NDLI: An Effective Dichotomy for the Counting Constraint Satisfaction Problem

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Dyer, Martin Richerby, David
Copyright Year	2013
Abstract	Bulatov [Proceedings of the $35$th 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.
Starting Page	1245
Ending Page	1274
Page Count	30
File Format	PDF
ISSN	00975397
DOI	10.1137/100811258
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	3 (Special Section on the Forty-Second Annual ACM Symposium on Theory of Computing (STOC 2010))
Volume Number	42
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2013-06-25
Access Restriction	Subscribed
Subject Keyword	Analysis of algorithms and problem complexity constraint satisfaction problem complexity dichotomy counting problems
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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