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A Trichotomy Theorem for the Approximate Counting of Complex-Weighted Bounded-Degree Boolean CSPs | | Tomoyuki Yamakami
; | | Date: |
16 Aug 2010 | | Abstract: | We determine the complexity of approximate counting of the total weight of
assignments for complex-weighted Boolean constraint satisfaction problems (or
CSPs), particularly, when degrees of instances are bounded from above by a
given constant, provided that all arity-1 constraints are freely available. All
degree-1 counting CSPs are solvable in polynomial time. When the degree is more
than 2, we present a trichotomy theorem that classifies all bounded-degree
counting CSPs into only three categories with a help of free arity-1
constraints. This classification extends to complex-weighted problems an
earlier result of Dyer, Goldberg, Jalsenius, and Richerby (2010) on the
complexity of the approximate counting of bounded-degree unweighted Boolean
CSPs. The framework of the proof of our trichotomy theorem is based on a theory
of signatures (Cai and Lu, 2007, 2008) used in Valiant’s holographic
algorithms. Despite the use of arbitrary complex-weight, our proof is rather
elementary and intuitive by an extensive use of a notion of limited
T-constructibility. For the degree-2 problems, we show that they are as hard to
approximate as complex Holant problems. | | Source: | arXiv, 1008.2688 | | Services: | Forum | Review | PDF | Favorites | |
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