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The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains

Authors Caterina Viola , Stanislav Živný

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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Constraint and logic programming
Keywords
  • promise constraint satisfaction
  • valued constraint satisfaction
  • convex relaxations
  • polymorphisms

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Abstract

Convex relaxations have been instrumental in solvability of constraint satisfaction problems (CSPs), as well as in the three different generalisations of CSPs: valued CSPs, infinite-domain CSPs, and most recently promise CSPs. In this work, we extend an existing tractability result to the three generalisations of CSPs combined: We give a sufficient condition for the combined basic linear programming and affine integer programming relaxation for exact solvability of promise valued CSPs over infinite-domains. This extends a result of Brakensiek and Guruswami [SODA'20] for promise (non-valued) CSPs (on finite domains).

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Caterina Viola and Stanislav Živný. The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 85:1-85:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.MFCS.2020.85

Author Details

Caterina Viola
  • Department of Computer Science, University of Oxford, UK
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.
  • Živný, Stanislav: Royal Society University Research Fellowship.

Acknowledgements

We thank the reviewers for their useful comments and suggestions.

References

  1. Per Austrin and Johan Håstad. On the usefulness of predicates. ACM Transactions on Computation Theory, 5(1):1:1-1:24, 2013. URL: https://doi.org/10.1145/2462896.2462897.
  2. Nikhil Bansal, Avrim Blum, and Shuchi Chawla. Correlation clustering. Machine Learning, 56(1-3):89-113, 2004. URL: https://doi.org/10.1023/B:MACH.0000033116.57574.95.
  3. Libor Barto, Jakub Bulín, Andrei Krokhin, and Jakub Opršal. Algebraic approach to promise constraint satisfaction. CoRR, 2019. URL: http://arxiv.org/abs/1811.00970v3.
  4. Libor Barto and Marcin Kozik. Robustly solvable constraint satisfaction problems. SIAM Journal on Computing, 45(4):1646-1669, 2016. URL: https://doi.org/10.1137/130915479.
  5. Libor Barto and Michael Pinsker. Topology Is Irrelevant (In a Dichotomy Conjecture for Infinite Domain Constraint Satisfaction Problems). SIAM Journal on Computing, 49(2):365–393, 2020. URL: https://doi.org/10.1137/18M1216213.
  6. Manuel Bodirsky and Martin Grohe. Non-dichotomies in constraint satisfaction complexity. In Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP), pages 184-196. Springer Verlag, 2008. URL: https://doi.org/10.1007/978-3-540-70583-3_16.
  7. Manuel Bodirsky, Peter Jonsson, and Trung Van Pham. The complexity of phylogeny constraint satisfaction problems. ACM Transactions on Computational Logic, 18(3), 2017. URL: https://doi.org/10.1145/3105907.
  8. Manuel Bodirsky, Peter Jonsson, and Timo Von Oertzen. Essential convexity and complexity of semi-algebraic constraints. Logical Methods in Computer Science, 8(4), 2012. URL: https://doi.org/10.2168/LMCS-8(4:5)2012.
  9. Manuel Bodirsky and Jan Kára. The complexity of temporal constraint satisfaction problems. Journal of the ACM, 57(2), 2010. Article 9. URL: https://doi.org/10.1145/1667053.1667058.
  10. Manuel Bodirsky, Marcello Mamino, and Caterina Viola. Piecewise linear valued CSPs solvable by linear programming relaxation. CoRR, 2019. URL: http://arxiv.org/abs/1912.09298.
  11. Manuel Bodirsky, Barnaby Martin, and Antoine Mottet. Discrete temporal constraint satisfaction problems. Journal of the ACM, 65(2):9:1-9:41, 2018. URL: https://doi.org/10.1145/3154832.
  12. Manuel Bodirsky, Barnaby Martin, Michael Pinsker, and András Pongrácz. Constraint satisfaction problems for reducts of homogeneous graphs. SIAM Journal on Computing, 48(4):1224-1264, 2019. URL: https://doi.org/10.1137/16M1082974.
