Structural Corrections to the Bethe Approximation of the Permanent

2026-08-31

AI Summary

研究通过改进Bethe近似算法识别并利用4-循环障碍，以提高对非负矩阵永久值的估计精度。

Abstract

We study deterministic approximation algorithms for the permanent of a nonnegative matrix through the Bethe permanent, an approximation computable in polynomial time. The tight analysis of Anari and Rezaei gives a universal comparison between the permanent and the Bethe permanent within a factor $(\sqrt 2)^n$. The simple example of the unweighted $4$-cycle $C_4$ (or a union of disjoint $C_4$'s) shows that this bound is tight. We show that such $4$-cycle obstructions can be identified and exploited algorithmically. Given a Bethe optimizer, our algorithm identifies nearly isolated weighted $2\times2$ blocks and peels off a vertex-disjoint family of them. If the total weighted correction is large, we can improve the Bethe approximation; if it is small, we show that the Bethe permanent is within a factor of $(\sqrt2 - \varepsilon)^n$ of the truth. Combining these facts, we obtain a deterministic polynomial time $(\sqrt2-\varepsilon)^n$-approximation algorithm for the permanent of an arbitrary nonnegative $n\times n$ matrix, where $\varepsilon>0$ is some absolute constant.

Problem

• permanent
• Bethe approximation
• nonnegative matrix

Innovation

• Bethe Approximation
• Permanent of Matrix
• Deterministic Polynomial Time Algorithm
• Isolated Weighted 2x2 Blocks