Faster FPRAS for the Permanent via Restricted Poincaré Inequalities and Coupled Flows

2026-08-27

AI Summary

本文通过引入受限Poincaré不等式和耦合流方法，改进了近似计算0/1矩阵永久值的算法，将时间复杂度从O(n^7 log^4 n)降低到O(n^6 log^5 n)。

Abstract

The permanent of an $n\times n$ $0/1$ matrix $A$ equals the number of perfect matchings in the bipartite graph with edges defined by $A$. Jerrum, Sinclair, and Vigoda (2004) presented an FPRAS for approximating the permanent of any nonnegative matrix using a novel simulated-annealing algorithm. The running time was improved by Bezáková, Štefankovič, Vazirani, and Vigoda (2008) to $O(n^7\log^4 n)$ for $0/1$ matrices, for any fixed approximation and success parameters. We present the first asymptotic improvement over this running time bound, obtaining an $O(n^6\log^5 n)$-time algorithm. As in the previous works, our algorithm extends to arbitrary nonnegative matrices. The analysis of Bezáková et al. yields an $O(n^4)$ relaxation time bound for the JSV Markov chain on perfect and near-perfect matchings with ideal hole weights, under which each hole pattern (the unmatched vertices, if any) is equally likely in the stationary distribution. We introduce a restricted Poincaré inequality for the partition into hole patterns and prove an $O(n^3)$ bound on the corresponding restricted relaxation time. Our proof uses a coupled multicommodity flow argument inspired by a recent transport-flow argument of Chen et al.~(2025) for the Jerrum-Sinclair chain on all matchings.

Problem

• FPRAS
• permanent
• perfect matchings
• running time
• relaxation time

Innovation

• Faster FPRAS
• Restricted Poincaré Inequalities
• Coupled Flows
• Permanent of Matrix
• Simulated Annealing