 Workshop on Tractable Reasoning. 35--39.Google ScholarGoogle Scholar
  17. Nikhil R. Devanur, Shaddin Dughmi, Roy Schwartz, Ankit Sharma, and Mohit Singh. 2013. On the approximation of submodular functions. (Apr. 2013). Retrieved from http://arxiv.org/abs/1304.4948.Google ScholarGoogle Scholar
  18. Tomás Feder and Moshe Y. Vardi. 1998. The computational structure of monotone monadic SNP and constraint satisfaction: A study through datalog and group theory. SIAM J. Comput. 28, 1 (1998), 57--104. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. Jiří Fiala and Jan Kratochvíl. 2008. Locally constrained graph homomorphisms - Structure, complexity, and applications. Comput. Sci. Rev. 2, 2 (2008), 97--111. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. Jiří Fiala and Daniël Paulusma. 2005. A complete complexity classification of the role assignment problem. Theor. Comput. Sci. 349, 1 (2005), 67--81. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. András Frank. 2011. Connections in Combinatorial Optimization. OUP Oxford. Retrieved from https://books.google.co.uk/books?id=acQZuWho7m8C.Google ScholarGoogle Scholar
  22. Peter Fulla and Stanislav Živný. 2016. A galois connection for valued constraint languages of infinite size. ACM Trans. Comput. Theory 8, 3 (2016). Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. Peter Fulla and Stanislav Živný. 2017. The complexity of ooleanBoolean surjective general-valued CSPs. In Proceedings of the 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS’17).Google ScholarGoogle Scholar
  24. Petr A. Golovach, Bernard Lidický, Barnaby Martin, and Daniël Paulusma. 2012. Finding vertex-surjective graph homomorphisms. Acta Informatica 49, 6 (2012), 381--394. Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. Petr A. Golovach, Daniël Paulusma, and Jian Song. 2012. Computing vertex-surjective homomorphisms to partially reflexive trees. Theor. Comput. Sci. 457 (2012), 86--100. Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. Anna Huber, Andrei Krokhin, and Robert Powell. 2014. Skew bisubmodularity and valued CSPs. SIAM J. Comput. 43, 3 (2014), 1064--1084.Google ScholarGoogle ScholarCross RefCross Ref
  27. David S. Johnson, Mihalis Yannakakis, and Christos H. Papadimitriou. 1988. On generating all maximal independent sets. Inform. Process. Lett. 27, 3 (1988), 119--123. Google ScholarGoogle ScholarDigital LibraryDigital Library
  28. Peter Jonsson, Mikael Klasson, and Andrei A. Krokhin. 2006. The approximability of three-valued MAX CSP. SIAM J. Comput. 35, 6 (2006), 1329--1349. Google ScholarGoogle ScholarDigital LibraryDigital Library
  29. David R. Karger. 1993. Global min-cuts in RNC, and other ramifications of a simple min-cut algorithm. In Proceedings of the 4th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’93). 21--30. Google ScholarGoogle ScholarDigital LibraryDigital Library
  30. Subhash Khot. 2010. On the unique games conjecture (invited survey). In Proceedings of the 25th Annual IEEE Conference on Computational Complexity (CCC’10). IEEE Computer Society, 99--121. Google ScholarGoogle ScholarDigital LibraryDigital Library
  31. Vladimir Kolmogorov, Andrei A. Krokhin, and Michal Rolínek. 2017. The complexity of general-valued CSPs. SIAM J. Comput. 46, 3 (2017), 1087--1110.Google ScholarGoogle ScholarDigital LibraryDigital Library
  32. Marcin Kozik and Joanna Ochremiak. 2015. Algebraic properties of valued constraint satisfaction problem. In Proceedings of the 42nd International Colloquium on Automata, Languages and Programming (ICALP’15) (Lecture Notes in Computer Science), Vol. 9134. Springer, 846--858.Google ScholarGoogle ScholarCross RefCross Ref
