AI 中文总结
本文提出一种基于电流启发多商品流界和空洞加权滑动链的FPRAS，将计算0/1矩阵永久式的运行时间改进到~O(n^{3.5})。
AI 中文摘要
我们给出了一个用于计算$n\times n$ $0/1$矩阵的永久式的FPRAS，其运行时间为$\widetilde{O}(n^{3.5}\varepsilon^{-2})$。我们的算法扩展到了任意非负矩阵的强多项式FPRAS，这与之前的工作一致。Jerrum、Sinclair和Vigoda（2004）首次给出了非负矩阵永久式的FPRAS。随后，Bezáková、Štefankovič、Vazirani和Vigoda（2008）将运行时间改进到$\widetilde{O}(n^7)$，最近Chen、Vigoda和Yang（2026）将其改进到$\widetilde{O}(n^6)$。我们引入了一种受电流流动启发的多商品流界，用路由能量取代了通常的路径长度因子。对于经典JSV链的增强版本，我们证明了松弛时间界为$O(n^3\log n)$，并表明该长度的平稳轨迹估计了所有平稳空洞模式概率，从而得到一个$\widetilde O(n^5)$时间的FPRAS算法。我们新的空洞加权滑动（HWS）链将两个界都改进到$O(n^2\log n)$，从而得到一个$\widetilde O(n^4)$时间的算法。最后，我们通过在一系列迭代的热启动序列中使用$\widetilde{O}(\sqrt{n})$个检查点温度子集来获得每个温度的初始化，从而实现了所声称的$\widetilde O(n^{3.5})$运行时间。
英文摘要
We give an FPRAS for the permanent of an $n\times n$ $0/1$ matrix with running time $\widetilde{O}(n^{3.5}\varepsilon^{-2})$. Our algorithm extends to a strongly polynomial FPRAS for arbitrary nonnegative matrices, as in previous works. Jerrum, Sinclair, and Vigoda (2004) gave the first FPRAS for the permanent of a nonnegative matrix. The running time was subsequently improved to $\widetilde{O}(n^7)$ by Bezáková, Štefankovič, Vazirani, and Vigoda (2008), and recently to $\widetilde{O}(n^6)$ by Chen, Vigoda, and Yang (2026). We introduce a multicommodity-flow bound inspired by electrical flows, replacing the usual path-length factor by routing energy. For a boosted version of the classical JSV chain, we prove a relaxation-time bound of $O(n^3\log n)$ and show that stationary trajectories of this length estimate all stationary hole-pattern probabilities, yielding an $\widetilde O(n^5)$-time FPRAS algorithm. Our new hole-weighted slide (HWS) chain improves both bounds to $O(n^2\log n)$, yielding an $\widetilde O(n^4)$-time algorithm. Finally, we obtain the claimed $\widetilde O(n^{3.5})$ running time by using a subset of $\widetilde{O}(\sqrt{n})$ checkpoint temperatures in an iterated sequence of warm-starts to obtain initializations at every temperature.
Comments44 pages
