SIAM Journal on Discrete Mathematics, Volume 35, Issue 2, Page 1503-1524, January 2021.
Jerrum, Sinclair, and Vigoda [J. ACM, 51 (2004), pp. 671--697] showed that the permanent of any square matrix can be estimated in polynomial time. This computation can be viewed as approximating the partition function of edge-weighted matchings in a bipartite graph. Equivalently, this may be viewed as approximating the partition function of vertex-weighted independent sets in the line graph of a bipartite graph. Line graphs of bipartite graphs are perfect graphs and are known to be precisely the class of (claw, diamond, odd hole)-free graphs. So how far does the result of Jerrum, Sinclair, and Vigoda extend? We first show that it extends to (claw, odd hole)-free graphs, and then show that it extends to the even larger class of (fork, odd hole)-free graphs. Our techniques are based on graph decompositions, which have been the focus of much recent work in structural graph theory, and on structural results of Chvátal and Sbihi [J. Combin. Theory Ser. B, 44 (1988)], Maffray and Reed [J. Combin. Theory Ser. B, 75 (1999)], and Lozin and Milanič [J. Discrete Algorithms, 6 (2008), pp. 595--604].

中文翻译：

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计算永久以外的加权独立集

SIAM 离散数学杂志，第 35 卷，第 2 期，第 1503-1524 页，2021 年 1 月。
Jerrum、Sinclair 和 Vigoda [J. ACM, 51 (2004), pp. 671--697] 表明可以在多项式时间内估计任何方阵的永久。这种计算可以看作是对二部图中边加权匹配的分配函数的近似。等效地，这可以看作是对二部图的折线图中顶点加权独立集的分区函数的逼近。二部图的折线图是完美图，并且已知是无（爪形、菱形、奇数孔）的图类。那么 Jerrum、Sinclair 和 Vigoda 的结果延伸到什么程度呢？我们首先证明它扩展到无（爪，奇孔）图，然后证明它扩展到更大的无（叉，奇孔）图。我们的技术基于图分解，这一直是结构图论领域近期工作的重点，以及 Chvátal 和 Sbihi 的结构结果 [J. 结合。理论系列 B, 44 (1988)]，Maffray 和 Reed [J. 结合。理论系列 B, 75 (1999)]，以及 Lozin 和 Milanič [J. 离散算法，6 (2008)，第 595--604 页]。