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Title:
A Complete Dichotomy Rises from the Capture of Vanishing Signatures
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Abstract:
Holant Problems are a broad framework to describe counting problems. The framework generalizes counting Constraint Satisfaction Problems and partition functions of Graph Homomorphisms.
We prove a complexity dichotomy theorem for Holant problems over an arbitrary set of complex-valued symmetric constraint functions $\mathcal{F}$, also called signatures, on Boolean variables. This extends and unifies all previous dichotomies for Holant problems on symmetric signatures (taking values without a finite modulus).The dichotomy theorem has an explicit tractability criterion.A Holant problem defined by $\mathcal{F}$ is solvable in polynomial time if it satisfies this tractability criterion, and is \#P-hard otherwise.The proof of this theorem utilizes many previous dichotomy theorems on Holant problems and Boolean \#CSP. Holographic transformations play an indispensable role, not only as a proof technique, but also in the statement of the dichotomy criterion.
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