Optimal girth-dependent bounds for the Bethe approximation of the permanent

2026-09-01

AI Summary

本文解决了非负矩阵的Bethe近似问题，通过提供一个依赖于图围长g的最优界限改进了永久值的估计方法。

Abstract

For an $n\times n$ nonnegative matrix $A$, the Bethe permanent, which is computable in deterministic polynomial time, satisfies the tight universal comparison \[\operatorname{Bethe}(A) \leq \operatorname{per}(A) \leq 2^{n/2}\operatorname{Bethe}(A).\] The lower bound, due to Gurvits, is attained on forests. The upper bound, due to Anari and Rezaei, is attained by the adjacency matrix of a disjoint union of $4$-cycles. Confirming a conjecture of Anari, we provide an optimal girth-dependent refinement of the above comparison. More precisely, we show that if the bipartite support graph of $A$ has girth at least an even integer $g \geq 4$, then \[\operatorname{Bethe}(A) \leq \operatorname{per}(A) \leq 2^{2n/g}\operatorname{Bethe}(A).\] The upper bound is attained by the adjacency matrix of a disjoint union of $g$-cycles.

Problem

• Bethe approximation
• permanent
• girth-dependent bounds
• nonnegative matrix

Innovation

• Bethe approximation
• permanent
• girth-dependent bounds
• nonnegative matrix
• bipartite support graph