The Exponential-Time Complexity of the complex weighted #CSP
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In this paper, I consider a fine-grained dichotomy of Boolean counting constraint satisfaction problem (#CSP), under the exponential time hypothesis of counting version (#ETH). Suppose ℱ is a finite set of algebraic complex-valued functions defined on Boolean domain. When ℱ is a subset of either two special function sets, I prove that #CSP(ℱ) is polynomial-time solvable, otherwise it can not be computed in sub-exponential time unless #ETH fails. I also improve the result by proving the same dichotomy holds for #CSP with bounded degree (every variable appears at most constant constraints), even for #R_3-CSP. An important preparation before proving the result is to argue that pinning (two special unary functions [1,0] and [0,1] are used to reduce arity) can also keep the sub-exponential lower bound of a Boolean #CSP problem. I discuss this issue by utilizing some common methods in proving #P-hardness of counting problems. The proof illustrates the internal correlation among these commonly used methods.
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