AI 中文总结
该研究针对永久值的FPRAS算法，通过受限庞加莱不等式与耦合流论证，将0/1矩阵的近似运行时间从O(n^7 log^4 n)渐近改进至O(n^6 log^5 n)，且可扩展至任意非负矩阵。
AI 中文摘要
n×n的0/1矩阵A的永久值等于由A定义边的二分图中的完美匹配数。Jerrum、Sinclair和Vigoda（2004）提出了一种FPRAS，用于近似任意非负矩阵的永久值，采用了一种新型模拟退火算法。Bezáková等人（2008）将0/1矩阵的运行时间改进为O(n^7 log^4 n)，适用于任意固定的近似和成功参数。本文首次在该运行时间界上取得渐近改进，得到了O(n^6 log^5 n)时间的算法，且与此前工作一样，该算法可扩展至任意非负矩阵。Bezáková等人的分析在理想洞权重下，对完美匹配和近完美匹配上的JSV马尔可夫链给出了O(n^4)的松弛时间界，其中洞模式（即未匹配顶点，若存在）在平稳分布中概率相等。本文引入了针对洞模式划分的受限庞加莱不等式，并证明了相应受限松弛时间的O(n^3)界。证明采用了耦合多商品流论证，该论证受Chen等人（2025）针对所有匹配上的Jerrum-Sinclair链的近期传输流论证启发。
英文摘要
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.
