Faster FPRAS for the Permanent via Restricted Poincar\'e Inequalities and Coupled Flows

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 Bez\'akov\'a, \v{S}tefankovi\v{c}, 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 Bez\'akov\'a 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 Poincar\'e 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.