Abstract
In this article, we study the complexity of counting Constraint Satisfaction Problems (CSPs) of the form #CSP(C, -), in which the goal is, given a relational structure A from a class C of structures and an arbitrary structure B, to find the number of homomorphisms from A to B. Flum and Grohe showed that #CSP(C, -) is solvable in polynomial time if C has bounded treewidth [FOCS’02]. Building on the work of Grohe [JACM’07] on decision CSPs, Dalmau and Jonsson then showed that if C is a recursively enumerable class of relational structures of bounded arity, then, assuming FPT≠ #W[1], there are no other cases of #CSP(C, -) solvable exactly in polynomial time (or even fixed-parameter time) [TCS’04].
We show that, assuming FPT ≠ W[1] (under randomised parameterised reductions) and for C satisfying certain general conditions, #CSP(C,-) is not solvable even approximately for C of unbounded treewidth; that is, there is no fixed parameter tractable (and thus also not fully polynomial) randomised approximation scheme for #CSP(C, -). In particular, our condition generalises the case when C is closed undertaking minors.
- Vikraman Arvind and Venkatesh Raman. 2002. Approximation algorithms for some parameterized counting problems. In Proceedings of the 13th International Symposium on Algorithms and Computation (ISAAC’02) (Lecture Notes in Computer Science), Vol. 2518. Springer, 453--464. DOI:https://doi.org/10.1007/3-540-36136-7_40Google Scholar
Cross Ref
- Johann Brault-Baron, Florent Capelli, and Stefan Mengel. 2015. Understanding model counting for beta-acyclic CNF-formulas. In Proceedings of the 32nd International Symposium on Theoretical Aspects of Computer Science (STACS’15). 143--156. DOI:https://doi.org/10.4230/LIPIcs.STACS.2015.143Google Scholar
- 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. DOI:https://doi.org/10.1109/FOCS.2017.37Google Scholar
Cross Ref
- Andrei Bulatov, Peter Jeavons, and Andrei Krokhin. 2005. Classifying the complexity of constraints using finite algebras. SIAM Journal on Computing 34, 3 (2005), 720--742. DOI:https://doi.org/10.1137/S0097539700376676Google Scholar
Digital Library
- Andrei A. Bulatov. 2013. The complexity of the counting constraint satisfaction problem. Journal of the ACM 60, 5 (2013), 34. DOI:https://doi.org/10.1145/2528400Google Scholar
Digital Library
- Andrei A. Bulatov and Stanislav Živný. 2019. Approximate counting CSP seen from the other side. In Proceedings of the 44th International Symposium on Mathematical Foundations of Computer Science (MFCS’19) (LIPIcs), Vol. 138. Schloss Dagstuhl—Leibniz-Zentrum für Informatik, 60:1–60:14. DOI:https://doi.org/10.4230/LIPIcs.MFCS.2019.60Google Scholar
- Jin-Yi Cai and Xi Chen. 2017. Complexity of counting CSP with complex weights. Journal of the ACM 64, 3 (2017), 19:1–19:39. DOI:https://doi.org/10.1145/2822891Google Scholar
Digital Library
- Xi Chen, Martin E. Dyer, Leslie Ann Goldberg, Mark Jerrum, Pinyan Lu, Colin McQuillan, and David Richerby. 2015. The complexity of approximating conservative counting CSPs. Journal of Computer and System Sciences 81, 1 (2015), 311--329. DOI:https://doi.org/10.1109/FOCS.2017.37Google Scholar
Digital Library
- Nadia Creignou and Miki Hermann. 1996. Complexity of generalized satisfiability counting problems. Information and Computation 125, 1 (1996), 1--12. DOI:https://doi.org/10.1006/inco.1996.0016Google Scholar
Digital Library
- Víctor Dalmau and Peter Jonsson. 2004. The complexity of counting homomorphisms seen from the other side. Theoretical Computer Science 329, 1--3 (2004), 315--323. DOI:https://doi.org/10.1016/j.tcs.2004.08.008Google Scholar
Digital Library