  13. Manuel Bodirsky and Michael Pinsker. Schaefer’s theorem for graphs. Journal of the ACM, 62(3):19:1-19:52, 2015. URL: https://doi.org/10.1145/2764899.
  14. Stephen P. Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004. URL: https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf.
  15. Joshua Brakensiek and Venkatesan Guruswami. Promise constraint satisfaction: Structure theory and a symmetric boolean dichotomy. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'18), pages 1782-1801. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.117.
  16. Joshua Brakensiek and Venkatesan Guruswami. An Algorithmic Blend of LPs and Ring Equations for Promise CSPs. In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'19), pages 436-455. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.28.
  17. Joshua Brakensiek and Venkatesan Guruswami. Symmetric polymorphisms and efficient decidability of promise CSPs. In Proceedings of the 31st ACM-SIAM Symposium on Discrete Algorithms (SODA'20), pages 297-304, 2020. URL: https://doi.org/10.1137/1.9781611975994.18.
  18. Joshua Brakensiek, Venkatesan Guruswami, Marcin Wrochna, and Stanislav Živný. The power of the combined basic LP and affine relaxation for promise CSPs. eccc, 2020. URL: http://eccc.hpi-web.de/report/2020/004/.
  19. Andrei A. Bulatov. A dichotomy theorem for nonuniform CSPs. In Proceedings of the 58th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 319-330, 2017. URL: https://doi.org/10.1109/FOCS.2017.37.
  20. Andrei A. Bulatov, Andrei A. Krokhin, and Peter G. Jeavons. Classifying the complexity of constraints using finite algebras. SIAM Journal on Computing, 34(3):720-742, 2005. URL: https://doi.org/10.1137/S0097539700376676.
  21. Jakub Bulín, Andrei Krokhin, and Jakub Opršal. Algebraic approach to promise constraint satisfaction. In Proceedings of the 51st ACM Annual Symposium on Theory of Computing (STOC'19), page 602–613. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316300.
  22. Siu On Chan. Approximation resistance from pairwise-independent subgroups. Journal of the ACM, 63(3), 2016. Article No. 27. URL: https://doi.org/10.1145/2873054.
  23. Siu On Chan, James R. Lee, Prasad Raghavendra, and David Steurer. Approximate constraint satisfaction requires large LP relaxations. Journal of the ACM, 63(4):34:1-34:22, 2016. URL: https://doi.org/10.1145/2811255.
  24. Tsu-Wu J. Chou and George E. Collins. Algorithms for the solution of systems of linear Diophantine equations. SIAM Journal on Computing, 11:687-708, 1982. URL: https://doi.org/10.1137/0211057.
  25. Víctor Dalmau, Marcin Kozik, Andrei A. Krokhin, Konstantin Makarychev, Yury Makarychev, and Jakub Opršal. Robust algorithms with polynomial loss for near-unanimity csps. SIAM Journal on Computing, 48(6):1763-1795, 2019. URL: https://doi.org/10.1137/18M1163932.
  26. Víctor Dalmau and Andrei A. Krokhin. Robust satisfiability for csps: Hardness and algorithmic results. ACM Transactions on Computation Theory, 5(4):15:1-15:25, 2013. URL: https://doi.org/10.1145/2540090.
  27. 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, 1999. URL: https://doi.org/10.1137/S0097539794266766.
  28. Miron Ficak, Marcin Kozik, Miroslav Olšák, and Szymon Stankiewicz. Dichotomy for symmetric Boolean PCSPs. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP'19), volume 132 of LIPIcs, pages 57:1-57:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.57.
  29. Mrinalkanti Ghosh and Madhur Tulsiani. From weak to strong linear programming gaps for all constraint satisfaction problems. Theory of Computing, 14(1):1-33, 2018. URL: https://doi.org/10.4086/toc.2018.v014a010.
  30. Michel X. Goemans and David P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM, 42(6):1115-1145, 1995. URL: https://doi.org/10.1145/227683.227684.
  31. Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric Algorithms andCombinatorial Optimization, volume 2. Springer-Verlag Berlin Heidelberg, 1993. URL: https://doi.org/10.1007/978-3-642-78240-4.
