A Tight Analysis of Bethe Approximation for Permanent
Nima Anari, Alireza Rezaei · SIAM Journal on Computing · 2019
Abstract. We prove that the permanent of nonnegative matrices can be deterministically approximated within a factor of [Formula: see text] in polynomial time, improving upon 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 [Formula: see text]. This resolves a conjecture of [L. Gurvits, Unleashing the Power of Schrijver’s Permanental Inequality with the Help of the Bethe Approximation, preprint, arxiv 1106.2844, 2011]. Our bound is tight and, when combined with previously known inequalities lower bounding the permanent, fully resolves the quality of Bethe approximation for the permanent. As an additional corollary of our methods, we resolve a conjecture of [M. Chertkov and A. B. Yedidia, J. Mach. Learn. Res., 14 (2013), pp. 2029–2066], proving that fractional belief propagation with fractional parameter [Formula: see text] yields an upper bound on the permanent.