Unleashing the power of Schrijver's permanental inequality with the help of the Bethe Approximation

TL;DR

This paper leverages Schrijver's permanental inequality and the Bethe approximation to establish new bounds on the permanent of doubly-stochastic matrices, proving a conjecture related to the monomer-dimer problem and proposing a potential polynomial-time approximation algorithm.

Contribution

It introduces a novel lower bound on the permanent using Schrijver's inequality and the Bethe approximation, and applies it to prove the Asymptotic Lower Matching Conjecture, while also exploring implications for permanent approximation.

Findings

Established a new lower bound for the permanent of doubly-stochastic matrices.

Proved the Asymptotic Lower Matching Conjecture (LAMC).

Suggested a potential polynomial-time approximation algorithm with a factor of (.5)^{n}.

Abstract

Let $A \in \Omega_n$ be doubly-stochastic $n \times n$ matrix. Alexander Schrijver proved in 1998 the following remarkable inequality per(\widetilde{A}) \geq \prod_{1 \leq i,j \leq n} (1- A(i,j)); \widetilde{A}(i,j) =: A(i,j)(1-A(i,j)), 1 \leq i,j \leq n. We use the above Shrijver's inequality to prove the following lower bound: \frac{per(A)}{F(A)} \geq 1; F(A) =: \prod_{1 \leq i,j \leq n} (1- A(i,j))^{1- A(i,j)}. We use this new lower bound to prove S.Friedland's Asymptotic Lower Matching Conjecture(LAMC) on monomer-dimer problem. We use some ideas of our proof of (LAMC) to disprove [Lu,Mohr,Szekely] positive correlation conjecture. We present explicit doubly-stochastic $n \times n$ matrices $A$ with the ratio $\frac{per(A)}{F(A)} = \sqrt{2}^{n}$; conjecture that \max_{A \in \Omega_n}\frac{per(A)}{F(A)} \approx (\sqrt{2})^{n} and give some examples supporting the conjecture. If true, the conjecture (and other ones stated in the paper) would imply a deterministic poly-time algorithm to approximate the permanent of $n \times n$ nonnegative matrices within the relative factor $(\sqrt{2})^{n}$. The best current such factor is $e^n$.