Journal
SIAM JOURNAL ON COMPUTING
Volume 42, Issue 3, Pages 1245-1274Publisher
SIAM PUBLICATIONS
DOI: 10.1137/100811258
Keywords
constraint satisfaction problem; counting problems; complexity dichotomy
Funding
- EPSRC [EP/E062172/1, EP/I012087/1]
- Engineering and Physical Sciences Research Council [EP/I012087/1, EP/E062172/1] Funding Source: researchfish
- EPSRC [EP/I012087/1, EP/E062172/1] Funding Source: UKRI
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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 Gamma, 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.
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