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Beyond Boolean Surjective VCSPs

Authors Gregor Matl, Stanislav Živný

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LIPIcs.STACS.2019.52.pdf
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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • constraint satisfaction problems
  • valued constraint satisfaction
  • surjective constraint satisfaction
  • graph cuts

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Abstract

Fulla, Uppman, and Živný [ACM ToCT'18] established a dichotomy theorem for Boolean surjective general-valued constraint satisfaction problems (VCSPs), i.e., VCSPs on two-element domains in which both labels have to be used in a solution. This result, in addition to identifying the complexity frontier, features the discovery of a new non-trivial tractable case (called EDS) that does not appear in the non-surjective setting.
In this work, we go beyond Boolean domains. As our main result, we introduce a generalisation of EDS to arbitrary finite domains called SEDS (similar to EDS) and establish a conditional complexity classification of SEDS VCSPs based on a reduction to smaller domains. This gives a complete classification of SEDS VCSPs on three-element domains. The basis of our tractability result is a natural generalisation of the Min-Cut problem, in which only solutions of certain size (given by a lower and upper bound) are permitted. We show that all near-optimal solutions to this problem can be enumerated in polynomial time, which might be of independent interest.

Cite As Get BibTex

Gregor Matl and Stanislav Živný. Beyond Boolean Surjective VCSPs. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.STACS.2019.52

Author Details

Gregor Matl
  • Department of Informatics, Technical University of Munich, Germany
Stanislav Živný
  • Department of Computer Science, University of Oxford, UK

Funding

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 714532). The paper reflects only the authors' views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein.
  • Matl, Gregor: Work done while at the University of Oxford.
  • Živný, Stanislav: Supported by a Royal Society University Research Fellowship.

