Abstract
We study the constraint satisfaction problem (CSP) parameterized by a constraint language Γ (CSPΓ) and how the choice of Γ affects its worst-case time complexity. Under the exponential-time hypothesis (ETH), we rule out the existence of subexponential algorithms for finite-domain NP-complete CSPΓ problems. This extends to certain infinite-domain CSPs and structurally restricted problems. For CSPs with finite domain D and where all unary relations are available, we identify a relation SD such that the time complexity of the NP-complete problem CSP({SD}) is a lower bound for all NP-complete CSPs of this kind. We also prove that the time complexity of CSP({SD}) strictly decreases when |D| increases (unless the ETH is false) and provide stronger complexity results in the special case when |D|=3.
- V. B. Alekseev and A. A. Voronenko. 1994. On some closed classes in partial two-valued logic. Discrete Mathematics and Applications 4, 5 (1994), 401--419Google Scholar
Cross Ref
- L. Barto. 2014. The constraint satisfaction problem and universal algebra. ACM SIGLOG News 1, 2 (Oct. 2014), 14--24. Google Scholar
Digital Library
- L. Barto and M. Pinsker. 2016. 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’16). ACM, New York, NY, 615--622. Google Scholar
Digital Library
- M. Behrisch, M. Hermann, S. Mengel, and G. Salzer. 2019. Minimal distance of propositional models. Theory of Computing Systems 63, 6 (Aug. 2019), 1131--1184. Google Scholar
Digital Library
- M. Bodirsky. 2012. Complexity classification in infinite-domain constraint satisfaction. Mémoire d’habilitation à diriger des recherches, Université Diderot -- Paris 7. Available at arXiv:1201.0856.Google Scholar
- M. Bodirsky, P. Jonsson, and T. V. Pham. 2017. The complexity of phylogeny constraint satisfaction problems. ACM Transactions on Computational Logic 18, 3 (2017), 23:1--23:42. Google Scholar
Digital Library
- M. Bodirsky and J. Kára. 2008. The complexity of equality constraint languages. Theory of Computing Systems 43, 2 (Aug. 2008), 136--158. Google Scholar
Cross Ref
- M. Bodirsky and J. Kára. 2010. The complexity of temporal constraint satisfaction problems. Journal of the ACM 57, 2, Article 9 (2010), 41 pages. Google Scholar
Digital Library
- M. Bodirsky and M. Pinsker. 2015. Schaefer’s theorem for graphs. Journal of the ACM 62, 3, Article 19 (June 2015), 52 pages. Google Scholar
Digital Library
- V. G. Bodnarchuk, L. A. Kaluzhnin, V. N. Kotov, and B. A. Romov. 1969. Galois theory for Post algebras. I. Cybernetics 5, 3 (1969), 243--252.Google Scholar
Cross Ref
- V. G. Bodnarchuk, L. A. Kaluzhnin, V. N. Kotov, and B. A. Romov. 1969. Galois theory for Post algebras. II. Cybernetics 5, 5 (1969), 531--539.Google Scholar
Cross Ref
- E. Böhler, E. Hemaspaandra, S. Reith, and H. Vollmer. 2002. Equivalence and isomorphism for Boolean constraint satisfaction. In Proceedings of the 16th International Workshop on Computer Science Logic (CSL’02). Springer Berlin, Berlin, 412--426. Google Scholar
Digital Library
- F. Börner. 2008. Basics of Galois connections. In Complexity of Constraints, N. Creignou, P. G. Kolaitis, and H. Vollmer (Eds.). Lecture Notes in Computer Science, Vol. 5250. Springer Berlin, 38--67. Google Scholar
Digital Library
- A. Bulatov. 2011. Complexity of conservative constraint satisfaction problems. ACM Transactions on Computational Logic 12, 4, Article 24 (July 2011), 66 pages. Google Scholar
Digital Library
- A. Bulatov. 2017. A dichotomy theorem for nonuniform CSPs. In Proceedings of the 58th Annual Symposium on Foundations of Computer Science (FOCS’17). IEEE Computer Society, 319--330.Google Scholar
Cross Ref
- A. Bulatov and A. Hedayaty. 2012. Counting problems and clones of functions. Multiple-Valued Logic and Soft Computing 18, 2 (2012), 117--138.Google Scholar
- A. Bulatov, P. Jeavons, and A. Krokhin. 2005. Classifying the complexity of constraints using finite algebras. SIAM Journal on Computing 34, 3 (March 2005), 720--742. Google Scholar
Digital Library
- C. Calabro, R. Impagliazzo, and R. Paturi. 2009. The complexity of satisfiability of small depth circuits. In Parameterized and Exact Computation, J. Chen and F. V. Fomin (Eds.). Lecture Notes in Computer Science, Vol. 5917. Springer Berlin, 75--85. Google Scholar
Digital Library
- C. Carbonnel and M. C. Cooper. 2016. Tractability in constraint satisfaction problems: a survey. Constraints 21, 2 (Apr. 2016), 115--144. Google Scholar
Digital Library
- M. C. Cooper and S. Zǐvný. 2017. Hybrid tractable classes of constraint problems. In The Constraint Satisfaction Problem: Complexity and Approximability, Andrei Krokhin and Stanislav Zǐvný (Eds.). Dagstuhl Follow-Ups, Vol. 7. Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 113--135.Google Scholar
- N. Creignou, U. Egly, and J. Schmidt. 2014. Complexity Classifications for Logic-Based Argumentation. ACM Transactions on Computational Logic 15, 3 (2014), 19:1--19:20. Google Scholar
Digital Library
