A Tight Analysis of Bethe Approximation for Permanent

11/07/2018

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by   Nima Anari, et al.

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We prove that the permanent of nonnegative matrices can be deterministically approximated within a factor of √(2)^n in polynomial time, improving upon the previous deterministic approximations. We show this by proving that the Bethe approximation of the permanent, a quantity computable in polynomial time, is at least as large as the permanent divided by √(2)^n. This resolves a conjecture of Gurvits. Our bound is tight, and when combined with previously known inequalities lower bounding the permanent, fully resolves the quality of Bethe approximation for permanent.