Smoothed counting of 0-1 points in polyhedra
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Given a system of linear equations ℓ_i(x)=β_i in an n-vector x of 0-1 variables, we compute the expectation of exp{- ∑_i γ_i (ℓ_i(x) - β_i)^2}, where x is a vector of independent Bernoulli random variables and γ_i >0 are constants. The algorithm runs in quasi-polynomial n^O(ln n) time under some sparseness condition on the matrix of the system. The result is based on the absence of the zeros of the analytic continuation of the expectation for complex probabilities, which can also be interpreted as the absence of a phase transition in the Ising model with a sufficiently strong external field. As an example, we consider the problem of "smoothed counting" of perfect matchings in hypergraphs.
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