Complexity of generalized satisfiability counting problems
Information and Computation
The complexity of counting graph homomorphisms
Proceedings of the ninth international conference on on Random structures and algorithms
Complexity classifications of boolean constraint satisfaction problems
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STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
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FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
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Journal of the ACM (JACM)
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FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Holant problems and counting CSP
Proceedings of the forty-first annual ACM symposium on Theory of computing
The complexity of weighted Boolean #CSP with mixed signs
Theoretical Computer Science
The Complexity of Weighted Boolean CSP
SIAM Journal on Computing
Proceedings of the forty-second ACM symposium on Theory of computing
Graph homomorphisms with complex values: a dichotomy theorem
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
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A Complexity Dichotomy for Partition Functions with Mixed Signs
SIAM Journal on Computing
Non-negatively Weighted #CSP: An Effective Complexity Dichotomy
CCC '11 Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity
Computational Complexity of Holant Problems
SIAM Journal on Computing
Complexity of counting CSP with complex weights
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
A Dichotomy for Real Weighted Holant Problems
CCC '12 Proceedings of the 2012 IEEE Conference on Computational Complexity (CCC)
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We prove a complexity dichotomy theorem for the most general form of Boolean #CSP where every constraint function takes values in the field of complex numbers C. We first give a non-trivial tractable class of Boolean #CSP which was inspired by holographic reductions. The tractability crucially depends on algebraic cancelations which are absent for non-negative numbers. We then completely characterize all the tractable Boolean #CSP with complex-valued constraints and show that we have found all the tractable ones, and every remaining problem is #P-hard. We also improve our result by proving the same dichotomy theorem holds for Boolean #CSP with maximum degree 3 (every variable appears at most three times). The concept of Congruity and Semi-congruity provides a key insight and plays a decisive role in both the tractability and hardness proofs. We also introduce local holographic reductions as a technique in hardness proofs.