Time Complexity of Constraint Satisfaction via Universal Algebra

Authors Peter Jonsson, Victor Lagerkvist, Biman Roy



PDF
Thumbnail PDF

File

LIPIcs.MFCS.2017.17.pdf
  • Filesize: 0.56 MB
  • 15 pages

Document Identifiers

Author Details

Peter Jonsson
Victor Lagerkvist
Biman Roy

Cite AsGet BibTex

Peter Jonsson, Victor Lagerkvist, and Biman Roy. Time Complexity of Constraint Satisfaction via Universal Algebra. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.MFCS.2017.17

Abstract

The exponential-time hypothesis (ETH) states that 3-SAT is not solvable in subexponential time, i.e. not solvable in O(c^n) time for arbitrary c > 1, where n denotes the number of variables. Problems like k-SAT can be viewed as special cases of the constraint satisfaction problem (CSP), which is the problem of determining whether a set of constraints is satisfiable. In this paper we study the worst-case time complexity of NP-complete CSPs. Our main interest is in the CSP problem parameterized by a constraint language Gamma (CSP(Gamma)), and how the choice of Gamma affects the time complexity. It is believed that CSP(Gamma) is either tractable or NP-complete, and the algebraic CSP dichotomy conjecture gives a sharp delineation of these two classes based on algebraic properties of constraint languages. Under this conjecture and the ETH, we first rule out the existence of subexponential algorithms for finite domain NP-complete CSP(Gamma) problems. This result also extends to certain infinite-domain CSPs and structurally restricted CSP(Gamma) problems. We then begin a study of the complexity of NP-complete CSPs where one is allowed to arbitrarily restrict the values of individual variables, which is a very well-studied subclass of CSPs. For such CSPs with finite domain D, we identify a relation SD such that (1) CSP({SD}) is NP-complete and (2) if CSP(Gamma) over D is NP-complete and solvable in O(c^n) time, then CSP({SD}) is solvable in O(c^n) time, too. Hence, the time complexity of CSP({SD}) is a lower bound for all CSPs of this particular kind. We also prove that the complexity of CSP({SD}) is decreasing when |D| increases, unless the ETH is false. This implies, for instance, that for every c>1 there exists a finite-domain Gamma such that CSP(Gamma) is NP complete and solvable in O(c^n) time.
Keywords
  • Clone Theory
  • Universal Algebra
  • Constraint Satisfaction Problems

