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

Roxana Smarandache; Martin Haenggi

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arXiv · 1503.02217

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

Roxana Smarandache·Martin Haenggi

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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.

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Roxana Smarandache, Martin Haenggi. 2015-03-07. Bounding the Bethe and the Degree-$M$ Bethe Permanents. https://arxiv.org/abs/1503.02217

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