References

  1. Manuel Bodirsky, Jan Kára, and Barnaby Martin. The complexity of surjective homomorphism problems - a survey. Discrete Applied Mathematics, 160(12):1680-1690, 2012. URL: http://dx.doi.org/10.1016/j.dam.2012.03.029.
  2. Andrei Bulatov. A dichotomy theorem for constraint satisfaction problems on a 3-element set. Journal of the ACM, 53(1):66-120, 2006. URL: http://dx.doi.org/10.1145/1120582.1120584.
  3. Andrei Bulatov, Peter Jeavons, and Andrei Krokhin. Classifying the Complexity of Constraints using Finite Algebras. SIAM Journal on Computing, 34(3):720-742, 2005. URL: http://dx.doi.org/10.1137/S0097539700376676.
  4. Andrei A. Bulatov. A Dichotomy Theorem for Nonuniform CSPs. In Proceedings of the 58th Annual IEEE Symposium on Foundations of Computer Science (FOCS'17), pages 319-330, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.37.
  5. Andrei A. Bulatov and Dániel Marx. The complexity of global cardinality constraints. Logical Methods in Computer Science, Volume 6, Issue 4, 2010. URL: http://dx.doi.org/10.2168/LMCS-6(4:4)2010.
  6. David A. Cohen, Martin C. Cooper, Peter G. Jeavons, and Andrei A. Krokhin. The Complexity of Soft Constraint Satisfaction. Artificial Intelligence, 170(11):983-1016, 2006. URL: http://dx.doi.org/10.1016/j.artint.2006.04.002.
  7. Nadia Creignou and Jean-Jacques Hébrard. On Generating All Solutions of Generalized Satisfiability Problems. Informatique Théorique et Applications, 31:499-511, November 1997. URL: http://www.numdam.org/article/ITA_1997__31_6_499_0.pdf.
  8. Elias Dahlhaus, David S. Johnson, Christos H. Papadimitriou, Paul D. Seymour, and Mihalis Yannakakis. The complexity of multiterminal cuts. SIAM Journal on Computing, 23(4):864-894, 1994. URL: http://dx.doi.org/10.1137/S0097539792225297.
  9. Tomás Feder and Moshe Y Vardi. The computational structure of monotone monadic SNP and constraint satisfaction: A study through Datalog and group theory. SIAM Journal on Computing, 28(1):57-104, 1998. URL: http://dx.doi.org/10.1137/S0097539794266766.
  10. Peter Fulla, Hannes Uppman, and Stanislav Živný. The complexity of Boolean surjective general-valued CSPs. ACM Transactions on Computation Theory, 11(1), 2018. Article No. 4. URL: http://dx.doi.org/10.1145/3282429.
  11. Olivier Goldschmidt and Dorit S. Hochbaum. A Polynomial Algorithm for the k-cut Problem for Fixed k. Mathematics of Operations Research, 19(1):24-37, 1994. URL: http://dx.doi.org/10.1287/moor.19.1.24.
  12. Pavol Hell and Jaroslav Nešetřil. On the complexity of H-coloring. Journal of Combinatorial Theory, Series B, 48(1):92-110, 1990. URL: http://dx.doi.org/10.1016/0095-8956(90)90132-J.
  13. Pavol Hell and Jaroslav Nešetřil. Colouring, constraint satisfaction, and complexity. Computer Science Review, 2(3):143-163, 2008. URL: http://dx.doi.org/10.1016/j.cosrev.2008.10.003.
  14. Anna Huber and Vladimir Kolmogorov. Towards Minimizing k-Submodular Functions. In Proceedings of the 2nd International Symposium on Combinatorial Optimization (ISCO'12), volume 7422 of Lecture Notes in Computer Science, pages 451-462. Springer, 2012. URL: http://dx.doi.org/10.1007/978-3-642-32147-4_40.
  15. Peter Jeavons, David Cohen, and Marc Gyssens. Closure Properties of Constraints. Journal of the ACM, 44(4):527-548, July 1997. URL: http://dx.doi.org/10.1145/263867.263489.
  16. Vladimir Kolmogorov, Andrei Krokhin, and Michal Rolínek. The complexity of general-valued CSPs. SIAM Journal on Computing, 46(3):1087-1110, 2017. URL: http://dx.doi.org/10.1137/16M1091836.
  17. Vladimir Kolmogorov, Johan Thapper, and Stanislav Živný. The power of linear programming for general-valued CSPs. SIAM Journal on Computing, 44(1):1-36, 2015. URL: http://dx.doi.org/10.1137/130945648.
  18. Marcin Kozik and Joanna Ochremiak. Algebraic Properties of Valued Constraint Satisfaction Problem. In Proceedings of the 42nd International Colloquium on Automata, Languages and Programming (ICALP'15), volume 9134 of Lecture Notes in Computer Science, pages 846-858. Springer Berlin Heidelberg, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47672-7_69.
  19. Gregor Matl and Stanislav Živný. Beyond Boolean Surjective VCSPs. Technical report, arXiv, January 2019. URL: http://arxiv.org/abs/1901.07107.
  20. Thomas J. Schaefer. The Complexity of Satisfiability Problems. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing (STOC'78), pages 216-226. ACM, 1978. URL: http://dx.doi.org/10.1145/800133.804350.
  21. Thomas Schiex, Hélène Fargier, and Gérard Verfaillie. Valued Constraint Satisfaction Problems: Hard and Easy Problems. In Proceedings of the 14th International Joint Conference on Artificial Intelligence (IJCAI'95), pages 631-637, 1995. URL: http://ijcai.org/Proceedings/95-1/Papers/083.pdf.
  22. Alexander Schrijver. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer Science &Business Media, 2003. Google Scholar
  23. Johan Thapper and Stanislav Živný. The Complexity of Finite-Valued CSPs. Journal of the ACM, 63(4):37:1-37:33, September 2016. URL: http://dx.doi.org/10.1145/2974019.
  24. D. Zhuk. A Proof of CSP Dichotomy Conjecture. In Proceedings of the 58th Annual IEEE Symposium on Foundations of Computer Science (FOCS'17), pages 331-342, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.38.
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