- Víctor Dalmau, Phokion G. Kolaitis, and Moshe Y. Vardi. 2002. Constraint Satisfaction, Bounded Treewidth, and Finite-Variable Logics. In Proceedings of the 8th International Conference on Principles and Practice of Constraint Programming (CP’02) (Lecture Notes in Computer Science), Vol. 2470. Springer, 310--326. DOI:https://doi.org/10.1007/3-540-46135-3_21Google Scholar
- Reinhard Diestel. 2010. Graph Theory (4th ed.). Springer.Google Scholar
- Rodney G. Downey and Michael R. Fellows. 1995. Fixed-parameter tractability and completeness I: basic results. SIAM Journal on Computing Computing 24, 4 (1995), 873--921. DOI:https://doi.org/10.1137/S0097539792228228Google Scholar
Digital Library
- Rodney G. Downey and Michael R. Fellows. 1995. Fixed-parameter tractability and completeness II: On completeness for W[1]. Theoretical Computer Science 141, 1–2 (1995), 109--131. DOI:https://doi.org/10.1016/0304-3975(94)00097-3Google Scholar
Digital Library
- Rodney G. Downey, Michael R. Fellows, and Kenneth W. Regan. 1998. Parameterized circuit complexity and the W hierarchy. Theoretical Computer Science 191, 1--2 (1998), 97--115. DOI:https://doi.org/10.1016/S0304-3975(96)00317-9Google Scholar
Digital Library
- Martin E. Dyer, Leslie Ann Goldberg, Catherine S. Greenhill, and Mark Jerrum. 2004. The relative complexity of approximate counting problems. Algorithmica 38, 3 (2004), 471--500. DOI:https://doi.org/10.1007/s00453-003-1073-yGoogle Scholar
Cross Ref
- Martin E. Dyer, Leslie Ann Goldberg, and Mark Jerrum. 2010. An approximation trichotomy for Boolean #CSP. Journal of Computer and System Sciences 76, 3--4 (2010), 267--277. DOI:https://doi.org/10.1016/j.jcss.2009.08.003Google Scholar
Digital Library
- Martin E. Dyer and Catherine S. Greenhill. 2000. The complexity of counting graph homomorphisms. Random Structures and Algorithms 17, 3--4 (2000), 260--289. DOI:https://doi.org/10.1002/1098-2418(200010/12)17:3/4<260::AID-RSA5>3.0.CO;2-WGoogle Scholar
Cross Ref
- Martin E. Dyer and David Richerby. 2013. An effective dichotomy for the counting constraint satisfaction problem. SIAM Journal of Computing 42, 3 (2013), 1245--1274. DOI:https://doi.org/10.1137/100811258Google Scholar
Cross Ref
- Tommy Färnqvist. 2013. Exploiting Structure in CSP-related Problems. Ph.D. Dissertation. Department of Computer Science and Information Science, Linköping University. http://www.diva-portal.org/smash/get/diva2:576178/FULLTEXT01.pdf.Google Scholar
- Tomás Feder, Pavol Hell, Daniel Král’, and Jiří Sgall. 2005. Two algorithms for general list matrix partitions. In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’05). 870--876. http://dl.acm.org/citation.cfm?id=1070432.1070554Google Scholar
- Tomás Feder, Pavol Hell, and Wing Xie. 2007. Matrix partitions with finitely many obstructions. The Electronic Journal of Combinatorics 14, 1 (2007). http://www.combinatorics.org/Volume_14/Abstracts/v14i1r58.html.Google Scholar
Cross Ref
- 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 Journal on Computing 28, 1 (1998), 57--104. DOI:https://doi.org/10.1137/S0097539794266766Google Scholar
Digital Library
- Jörg Flum and Martin Grohe. 2002. The parameterized complexity of counting problems. In Proceedings of the 43rd Symposium on Foundations of Computer Science (FOCS’02). IEEE Computer Society, 538. DOI:https://doi.org/10.1109/SFCS.2002.1181978Google Scholar
Digital Library
- Jörg Flum and Martin Grohe. 2004. The parameterized complexity of counting problems. SIAM Journal on Computing 33, 4 (2004), 892--922. DOI:https://doi.org/10.1137/S0097539703427203Google Scholar
Digital Library
- Jörg Flum and Martin Grohe. 2006. Parametrized Complexity Theory. Springer.Google Scholar
- Andreas Galanis, Leslie Ann Goldberg, and Mark Jerrum. 2016. Approximately counting H-colorings is #BIS-Hard. SIAM Journal on Computing 45, 3 (2016), 680--711. DOI:https://doi.org/10.1137/15M1020551Google Scholar
Cross Ref
