AI 中文总结
提出首个确定性多项式时间算法，通过凸优化和相关性衰减，在次指数因子内近似任意非负矩阵的积和式，显著改进先前指数因子。
AI 中文摘要
我们给出了第一个确定性多项式时间算法，该算法在次指数因子内近似任意非负有理矩阵的积和式。对于阶数为 $n$ 的矩阵，近似因子为 \\[ \exp\\!\left(O\\!\left(\frac{n(\log\log n)^2}{\log n}\right)\right)=\exp(o(n)). \\] 所有先前已知的针对无限制输入的确定性多项式时间保证的近似因子均为 $\exp(\Omega(n))$。我们的证明使用凸优化来收紧积和式的上界。该上界基于表示矩阵的二部图中所有匹配的加权和，而未匹配顶点之间的相关性控制其误差。我们使用相关性衰减和顶点删除影响的界来确定性近似这些和。
英文摘要
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.
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