We consider the maximum weight $b$-matching problem in the random-order semi-streaming model. Assuming all weights are small integers drawn from $[1,W]$, we present a $2 - \frac{1}{2W} + \varepsilon$ approximation algorithm, using a memory of $O(\max(|M_G|, n) \cdot poly(\log(m),W,1/\varepsilon))$, where $|M_G|$ denotes the cardinality of the optimal matching. Our result generalizes that of Bernstein [Bernstein, 2015], which achieves a $3/2 + \varepsilon$ approximation for the maximum cardinality simple matching. When $W$ is small, our result also improves upon that of Gamlath et al. [Gamlath et al., 2019], which obtains a $2 - \delta$ approximation (for some small constant $\delta \sim 10^{-17}$) for the maximum weight simple matching. In particular, for the weighted $b$-matching problem, ours is the first result beating the approximation ratio of $2$. Our technique hinges on a generalized weighted version of edge-degree constrained subgraphs, originally developed by Bernstein and Stein [Bernstein and Stein, 2015]. Such a subgraph has bounded vertex degree (hence uses only a small number of edges), and can be easily computed. The fact that it contains a $2 - \frac{1}{2W} + \varepsilon$ approximation of the maximum weight matching is proved using the classical K\H{o}nig-Egerv\'ary's duality theorem.

翻译：我们考虑随机顺序半流模型中的最大权重$b$-匹配问题。假设所有权重均为取自$[1,W]$的小整数，我们提出一种$2 - \frac{1}{2W} + \varepsilon$近似算法，其内存使用量为$O(\max(|M_G|, n) \cdot poly(\log(m),W,1/\varepsilon))$，其中$|M_G|$表示最优匹配的基数。该结果推广了Bernstein [Bernstein, 2015]的结论——后者针对最大基数简单匹配实现了$3/2 + \varepsilon$近似。当$W$较小时，我们的结果也优于Gamlath等人 [Gamlath et al., 2019]的结论——其对最大权重简单匹配可达到$2 - \delta$近似（其中$\delta \sim 10^{-17}$为小常数）。特别地，对加权$b$-匹配问题而言，这是首个突破近似比$2$的结果。我们的技术核心在于对Bernstein和Stein [Bernstein and Stein, 2015]提出的边度约束子图进行加权版本的推广。此类子图顶点度数有界（因此仅使用少量边），且易于计算。利用经典Kőnig-Egerváry对偶定理可证明该子图包含最大权重匹配的$2 - \frac{1}{2W} + \varepsilon$近似。