- Andreas Galanis, Leslie Ann Goldberg, and Mark Jerrum. 2017. A complexity trichotomy for approximately counting list H-Colorings. ACM Transactions on Computation Theory 9, 2 (2017), 9:1–9:22. DOI:https://doi.org/10.1145/3037381Google Scholar
Digital Library
- Michael R. Garey and David S. Johnson. 1979. Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman.Google Scholar
Digital Library
- Georg Gottlob, Nicola Leone, and Francesco Scarcello. 2002. Hypertree decomposition and tractable queries. Journal of Computer and System Sciences 64, 3 (2002), 579--627. DOI:https://doi.org/10.1006/jcss.2001.1809Google Scholar
Digital Library
- Martin Grohe. 2007. The complexity of homomorphism and constraint satisfaction problems seen from the other side. Journal of the ACM 54, 1 (2007), 1--24. DOI:https://doi.org/10.1145/1206035.1206036Google Scholar
Digital Library
- Martin Grohe and Dániel Marx. 2014. Constraint solving via fractional edge covers. ACM Transactions on Algorithms 11, 1 (2014), 4:1–4:20. DOI:https://doi.org/10.1145/2636918Google Scholar
Digital Library
- Martin Grohe, Thomas Schwentick, and Luc Segoufin. 2001. When is the evaluation of conjunctive queries tractable? In Proceedings of the 33rd ACM Symposium on Theory of Computing (STOC’01). ACM, 657--666. DOI:https://doi.org/10.1145/380752.380867Google Scholar
- Pavol Hell and Jaroslav Nešetřil. 1990. On the Complexity of -coloring. Journal of Combinatorial Theory, Series B 48, 1 (1990), 92--110. DOI:https://doi.org/10.1016/0095-8956(90)90132-JGoogle Scholar
Digital Library
- Phokion G. Kolaitis and Moshe Y. Vardi. 2000. Conjunctive-Query Containment and Constraint Satisfaction. Journal of Computer and System Sciences 61, 2 (2000), 302--332. DOI:https://doi.org/10.1006/jcss.2000.1713Google Scholar
Digital Library
- Dániel Marx. 2013. Tractable hypergraph properties for constraint satisfaction and conjunctive queries. J. ACM 60, 6, Article 42 (2013). DOI:https://doi.org/10.1145/2535926Google Scholar
Digital Library
- Kitty Meeks. 2016. The challenges of unbounded treewidth in parameterised subgraph counting problems. Discrete Applied Mathematics 198 (2016), 170--194. DOI:https://doi.org/10.1016/j.dam.2015.06.019Google Scholar
Digital Library
- Stefan Mengel. 2013. Conjunctive Queries, Arithmetic Circuits and Counting Complexity. Ph.D. Dissertation. University of Paderborn. http://nbn-resolving.de/urn:nbn:de:hbz:466:2-11944Google Scholar
- Michael Mitzenmacher and Eli Upfal. 2017. Probability and Computing (2nd ed.). Cambridge University Press, Cambridge. xx+467 pages. Randomization and Probabilistic Techniques in Algorithms and Data Analysis.Google Scholar
- Reinhard Pichler and Sebastian Skritek. 2013. Tractable counting of the answers to conjunctive queries. Journal of Computer and System Sciences 79, 6 (2013), 984--1001. DOI:https://doi.org/10.1016/j.jcss.2013.01.012Google Scholar
Digital Library
- Neil Robertson and Paul D. Seymour. 1984. Graph minors. III. Planar tree-width. Journal of Combinatorial Theory, Series B 36, 1 (1984), 49--64. DOI:https://doi.org/10.1016/0095-8956(84)90013-3Google Scholar
Cross Ref
- Neil Robertson and Paul D. Seymour. 1986. Graph minors. V. Excluding a planar graph. Journal of Combinatorial Theory, Series B 41, 1 (1986), 92--114. DOI:https://doi.org/10.1016/0095-8956(86)90030-4Google Scholar
Digital Library
- Thomas J. Schaefer. 1978. The complexity of satisfiability problems. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing (STOCS’78). 216--226. DOI:https://doi.org/10.1145/800133.804350Google Scholar
Digital Library
- 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. DOI:https://doi.org/10.1109/FOCS.2017.38Google Scholar
Cross Ref
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Approximate Counting CSP Seen from the Other Side
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