  33. Andrei A. Krokhin and Stanislav Živný (Eds.). 2017. The Constraint Satisfaction Problem: Complexity and Approximability. Dagstuhl Follow-Ups, Vol. 7. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.Google ScholarGoogle Scholar
  34. Konstantin Makarychev and Yury Makarychev. 2017. Approximation algorithms for CSPs. See Reference Krokhin and Živný {33}, 287--325.Google ScholarGoogle Scholar
  35. Barnaby Martin and Daniël Paulusma. 2015. The computational complexity of disconnected cut and 2K-partition. J. Combinator. Theory, Ser. B 111 (2015), 17--37. Google ScholarGoogle ScholarDigital LibraryDigital Library
  36. Hiroshi Nagamochi and Toshihide Ibaraki. 2008. Algorithmic Aspects of Graph Connectivity. Vol. 123. Cambridge University Press, New York. Google ScholarGoogle ScholarDigital LibraryDigital Library
  37. Prasad Raghavendra. 2008. Optimal algorithms and inapproximability results for every CSP? In Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC’08). ACM, 245--254. Google ScholarGoogle ScholarDigital LibraryDigital Library
  38. Prasad Raghavendra. 2009. Approximating NP-hard problems: Efficient algorithms and their limits. Ph.D. Thesis. University of Washington. Google ScholarGoogle ScholarDigital LibraryDigital Library
  39. Thomas J. Schaefer. 1978. The complexity of satisfiability problems. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing (STOC’78). ACM, 216--226. Google ScholarGoogle ScholarDigital LibraryDigital Library
  40. Alexander Schrijver. 2003. Combinatorial Optimization: Polyhedra and Efficiency. Algorithms and Combinatorics, Vol. 24. Springer.Google ScholarGoogle Scholar
  41. Mechthild Stoer and Frank Wagner. 1997. A simple min-cut algorithm. J. ACM 44, 4 (1997), 585--591. Google ScholarGoogle ScholarDigital LibraryDigital Library
  42. Johan Thapper and Stanislav Živný. 2016. The complexity of finite-valued CSPs. J. ACM 63, 4 (2016). Google ScholarGoogle ScholarDigital LibraryDigital Library
  43. Hannes Uppman. 2012. Max-sur-CSP on two elements. In Proceedings of the 18th International Conference on Principles and Practice of Constraint Programming (CP’12) (Lecture Notes in Computer Science), Vol. 7514. Springer, 38--54.Google ScholarGoogle ScholarCross RefCross Ref
  44. Leslie G. Valiant. 1979. The complexity of enumeration and reliability problems. SIAM J. Comput. 8, 3 (1979), 410--421.Google ScholarGoogle ScholarDigital LibraryDigital Library
  45. Alexander Vardy. 1997. Algorithmic complexity in coding theory and the minimum distance problem. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing (STOC’97). ACM, New York, NY, 92--109. Google ScholarGoogle ScholarDigital LibraryDigital Library
  46. Vijay V. Vazirani and Mihalis Yannakakis. 1992. Suboptimal cuts: Their enumeration, weight and number (extended abstract). In Proceedings of the 19th International Colloquium on Automata, Languages and Programming (ICALP’92). Springer-Verlag, 366--377. Retrieved from http://dl.acm.org/citation.cfm?id=646246.684856. Google ScholarGoogle ScholarDigital LibraryDigital Library
  47. Narayan Vikas. 2013. Algorithms for partition of some class of graphs under compaction and vertex-compaction. Algorithmica 67, 2 (2013), 180--206.Google ScholarGoogle ScholarCross RefCross Ref
  48. Dmitriy Zhuk. 2017. The proof of CSP dichotomy conjecture. In Proceedings of the 58th Annual IEEE Symposium on Foundations of Computer Science (FOCS’17). IEEE, 331--342.Google ScholarGoogle ScholarCross RefCross Ref

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