CSP Beyond Tractable Constraint Languages

Authors Jan Dreier , Sebastian Ordyniak , Stefan Szeider



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Jan Dreier
  • Algorithms and Complexity Group, TU Wien, Austria
Sebastian Ordyniak
  • Algorithms and Complexity Group, University of Leeds, UK
Stefan Szeider
  • Algorithms and Complexity Group, TU Wien, Austria

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Jan Dreier, Sebastian Ordyniak, and Stefan Szeider. CSP Beyond Tractable Constraint Languages. In 28th International Conference on Principles and Practice of Constraint Programming (CP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 235, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.CP.2022.20

Abstract

The constraint satisfaction problem (CSP) is among the most studied computational problems. While NP-hard, many tractable subproblems have been identified (Bulatov 2017, Zuk 2017). Backdoors, introduced by Williams, Gomes, and Selman (2003), gradually extend such a tractable class to all CSP instances of bounded distance to the class. Backdoor size provides a natural but rather crude distance measure between a CSP instance and a tractable class. Backdoor depth, introduced by Mählmann, Siebertz, and Vigny (2021) for SAT, is a more refined distance measure, which admits the parallel utilization of different backdoor variables. Bounded backdoor size implies bounded backdoor depth, but there are instances of constant backdoor depth and arbitrarily large backdoor size. Dreier, Ordyniak, and Szeider (2022) provided fixed-parameter algorithms for finding backdoors of small depth into the classes of Horn and Krom formulas. In this paper, we consider backdoor depth for CSP. We consider backdoors w.r.t. tractable subproblems C_Γ of the CSP defined by a constraint language Γ, i.e., where all the constraints use relations from the language Γ. Building upon Dreier et al.’s game-theoretic approach and their notion of separator obstructions, we show that for any finite, tractable, semi-conservative constraint language Γ, the CSP is fixed-parameter tractable parameterized by the backdoor depth into C_Γ plus the domain size. With backdoors of low depth, we reach classes of instances that require backdoors of arbitrary large size. Hence, our results strictly generalize several known results for CSP that are based on backdoor size.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • CSP
  • backdoor depth
  • constraint language
  • tractable class
  • recursive backdoor