- M. Cygan, F. V. Fomin, A. Golovnev, A. S. Kulikov, I. Mihajlin, J. Pachocki, and A. Socała. 2016. Tight bounds for graph homomorphism and subgraph isomorphism. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’16). Society for Industrial and Applied Mathematics, Philadelphia, PA, 1643--1649. Google Scholar
Digital Library
- R. de Haan, I. A. Kanj, and S. Szeider. 2015. On the subexponential-time complexity of CSP. Journal of Artificial Intelligence Research (JAIR) 52 (2015), 203--234. Google Scholar
Digital Library
- T. Feder and M. Y. Vardi. 1998. The computational structure of monotone monadic SNP and constraint satisfaction: A study through datalog and group theory. SIAM Journal on Computing 28, 1 (1998), 57--104. Google Scholar
Digital Library
- R. V. Freivald. 1966. A completeness criterion for partial functions of logic and many-valued logic algebras. Soviet Physics Doklady 11 (Oct. 1966), 288.Google Scholar
- D. Geiger. 1968. Closed systems of functions and predicates. Pacific Journal on Mathematics 27, 1 (1968), 95--100.Google Scholar
Cross Ref
- M. Grohe. 2006. The structure of tractable constraint satisfaction problems. In Proceedings of the 31st International Symposium on Mathematical Foundations of Computer Science (MFCS’06). Springer Berlin, Berlin,58--72. Google Scholar
Digital Library
- L. Ham. 2017. Gap theorems for robust satisfiability: Boolean CSPs and beyond. Theoretical Computer Science 676 (2017), 69--91.Google Scholar
Cross Ref
- T. Hertli. 2014. 3-SAT faster and simpler - Unique-SAT bounds for PPSZ hold in general. SIAM Journal on Computing 43, 2 (2014), 718--729.Google Scholar
Cross Ref
- R. Impagliazzo and R. Paturi. 2001. On the complexity of k-SAT. Journal of Computer and System Sciences 62, 2 (2001), 367--375. Google Scholar
Digital Library
- R. Impagliazzo, R. Paturi, and F. Zane. 2001. Which problems have strongly exponential complexity?Journal of Computer and System Sciences 63, 4 (2001), 512--530. Google Scholar
Digital Library
- P. Jeavons. 1998. On the algebraic structure of combinatorial problems. Theoretical Computer Science 200 (1998), 185--204. Google Scholar
Digital Library
- P. Jeavons, D. A. Cohen, and M. Gyssens. 1999. How to determine the expressive power of constraints. Constraints 4, 2 (1999), 113--131. Google Scholar
Digital Library
- P. Jonsson and V. Lagerkvist. 2017. An initial study of time complexity in infinite-domain constraint satisfaction. Artificial Intelligence 245 (2017), 115--133.Google Scholar
- P. Jonsson, V. Lagerkvist, G. Nordh, and B. Zanuttini. 2017. Strong partial clones and the time complexity of SAT problems. Journal of Computer and System Sciences 84 (2017), 52--78. Google Scholar
Digital Library
- V. Lagerkvist. 2015. Precise upper and lower bounds for the monotone constraint satisfaction problem. In Proceedings of the 40th International Symposium on Mathematical Foundations of Computer Science (MFCS’15). Springer Berlin, Berlin, 357--368.Google Scholar
Cross Ref
- R. Pöschel. 2004. Galois connections for operations and relations. In Galois Connections and Applications, K. Denecke, M. Erné, and S.L. Wismath (Eds.). Mathematics and Its Applications, Vol. 565. Springer Netherlands, 231--258.Google Scholar
- E. Post. 1941. The two-valued iterative systems of mathematical logic. Annals of Mathematical Studies 5 (1941), 1--122.Google Scholar
- B. A. Romov. 1981. The algebras of partial functions and their invariants. Cybernetics 17, 2 (1981), 157--167.Google Scholar
Cross Ref
- B. A. Romov. 2006. The completeness problem in partial hyperclones. Discrete Mathematics 306, 13 (July 2006), 1405--1414. Google Scholar
Digital Library
- F. Rossi, P. van Beek, and T. Walsh (Eds.). 2006. Handbook of Constraint Programming. Foundations of Artificial Intelligence, Vol. 2. Elsevier. Google Scholar
Digital Library
- S. J. Russell and P. Norvig. 2010. Artificial Intelligence - A Modern Approach (3rd internat. ed.). Pearson Education.Google Scholar
- T. Schaefer. 1978. The complexity of satisfiability problems. In Proceedings of the 10th Annual ACM Symposium on Theory Of Computing (STOC’78). ACM Press, 216--226. Google Scholar
Digital Library
- H. Schnoor and I. Schnoor. 2008. Partial polymorphisms and constraint satisfaction problems. In Complexity of Constraints, N. Creignou, P. G. Kolaitis, and H. Vollmer (Eds.). Lecture Notes in Computer Science, Vol. 5250. Springer Berlin Heidelberg, 229--254. Google Scholar
Digital Library
- K. Schölzel. 2015. Dichotomy on intervals of strong partial Boolean clones. Algebra Universalis 73, 3--4 (2015), 347--368.Google Scholar
Cross Ref
- M. Wahlström. 2007. Algorithms, Measures and Upper Bounds for Satisfiability and Related Problems. Ph.D. Dissertation. Linköping University.Google Scholar
- D. Zhuk. 2017. The proof of CSP dichotomy conjecture. In Proceedings of the 58th Annual Symposium on Foundations of Computer Science (FOCS’17). IEEE Computer Society, 331--342.Google Scholar
Cross Ref
Index Terms
Fine-Grained Time Complexity of Constraint Satisfaction Problems
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