A Tight Analysis of Bethe Approximation for Permanent [article]

Nima Anari, Alireza Rezaei
2019 arXiv   pre-print
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
more » ... nent, fully resolves the quality of Bethe approximation for permanent. As an additional corollary of our methods, we resolve a conjecture of Chertkov and Yedidia, proving that fractional belief propagation with fractional parameter γ=-1/2 yields an upper bound on the permanent.
arXiv:1811.02933v2 fatcat:7hxms2beffdynjxhgl5pygrthi