Complexity of generalized satisfiability counting problems
Information and Computation
Theoretical Computer Science
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
SIAM Journal on Computing
Signature Theory in Holographic Algorithms
ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
Holant problems and counting CSP
Proceedings of the forty-first annual ACM symposium on Theory of computing
The Complexity of Weighted Boolean CSP
SIAM Journal on Computing
An approximation trichotomy for Boolean #CSP
Journal of Computer and System Sciences
Holographic algorithms: From art to science
Journal of Computer and System Sciences
A trichotomy theorem for the approximate counting of complex-weighted bounded-degree boolean CSPs
COCOA'10 Proceedings of the 4th international conference on Combinatorial optimization and applications - Volume Part I
Approximate counting for complex-weighted Boolean constraint satisfaction problems
WAOA'10 Proceedings of the 8th international conference on Approximation and online algorithms
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Constraint satisfaction problems have been studied in numerous fields with practical and theoretical interests. In recent years, major breakthroughs have been made in a study of counting constraint satisfaction problems (or #CSPs). In particular, a computational complexity classification of bounded-degree #CSPs has been discovered for all degrees except for two, where the ''degree'' of an input instance is the maximal number of times that each input variable appears in a given set of constraints. Despite the efforts of recent studies, however, a complexity classification of degree-2 #CSPs has eluded from our understandings. This paper challenges this open problem and gives its partial solution by applying two novel proof techniques-T"2-constructibility and parametrized symmetrization-which are specifically designed to handle ''arbitrary'' constraints under randomized approximation-preserving reductions. We partition entire constraints into four sets and we classify the approximation complexity of all degree-2 #CSPs whose constraints are drawn from two of the four sets into two categories: problems computable in polynomial-time or problems that are at least as hard as #SAT. Our proof exploits a close relationship between complex-weighted degree-2 #CSPs and Holant problems, which are a natural generalization of complex-weighted #CSPs.