research-article

The Complexity of Boolean Surjective General-Valued CSPs

Authors:

Stanislav ŽivnýAuthors Info & Claims

ACM Transactions on Computation Theory (TOCT), Volume 11, Issue 1

Article No.: 4, Pages 1 - 31

https://doi.org/10.1145/3282429

Published: 21 November 2018 Publication History

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.

[2]

Libor Barto and Marcin Kozik. 2014. Constraint satisfaction problems solvable by local consistency methods. J. ACM 61, 1 (2014). Retrieved from

Digital Library

[3]

Libor Barto, Andrei Krokhin, and Ross Willard. 2017. Polymorphisms, and how to use them. See Reference Krokhin and Živný {33}, 1--44.

[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.

Digital Library

[5]

Andrei Bulatov. 2006. A dichotomy theorem for constraint satisfaction problems on a 3-element set. J. ACM 53, 1 (2006), 66--120.

Digital 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.

[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.

Digital Library

[8]

Andrei A. Bulatov, Víctor Dalmau, Martin Grohe, and Dániel Marx. 2012. Enumerating homomorphisms. J. Comput. Syst. Sci. 78, 2 (2012), 638--650.

Digital Library

[9]

Hubie Chen. 2014. An algebraic hardness criterion for surjective constraint satisfaction. Algebra Universalis 72, 4 (2014), 393--401.

[10]

David A. Cohen. 2004. Tractable decision for a constraint language implies tractable search. Constraints 9 (2004), 219--229.

Digital 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.

Digital 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.

Digital Library

[13]

Nadia Creignou. 1995. A dichotomy theorem for maximum generalized satisfiability problems. J. Comput. Syst. Sci. 51, 3 (1995), 511--522.

Digital 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.

[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.

Digital 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.

[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.

[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.

Digital Library

[19]

Jiří Fiala and Jan Kratochvíl. 2008. Locally constrained graph homomorphisms - Structure, complexity, and applications. Comput. Sci. Rev. 2, 2 (2008), 97--111.

Digital 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.

Digital Library

[21]

András Frank. 2011. Connections in Combinatorial Optimization. OUP Oxford. Retrieved from https://books.google.co.uk/books?id=acQZuWho7m8C.

[22]

Peter Fulla and Stanislav Živný. 2016. A galois connection for valued constraint languages of infinite size. ACM Trans. Comput. Theory 8, 3 (2016).

Digital 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).

[24]

Petr A. Golovach, Bernard Lidický, Barnaby Martin, and Daniël Paulusma. 2012. Finding vertex-surjective graph homomorphisms. Acta Informatica 49, 6 (2012), 381--394.

Digital 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.

Digital Library

[26]

Anna Huber, Andrei Krokhin, and Robert Powell. 2014. Skew bisubmodularity and valued CSPs. SIAM J. Comput. 43, 3 (2014), 1064--1084.

[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.

Digital 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.

Digital 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.

Digital 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.

Digital 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.

Digital 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.

[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.

[34]

Konstantin Makarychev and Yury Makarychev. 2017. Approximation algorithms for CSPs. See Reference Krokhin and Živný {33}, 287--325.

[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.

Digital Library

[36]

Hiroshi Nagamochi and Toshihide Ibaraki. 2008. Algorithmic Aspects of Graph Connectivity. Vol. 123. Cambridge University Press, New York.

Digital 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.

Digital Library

[38]

Prasad Raghavendra. 2009. Approximating NP-hard problems: Efficient algorithms and their limits. Ph.D. Thesis. University of Washington.

Digital 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.

Digital Library

[40]

Alexander Schrijver. 2003. Combinatorial Optimization: Polyhedra and Efficiency. Algorithms and Combinatorics, Vol. 24. Springer.

[41]

Mechthild Stoer and Frank Wagner. 1997. A simple min-cut algorithm. J. ACM 44, 4 (1997), 585--591.

Digital Library

[42]

Johan Thapper and Stanislav Živný. 2016. The complexity of finite-valued CSPs. J. ACM 63, 4 (2016).

Digital 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.

[44]

Leslie G. Valiant. 1979. The complexity of enumeration and reliability problems. SIAM J. Comput. 8, 3 (1979), 410--421.

Digital 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.

Digital 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.

Digital Library

[47]

Narayan Vikas. 2013. Algorithms for partition of some class of graphs under compaction and vertex-compaction. Algorithmica 67, 2 (2013), 180--206.

[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.

Cited By

Chen H(2024)Algebraic Global Gadgetry for Surjective Constraint Satisfactioncomputational complexity10.1007/s00037-024-00253-433:1Online publication date: 29-May-2024
https://doi.org/10.1007/s00037-024-00253-4
Mezei BWrochna MŽivný s(2023)PTAS for Sparse General-valued CSPsACM Transactions on Algorithms10.1145/356995619:2(1-31)Online publication date: 9-Mar-2023
https://dl.acm.org/doi/10.1145/3569956
Mezei BWrochna MŽivný SGorla D(2021)PTAS for sparse general-valued CSPsProceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science10.1109/LICS52264.2021.9470599(1-11)Online publication date: 29-Jun-2021
https://dl.acm.org/doi/10.1109/LICS52264.2021.9470599
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Index Terms

The Complexity of Boolean Surjective General-Valued CSPs
1. Theory of computation
  1. Computational complexity and cryptography
    1. Problems, reductions and completeness

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cover image ACM Transactions on Computation Theory

ACM Transactions on Computation Theory Volume 11, Issue 1

March 2019

145 pages

ISSN:1942-3454

EISSN:1942-3462

DOI:10.1145/3287761

Editor:
Venkatesan Guruswami
Carnegie Mellon University, USA

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Copyright © 2018 ACM.

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Publication History

Published: 21 November 2018

Accepted: 01 September 2018

Revised: 01 June 2018

Received: 01 November 2017

Published in TOCT Volume 11, Issue 1

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Cited By

Chen H(2024)Algebraic Global Gadgetry for Surjective Constraint Satisfactioncomputational complexity10.1007/s00037-024-00253-433:1Online publication date: 29-May-2024
https://doi.org/10.1007/s00037-024-00253-4
Mezei BWrochna MŽivný s(2023)PTAS for Sparse General-valued CSPsACM Transactions on Algorithms10.1145/356995619:2(1-31)Online publication date: 9-Mar-2023
https://dl.acm.org/doi/10.1145/3569956
Mezei BWrochna MŽivný SGorla D(2021)PTAS for sparse general-valued CSPsProceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science10.1109/LICS52264.2021.9470599(1-11)Online publication date: 29-Jun-2021
https://dl.acm.org/doi/10.1109/LICS52264.2021.9470599
Kawarabayashi KXu C(2020)Minimum Violation Vertex Maps and Their Applications to Cut ProblemsSIAM Journal on Discrete Mathematics10.1137/19M129089934:4(2183-2207)Online publication date: 22-Oct-2020
https://doi.org/10.1137/19M1290899
Matl GŽivný S(2020)Using a Min-Cut Generalisation to Go Beyond Boolean Surjective VCSPsAlgorithmica10.1007/s00453-020-00735-1Online publication date: 24-Jun-2020
https://doi.org/10.1007/s00453-020-00735-1

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