Given a bipartite graph $G(V= (A \cup B),E)$ with $n$ vertices and $m$ edges and a function $b \colon V \to \mathbb{Z}_+$, a $b$-matching is a subset of edges such that every vertex $v \in V$ is incident to at most $b(v)$ edges in the subset. When we are also given edge weights, the Max Weight $b$-Matching problem is to find a $b$-matching of maximum weight, which is a fundamental combinatorial optimization problem with many applications. Extending on the recent work of Zheng and Henzinger (IPCO, 2023) on standard bipartite matching problems, we develop a simple auction algorithm to approximately solve Max Weight $b$-Matching. Specifically, we present a multiplicative auction algorithm that gives a $(1 - \varepsilon)$-approximation in $O(m \varepsilon^{-1} \log \varepsilon^{-1} \log \beta)$ worst case time, where $\beta$ the maximum $b$-value. Although this is a $\log \beta$ factor greater than the current best approximation algorithm by Huang and Pettie (Algorithmica, 2022), it is considerably simpler to present, analyze, and implement.

翻译：给定一个二分图$G(V= (A \cup B),E)$，其包含$n$个顶点和$m$条边，以及函数$b \colon V \to \mathbb{Z}_+$，$b$-匹配是边的一个子集，其中每个顶点$v \in V$在该子集中最多与$b(v)$条边相关联。当给定边权重时，最大权重$b$-匹配问题旨在寻找一个权重最大的$b$-匹配，这是一个具有众多应用的基础组合优化问题。本文基于Zheng和Henzinger（IPCO, 2023）在标准二分图匹配问题上的近期工作，提出了一种简单的拍卖算法来近似求解最大权重$b$-匹配。具体而言，我们提出了一种乘性拍卖算法，在最坏情况下时间复杂度为$O(m \varepsilon^{-1} \log \varepsilon^{-1} \log \beta)$，可实现$(1 - \varepsilon)$-近似，其中$\beta$为最大$b$值。尽管该算法的时间复杂度比当前由Huang和Pettie（Algorithmica, 2022）提出的最佳近似算法多了一个$\log \beta$因子，但它在表述、分析和实现上均显著更为简单。