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. L. Barto. Constraint satisfaction problem and universal algebra. ACM SIGLOG News, 1(2):14-24, October 2014. Google Scholar
  2. L. Barto and M. Pinsker. The algebraic dichotomy conjecture for infinite domain constraint satisfaction problems. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2016), pages 615-622, New York, NY, USA, 2016. ACM. Google Scholar
  3. M. Behrisch, M. Hermann, S. Mengel, and G. Salzer. Give me another one! In Proceedings of the 26th International Symposium on Algorithms and Computation (ISAAC-2015), pages 664-676, 2015. Google Scholar
  4. M. Behrisch, M. Hermann, S. Mengel, and G. Salzer. As close as it gets. In Proceedings of the 10th International Workshop on Algorithms and Computation (WALCOM-2016), pages 222-235, 2016. Google Scholar
  5. M. Bodirsky. Complexity classification in infinite-domain constraint satisfaction. Mémoire d'habilitation à diriger des recherches, Université Diderot - Paris 7. Available at arXiv:1201.0856, 2012. Google Scholar
  6. M. Bodirsky, P. Jonsson, and T. V. Pham. The complexity of phylogeny constraint satisfaction. In 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016, February 17-20, 2016, Orléans, France, pages 20:1-20:13, 2016. Google Scholar
  7. M. Bodirsky and J. Kára. The complexity of temporal constraint satisfaction problems. Journal of the ACM, 57(2):9:1-9:41, 2010. Google Scholar
  8. M. Bodirsky and M. Pinsker. Schaefer’s theorem for graphs. J. ACM, 62(3):19:1-19:52, June 2015. Google Scholar
  9. V. G. Bodnarchuk, L. A. Kaluzhnin, V. N. Kotov, and B. A. Romov. Galois theory for Post algebras. I. Cybernetics, 5:243-252, 1969. Google Scholar
  10. V. G. Bodnarchuk, L. A. Kaluzhnin, V. N. Kotov, and B. A. Romov. Galois theory for Post algebras. II. Cybernetics, 5:531-539, 1969. Google Scholar
  11. E. Böhler, E. Hemaspaandra, S. Reith, and H. Vollmer. Equivalence and isomorphism for boolean constraint satisfaction. In In Proceedings of the 16th International Workshop on Computer Science Logic (CSL-2002), pages 412-426, Berlin, Heidelberg, 2002. Springer Berlin Heidelberg. Google Scholar
  12. A. Bulatov. Complexity of conservative constraint satisfaction problems. ACM Transactions on Computational Logic, 12(4):24:1-24:66, July 2011. Google Scholar
  13. A. Bulatov. A dichotomy theorem for nonuniform csps. CoRR, abs/1703.03021, 2017. URL: http://arxiv.org/abs/1703.03021.
  14. A. Bulatov and A. Hedayaty. Counting problems and clones of functions. Multiple-Valued Logic and Soft Computing, 18(2):117-138, 2012. Google Scholar
  15. A. Bulatov, P. Jeavons, and A. Krokhin. Classifying the complexity of constraints using finite algebras. SIAM Journal on Computing, 34(3):720-742, March 2005. URL: http://dx.doi.org/10.1137/S0097539700376676.
  16. N. Creignou, U. Egly, and J. Schmidt. Complexity classifications for logic-based argumentation. ACM Transactions on Computational Logic (TOCL), 15(3):19:1-19:20, 2014. Google Scholar
  17. R. de Haan, I. A. Kanj, and S. Szeider. On the subexponential-time complexity of CSP. Journal of Artificial Intelligence Research (JAIR), 52:203-234, 2015. URL: http://dx.doi.org/10.1613/jair.4540.
  18. T. Feder and M.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. Google Scholar
  19. D. Geiger. Closed systems of functions and predicates. Pacific Journal of Mathematics, 27(1):95-100, 1968. Google Scholar
  20. M. Grohe. The structure of tractable constraint satisfaction problems. In Proceedings of the 31st International Symposium on Mathematical Foundations of Computer Science (MFCS 2006), pages 58-72, Berlin, Heidelberg, 2006. Springer Berlin Heidelberg. Google Scholar
  21. L. Ham. Gap theorems for robust satisfiability: Boolean CSPs and beyond. To appear in Theoretical Computer Science, 2017. URL: http://dx.doi.org/10.1016/j.tcs.2017.03.006.
  22. T. Hertli. 3-SAT faster and simpler - unique-SAT bounds for PPSZ hold in general. SIAM Journal on Computing, 43(2):718-729, 2014. URL: http://dx.doi.org/10.1137/120868177.
  23. R. Impagliazzo and R. Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001. Google Scholar
  24. R. Impagliazzo, R. Paturi, and F. Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63:512-530, 2001. Google Scholar
  25. P. Jeavons. On the algebraic structure of combinatorial problems. Theoretical Computer Science, 200:185-204, 1998. Google Scholar
  26. P. Jeavons, D. Cohen, and M. Gyssens. Closure properties of constraints. Journal of the ACM, 44(4):527-548, July 1997. URL: http://dx.doi.org/10.1145/263867.263489.
  27. P. Jonsson, V. Lagerkvist, G. Nordh, and B. Zanuttini. Strong partial clones and the time complexity of SAT problems. Journal of Computer and System Sciences, 84:52-78, 2017. Google Scholar
  28. P. Jonsson, V. Lagerkvist, and B. Roy. Time Complexity of Constraint Satisfaction via Universal Algebra. ArXiv e-prints, June 2017. URL: http://arxiv.org/abs/1706.05902.
  29. E. Post. The two-valued iterative systems of mathematical logic. Annals of Mathematical Studies, 5:1-122, 1941. Google Scholar
  30. A. Rafiey, J. Kinne, and T. Feder. Dichotomy for digraph homomorphism problems. CoRR, abs/1701.02409, 2017. URL: http://arxiv.org/abs/1701.02409.
  31. B.A. Romov. The algebras of partial functions and their invariants. Cybernetics, 17(2):157-167, 1981. Google Scholar
  32. F. Rossi, P. van Beek, and T. Walsh, editors. Handbook of Constraint Programming, volume 2 of Foundations of Artificial Intelligence. Elsevier, 2006. Google Scholar
  33. S. J. Russell and P. Norvig. Artificial Intelligence - A Modern Approach (3. internat. ed.). Pearson Education, 2010. Google Scholar
  34. T. Schaefer. The complexity of satisfiability problems. In Proceedings of the 10th Annual ACM Symposium on Theory Of Computing (STOC-78), pages 216-226. ACM Press, 1978. Google Scholar
  35. H. Schnoor and I. Schnoor. Partial polymorphisms and constraint satisfaction problems. In N. Creignou, P. G. Kolaitis, and H. Vollmer, editors, Complexity of Constraints, volume 5250 of Lecture Notes in Computer Science, pages 229-254. Springer Berlin Heidelberg, 2008. Google Scholar
  36. M. Wahlström. Algorithms, measures and upper bounds for satisfiability and related problems. PhD thesis, Linköping University, TCSLAB - Theoretical Computer Science Laboratory, The Institute of Technology, 2007. Google Scholar
  37. D. Zhuk. The proof of csp dichotomy conjecture. CoRR, abs/1704.01914, 2017. URL: https://arxiv.org/abs/1704.01914.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail