A Tight Analysis of Bethe Approximation for Permanent

We prove that the permanent of nonnegative matrices can be deterministically approximated within a factor of √2
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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
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. This resolves a conjecture of Gurvits [21]. Our bound is tight, and when combined with previously known inequalities lower bounding the permanent, fully resolves the quality of Bethe approximation for permanent.