Complexity theory of real functions
Complexity theory of real functions
Complexity of generalized satisfiability counting problems
Information and Computation
On the Structure of Polynomial Time Reducibility
Journal of the ACM (JACM)
Information Processing Letters
Fanout limitations on constraint systems
Theoretical Computer Science
On Counting Independent Sets in Sparse Graphs
SIAM Journal on Computing
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
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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We determine the computational complexity of approximately counting the total weight of variable assignments for every complex-weighted Boolean constraint satisfaction problem (or CSP) with any number of additional unary (i.e., arity 1) constraints, particularly, when degrees of input instances are bounded from above by a fixed constant. All degree-1 counting CSPs are obviously solvable in polynomial time. When the instance's degree is more than two, we present a dichotomy theorem that classifies all counting CSPs admitting free unary constraints into exactly two categories. This classification theorem extends, to complex-weighted problems, an earlier result on the approximation complexity of unweighted counting Boolean CSPs of bounded degree. The framework of the proof of our theorem is based on a theory of signature developed from Valiant's holographic algorithms that can efficiently solve seemingly intractable counting CSPs. Despite the use of arbitrary complex weight, our proof of the classification theorem is rather elementary and intuitive due to an extensive use of a novel notion of limited T-constructibility. For the remaining degree-2 problems, in contrast, they are as hard to approximate as Holant problems, which are a generalization of counting CSPs.