Dyer, M and Richerby, D (2013) An effective dichotomy for the counting constraint satisfaction problem. SIAM Journal on Computing, 42 (3). 1245 - 1274. ISSN 0097-5397
Abstract
Bulatov [Proceedings of the 35th International Colloquium on Automata, Languages and Programming (Part 1), Lecture Notes in Comput. Sci. 5125, Springer, New York, 2008, pp. 646- 661] gave a dichotomy for the counting constraint satisfaction problem #CSP. A problem from #CSP is characterized by a constraint language Ã, a fixed, finite set of relations over a finite domain D. An instance of the problem uses these relations to constrain an arbitrarily large finite set of variables. Bulatov showed that the problem of counting the satisfying assignments of instances of any problem from #CSP is either in polynomial time (FP) or is #P-complete. His proof draws heavily on techniques from universal algebra and cannot be understood without a secure grasp of that field. We give an elementary proof of Bulatov's dichotomy, based on succinct representations, which we call frames, of a class of highly structured relations, which we call strongly rectangular. We show that these are precisely the relations which are invariant under a Mal'tsev polymorphism. En route, we give a simplification of a decision algorithm for strongly rectangular constraint languages due to Bulatov and Dalmau [SIAM J. Comput., 36 (2006), pp. 16-27]. We establish a new criterion for the #CSP dichotomy, which we call strong balance, and we prove that this property is decidable. In fact, we establish membership in NP. Thus, we show that the dichotomy is effective, resolving the most important open question concerning the #CSP dichotomy.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | (c) 2013, Society for Industrial and Applied Mathematics. This is an author produced version of a paper published in the SIAM Journal on Computing. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Complexity dichotomy; Constraint satisfaction problem; Counting problems |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) > Institute for Computational and Systems Science (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 17 Sep 2013 11:14 |
Last Modified: | 23 Jun 2023 21:34 |
Published Version: | http://dx.doi.org/10.1137/100811258 |
Status: | Published |
Publisher: | Society for Industrial and Applied Mathematics |
Identification Number: | 10.1137/100811258 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:76432 |