For an integer $b\ge 0$, a $b$-matching in a graph $G=(V,E)$ is a set $S\subseteq E$ such that each vertex $v\in V$ is incident to at most $b$ edges in $S$. We design a fully polynomial-time approximation scheme (FPTAS) for counting the number of $b$-matchings in graphs with bounded degrees. Our FPTAS also applies to a broader family of counting problems, namely Holant problems with log-concave signatures. Our algorithm is based on Moitra's linear programming approach (JACM'19). Using a novel construction called the extended coupling tree, we derandomize the coupling designed by Chen and Gu (SODA'24).

翻译：对于整数$b\ge 0$，图$G=(V,E)$中的$b$-匹配是指边集的一个子集$S\subseteq E$，使得每个顶点$v\in V$在$S$中最多关联$b$条边。我们为度数有界图中$b$-匹配的计数问题设计了一个完全多项式时间近似方案（FPTAS）。我们的FPTAS也适用于更广泛的计数问题族，即具有对数凹符号的Holant问题。我们的算法基于Moitra的线性规划方法（JACM'19）。通过使用一种称为扩展耦合树的新颖构造，我们对Chen和Gu（SODA'24）设计的耦合进行了去随机化处理。