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References

  1. Andrei A. Bulatov. Complexity of conservative constraint satisfaction problems. ACM Trans. Comput. Log., 12(4):24:1-24:66, 2011. URL: https://doi.org/10.1145/1970398.1970400.
  2. Andrei A. Bulatov. The complexity of the counting constraint satisfaction problem. J. of the ACM, 60(5):34:1-34:41, 2013. URL: https://doi.org/10.1145/2528400.
  3. Andrei A. Bulatov. A dichotomy theorem for nonuniform CSPs. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 319-330. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.37.
  4. Jannis Bulian and Anuj Dawar. Graph isomorphism parameterized by elimination distance to bounded degree. Algorithmica, 75(2):363-382, 2016. Google Scholar
  5. Clément Carbonnel and Martin C. Cooper. Tractability in constraint satisfaction problems: a survey. Constraints, 21(2):115-144, 2016. Google Scholar
  6. David Cohen and Peter Jeavons. The complexity of constraint languages. In Francesca Rossi, Peter van Beek, and Toby Walsh, editors, Handbook of Constraint Programming, volume I, chapter 8, pages 245-280. Elsevier, 2006. Google Scholar
  7. David Cohen, Peter Jeavons, and Marc Gyssens. A unified theory of structural tractability for constraint satisfaction and spread cut decomposition. In International Joint Conferences on Artificial Intelligence (IJCAI-05), pages 72-77, 2005. Google Scholar
  8. David A. Cohen, Martin C. Cooper, Páidí Creed, Dániel Marx, and András Z. Salamon. The tractability of CSP classes defined by forbidden patterns. J. Artif. Intell. Res., 45:47-78, 2012. Google Scholar
  9. David A. Cohen, Martin C. Cooper, Peter G. Jeavons, and Stanislav Zivný. Tractable classes of binary CSPs defined by excluded topological minors. In Qiang Yang and Michael J. Wooldridge, editors, Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence, IJCAI 2015, Buenos Aires, Argentina, July 25-31, 2015, pages 1945-1951. AAAI Press, 2015. Google Scholar
  10. Martin C. Cooper, David A. Cohen, and Peter Jeavons. Characterising tractable constraints. Artificial Intelligence, 65(2):347-361, 1994. URL: https://doi.org/10.1016/0004-3702(94)90021-3.
  11. Martin C. Cooper, Philippe Jégou, and Cyril Terrioux. A microstructure-based family of tractable classes for CSPs. In Gilles Pesant, editor, Principles and Practice of Constraint Programming - 21st International Conference, CP 2015, Cork, Ireland, August 31 - September 4, 2015, Proceedings, volume 9255 of Lecture Notes in Computer Science, pages 74-88. Springer Verlag, 2015. Google Scholar
  12. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  13. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer Verlag, 2013. Google Scholar
  14. Jan Dreier, Sebastian Ordyniak, and Stefan Szeider. SAT backdoors: Depth beats size. In 30th Annual European Symposium on Algorithms (ESA 2022), volume 244 of LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. to appear. URL: http://arxiv.org/abs/2202.08326.
  15. Jörg Flum and Martin Grohe. Parameterized Complexity Theory, volume XIV of Texts in Theoretical Computer Science. An EATCS Series. Springer Verlag, Berlin, 2006. Google Scholar
  16. Fedor V Fomin, Petr A Golovach, and Dimitrios M Thilikos. Parameterized complexity of elimination distance to first-order logic properties. arXiv preprint arXiv:2104.02998, 2021. Google Scholar
  17. Robert Ganian, M. S. Ramanujan, and Stefan Szeider. Discovering archipelagos of tractability for constraint satisfaction and counting. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1670-1681. SIAM, 2016. Google Scholar
  18. Robert Ganian, M. S. Ramanujan, and Stefan Szeider. Combining treewidth and backdoors for CSP. In Heribert Vollmer and Brigitte Vallée, editors, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017), volume 66 of Leibniz International Proceedings in Informatics (LIPIcs), pages 36:1-36:17, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.STACS.2017.36.
  19. Robert Ganian, M. S. Ramanujan, and Stefan Szeider. Discovering archipelagos of tractability for constraint satisfaction and counting. ACM Transactions on Algorithms, 13(2):29:1-29:32, 2017. Full version of a SODA'16 paper. URL: https://doi.org/10.1145/3014587.
  20. Serge Gaspers, Neeldhara Misra, Sebastian Ordyniak, Stefan Szeider, and Stanislav Zivny. Backdoors into heterogeneous classes of SAT and CSP. J. of Computer and System Sciences, 85:38-56, 2017. URL: https://doi.org/10.1016/j.jcss.2016.10.007.
  21. Georg Gottlob, Nicola Leone, and Francesco Scarcello. A comparison of structural CSP decomposition methods. Artificial Intelligence, 124(2):243-282, 2000. Google Scholar
  22. Georg Gottlob, Nicola Leone, and Francesco Scarcello. Hypertree decompositions and tractable queries. J. of Computer and System Sciences, 64(3):579-627, 2002. Google Scholar
  23. Nikolas Mählmann, Sebastian Siebertz, and Alexandre Vigny. Recursive backdoors for SAT. In Filippo Bonchi and Simon J. Puglisi, editors, 46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021, August 23-27, 2021, Tallinn, Estonia, volume 202 of LIPIcs, pages 73:1-73:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.MFCS.2021.73.
  24. Jaroslav Nesetril and Patrice Ossona de Mendez. Tree-depth, subgraph coloring and homomorphism bounds. European J. Combin., 27(6):1022-1041, 2006. Google Scholar
  25. Jaroslav Nešetřil and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012. Google Scholar
  26. Rolf Niedermeier. Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2006. Google Scholar
  27. Sebastian Ordyniak, Andre Schidler, and Stefan Szeider. Backdoor DNFs. In Zhi-Hua Zhou, editor, Proceeding of IJCAI-2021, the 30th International Joint Conference on Artificial Intelligence, pages 1403-1409, 2021. URL: https://doi.org/10.24963/ijcai.2021/194.
  28. Marko Samer and Stefan Szeider. Backdoor trees. In AAAI 08, Twenty-Third Conference on Artificial Intelligence, Chicago, Illinois, July 13-17, 2008, pages 363-368. AAAI Press, 2008. Google Scholar
  29. Marko Samer and Stefan Szeider. Fixed-parameter tractability. In Armin Biere, Marijn Heule, Hans van Maaren, and Toby Walsh, editors, Handbook of Satisfiability, 2nd Edition, chapter 17, pages 693-736. IOS Press, 2021. URL: https://doi.org/10.3233/FAIA201000.
  30. 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, 1978. Google Scholar
  31. Ryan Williams, Carla Gomes, and Bart Selman. Backdoors to typical case complexity. In Georg Gottlob and Toby Walsh, editors, Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence, IJCAI 2003, pages 1173-1178. Morgan Kaufmann, 2003. Google Scholar
  32. Ryan Williams, Carla Gomes, and Bart Selman. On the connections between backdoors, restarts, and heavy-tailedness in combinatorial search. In Informal Proc. of the Sixth International Conference on Theory and Applications of Satisfiability Testing, S. Margherita Ligure - Portofino, Italy, May 5-8, 2003 (SAT 2003), pages 222-230, 2003. Google Scholar
  33. Dmitriy Zhuk. A proof of CSP dichotomy conjecture. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 331-342. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.38.
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