Bounding the Bethe and the Degree-$M$ Bethe Permanents

TL;DR

This paper proves conjectures relating the permanents of lifted matrices and their Bethe permanents, establishing inequalities that deepen understanding of matrix permanents and their bounds.

Contribution

It provides a proof of conjectured inequalities between permanents of lifted matrices and their Bethe permanents, and explores properties of permanents of block matrices.

Findings

Proved perm$( heta^{up P}) \u2264 up M$ perm$( heta)$ for lifted matrices.

Established perm$_{M, ext{B}}( heta) \u2264 ext{perm}( heta)$ for Bethe permanents.

Provided an alternative combinatorial proof of the Bethe permanent inequality.

Abstract

It was recently conjectured that the permanent of a ${P}$-lifting $\theta^{\uparrow{P}}$ of a matrix $\theta$ of degree $M$ is less than or equal to the $M$th power of the permanent perm$(\theta)$, i.e., perm$(\theta^{\uparrow{P}})\leq(\text{perm}(\theta))^M$ and, consequently, that the degree-$M$ Bethe permanent $\text{perm}_{M,\mathrm{B}} (\theta)$ of a matrix $\theta$ is less than or equal to the permanent perm$(\theta)$ of $\theta$, i.e., perm$_{M, \mathrm{B}} (\theta)\leq \text{perm}(\theta)$. In this paper, we prove these related conjectures and show in addition a few properties of the permanent of block matrices that are lifts of a matrix. As a corollary, we obtain an alternative proof of the inequality perm$_{\mathrm{B}} (\theta)\leq \text{perm}(\theta)$ on the Bethe permanent of the base matrix $\theta$ that uses only the combinatorial definition of the Bethe permanent.