Subexponential Approximation of the Permanent in Deterministic Polynomial Time

2026-09-09

AI Summary

本文提出了首个在确定性多项式时间内以次指数因子近似任意非负有理矩阵的permanent值的方法，使用了凸优化和相关衰减技术。

Abstract

We give the first deterministic polynomial time algorithm that approximates the permanent of arbitrary nonnegative rational matrices within a subexponential factor. For a matrix of order $n$, the approximation factor is \[ \exp\!\left(O\!\left(\frac{n(\log\log n)^2}{\log n}\right)\right)=\exp(o(n)). \] All previously known deterministic polynomial time guarantees for unrestricted inputs had approximation factors $\exp(Ω(n))$. Our proof uses convex optimization to tighten an upper bound on the permanent. The bound is based on weighted sums over all matchings in a bipartite graph representing the matrix, and correlations between unmatched vertices control its error. We approximate these sums deterministically using correlation decay and a bound on the effect of vertex deletion.

Problem

• deterministic polynomial time
• approximation
• permanent
• subexponential factor

Innovation

• subexponential approximation
• deterministic polynomial time
• convex optimization
• correlation decay