发表机构
California Institute of Technology; University of Illinois Chicago（加州理工学院; 伊利诺伊大学芝加哥分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对非负矩阵永久值的Bethe近似，研究人员证实Anari的猜想，给出了基于二分支撑图围长g（g≥4）的最优界，细化了原有通用比较关系，该上界在g-环不交并邻接矩阵上达到。
AI 中文摘要
对于一个n×n非负矩阵A，可在确定性多项式时间内计算的Bethe永久值满足严格的通用比较关系：Bethe(A) ≤ per(A) ≤ 2^{n/2} Bethe(A)。由Gurvits给出的下界在森林上达到，由Anari和Rezaei给出的上界在4-环不交并的邻接矩阵上达到。我们证实了Anari的一个猜想，对上述比较关系提供了最优的围长相关细化。更准确地说，我们证明若A的二分支撑图的围长至少为偶整数g≥4，则Bethe(A) ≤ per(A) ≤ 2^{2n/g} Bethe(A)，该上界在g-环不交并的邻接矩阵上达到。
英文摘要
For an $n\times n$ nonnegative matrix $A$, the Bethe permanent, which is computable in deterministic polynomial time, satisfies the tight universal comparison \[\operatorname{Bethe}(A) \leq \operatorname{per}(A) \leq 2^{n/2}\operatorname{Bethe}(A).\] The lower bound, due to Gurvits, is attained on forests. The upper bound, due to Anari and Rezaei, is attained by the adjacency matrix of a disjoint union of $4$-cycles. Confirming a conjecture of Anari, we provide an optimal girth-dependent refinement of the above comparison. More precisely, we show that if the bipartite support graph of $A$ has girth at least an even integer $g \geq 4$, then \[\operatorname{Bethe}(A) \leq \operatorname{per}(A) \leq 2^{2n/g}\operatorname{Bethe}(A).\] The upper bound is attained by the adjacency matrix of a disjoint union of $g$-cycles.
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