  32. Peter Jonsson and Johan Thapper. Constraint satisfaction and semilinear expansions of addition over the rationals and the reals. Journal of Computer and System Sciences, 82(5):912-928, 2016. URL: https://doi.org/10.1016/j.jcss.2016.03.002.
  33. Ravindran Kannan and Achim Bachem. Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix. SIAM Journal on Computing, 8(4):499-507, 1979. URL: https://doi.org/10.1137/0208040.
  34. Alexander Kazda. Minion homomorphisms give reductions between promise valued CSPs, 2020. In preparation. Google Scholar
  35. Vladimir Kolmogorov, Andrei A. Krokhin, and Michal Rolínek. The complexity of general-valued csps. SIAM Journal on Computing, 46(3):1087-1110, 2017. URL: https://doi.org/10.1137/16M1091836.
  36. 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: https://doi.org/10.1137/130945648.
  37. Pravesh K. Kothari, Raghu Meka, and Prasad Raghavendra. Approximating rectangles by juntas and weakly-exponential lower bounds for LP relaxations of CSPs. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC'17), pages 590-603, 2017. URL: https://doi.org/10.1145/3055399.3055438.
  38. Dexter Kozen. Results on the propositional μ-calculus. Theoretical Computer Science, 27:333-354, 1983. URL: https://doi.org/10.1016/0304-3975(82)90125-6.
  39. 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, 2015. URL: https://doi.org/10.1007/978-3-662-47672-7_69.
  40. Gábor Kun, Ryan O'Donnell, Suguru Tamaki, Yuichi Yoshida, and Yuan Zhou. Linear programming, width-1 CSPs, and robust satisfaction. In Proceedings of the 3rd Innovations in Theoretical Computer Science (ITCS'12), pages 484-495. ACM, 2012. URL: https://doi.org/10.1145/2090236.2090274.
  41. James R. Lee, Prasad Raghavendra, and David Steurer. Lower bounds on the size of semidefinite programming relaxations. In Proceedings of the 47th Symposium on Theory of Computing (STOC'15), pages 567-576, 2015. URL: https://doi.org/10.1145/2746539.2746599.
  42. Prasad Raghavendra. Optimal algorithms and inapproximability results for every CSP? In Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC'08), pages 245-254, 2008. URL: https://doi.org/10.1145/1374376.1374414.
  43. 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: https://doi.org/10.1145/800133.804350.
  44. Johan Thapper and Stanislav Živný. The power of linear programming for valued CSPs. In Proceedings of the 53rd IEEE Annual Symposium on Foundations of Computer Science (FOCS'12), pages 669-678. IEEE Computer Society, 2012. URL: https://doi.org/10.1109/FOCS.2012.25.
  45. Johan Thapper and Stanislav Živný. The complexity of finite-valued CSPs. Journal of the ACM, 63(4), 2016. Article No. 37. URL: https://doi.org/10.1145/2974019.
  46. Johan Thapper and Stanislav Živný. The Power of Sherali-Adams Relaxations for General-Valued CSPs. SIAM Journal on Computing, 46(4):1241-1279, 2017. URL: https://doi.org/10.1137/16M1079245.
  47. Johan Thapper and Stanislav Živný. The limits of SDP relaxations for general-valued csps. ACM Transactions on Computation Theory, 10(3):12:1-12:22, 2018. URL: https://doi.org/10.1145/3201777.
  48. Madhur Tulsiani. CSP gaps and reductions in the Lasserre hierarchy. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC'09), pages 303-312. ACM, 2009. URL: https://doi.org/10.1145/1536414.1536457.
  49. Caterina Viola. Valued Constraint Satisfaction Problems over Infinite Domains. PhD thesis, Technische Universität Dresden, Dresden, Germany, 2020. Google Scholar
  50. Caterina Viola and Stanislav Živný. The combined basic LP and affine IP relaxation for promise VCSPs on infinite domains. CoRR, 2020. URL: http://arxiv.org/abs/2007.01779.
  51. Dmitriy Zhuk. A proof of CSP dichotomy conjecture. In Proceedings of the 58th IEEE Annual Symposium on Foundations of Computer Science (FOCS'17), pages 331-342, 2017. URL: https://doi.org/10.1109/FOCS.2017.38.
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