AI 中文总结
该研究针对非负矩阵永久值，通过识别并剥离4-环等结构修正Bethe近似，得到了确定性多项式时间的(√2−ε)^n近似算法，突破了原有(√2)^n的界。
AI 中文摘要
我们研究通过Bethe永久值（一种可在多项式时间内计算的近似值）对非负矩阵永久值的确定性近似算法。Anari和Rezaei的紧分析给出了永久值与Bethe永久值在因子(√2)^n内的通用比较。未加权4-环C₄（或不相交C₄的并）的简单例子表明该界是紧的。我们证明这类4-环障碍可通过算法识别和利用。给定一个Bethe优化器，我们的算法识别近乎孤立的加权2×2块，并剥离其中顶点不相交的族。若总加权修正量较大，可改进Bethe近似；若较小，则证明Bethe永久值与真实值的因子在(√2−ε)^n内，其中ε>0为某绝对常数。结合这些结论，我们得到针对任意n×n非负矩阵永久值的确定性多项式时间(√2−ε)^n近似算法。
英文摘要
We study deterministic approximation algorithms for the permanent of a nonnegative matrix through the Bethe permanent, an approximation computable in polynomial time. The tight analysis of Anari and Rezaei gives a universal comparison between the permanent and the Bethe permanent within a factor $(\sqrt 2)^n$. The simple example of the unweighted $4$-cycle $C_4$ (or a union of disjoint $C_4$'s) shows that this bound is tight. We show that such $4$-cycle obstructions can be identified and exploited algorithmically. Given a Bethe optimizer, our algorithm identifies nearly isolated weighted $2\times2$ blocks and peels off a vertex-disjoint family of them. If the total weighted correction is large, we can improve the Bethe approximation; if it is small, we show that the Bethe permanent is within a factor of $(\sqrt2 - \varepsilon)^n$ of the truth. Combining these facts, we obtain a deterministic polynomial time $(\sqrt2-\varepsilon)^n$-approximation algorithm for the permanent of an arbitrary nonnegative $n\times n$ matrix, where $\varepsilon>0$ is some absolute constant.
