A Tight Analysis of Bethe Approximation for Permanent.

Nima Anari, Alireza Rezaei

Abstract: We prove that the permanent of nonnegative matrices can be deterministically approximated within a factor of √2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> 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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> . 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.