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2011 IEEE 26th Annual Conference on Computational Complexity (2011)
San Jose, California USA
June 8, 2011 to June 11, 2011
ISSN: 1093-0159
ISBN: 978-0-7695-4411-3
pp: 45-54
We prove a complexity dichotomy theorem for all non-negatively weighted counting Constraint Satisfaction Problems (#CSP). This caps a long series of important results on counting problems, including unweighted and weighted graph homomorphisms and the celebrated dichotomy theorem for unweighted #CSP. Our dichotomy theorem gives a succinct criterion for tractability. If a set F of constraint functions satisfies the criterion, then the #CSP problem defined by F is solvable in polynomial time; if it does not satisfy the criterion, then the problem is #P-hard. We furthermore show that the question of whether F satisfies the criterion is decidable in NP. Surprisingly, our tractability criterion is simpler than the previous tractability criteria for the more restricted classes of problems, although when specialized to those cases, they are logically equivalent. Our proof mainly uses Linear Algebra and represents a departure from Universal Algebra, the dominant methodology in recent years.

P. Lu, J. Cai and X. Chen, "Non-negatively Weighted #CSP: An Effective Complexity Dichotomy," 2011 IEEE 26th Annual Conference on Computational Complexity(CCC), San Jose, California USA, 2011, pp. 45-54.
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