The maximum weighted matching (MWM) problem is one of the most well-studied combinatorial optimization problems in distributed graph algorithms. Despite a long development on the problem, and the recent progress of Fischer, Mitrovic, and Uitto [FMU22] who gave a $\text{poly}(1/\epsilon, \log n)$-round algorithm for obtaining a $(1-\epsilon)$-approximate solution for unweighted maximum matching, it had been an open problem whether a $(1-\epsilon)$-approximate MWM can be obtained in $\text{poly}(1/\epsilon, \log n)$ rounds in the CONGEST model. Algorithms with such running times were only known for special graph classes such as bipartite graphs [AKO18] and minor-free graphs [CS22]. For general graphs, the previously known algorithms require exponential in $(1/\epsilon)$ rounds for obtaining a $(1-\epsilon)$-approximate solution [FFK21] or achieve an approximation factor of at most 2/3 [AKO18]. In this work, we settle this open problem by giving a deterministic $\text{poly}(1/\epsilon, \log n)$-round algorithm for computing a $(1-\epsilon)$-approximate MWM for general graphs in the CONGEST model. Our proposed solution extends the algorithm of Fischer, Mitrovic, and Uitto [FMU22], blends in the sequential algorithm from Duan and Pettie [DP14] and the work of Faour, Fuchs, and Kuhn [FFK21]. Interestingly, this solution also implies a CREW PRAM algorithm with $\text{poly}(1/\epsilon, \log n)$ span using only $O(m)$ processors. In addition, with the reduction from Gupta and Peng [GP13], we further obtain a semi-streaming algorithm with $\text{poly}(1/\epsilon)$ passes. When $\epsilon$ is smaller than a constant $o(1)$ but at least $1/\log^{o(1)} n$, our algorithm is more efficient than both Ahn and Guha's $\text{poly}(1/\epsilon, \log n)$-passes algorithm [AG13] and Gamlath, Kale, Mitrovic, and Svensson's $(1/\epsilon)^{O(1/\epsilon^2)}$-passes algorithm [GKMS19].

翻译：最大权重匹配（MWM）问题是分布式图算法中研究最深入的组合优化问题之一。尽管该问题已有长期研究，且Fischer、Mitrovic和Uitto [FMU22]近期提出了在未加权最大匹配问题上获得$(1-\epsilon)$近似解的$\text{poly}(1/\epsilon, \log n)$轮算法，但在CONGEST模型中能否用$\text{poly}(1/\epsilon, \log n)$轮实现$(1-\epsilon)$近似MWM仍是一个未解决的问题。此前仅对二部图[AKO18]和无minor图[CS22]等特殊图类存在此类运行时间的算法。对于一般图，已知算法要么需要指数级$(1/\epsilon)$轮才能获得$(1-\epsilon)$近似解[FFK21]，要么最多达到2/3的近似因子[AKO18]。本文通过提出一个确定性$\text{poly}(1/\epsilon, \log n)$轮算法，解决了CONGEST模型下一般图$(1-\epsilon)$近似MWM计算这一开放问题。我们提出的方案扩展了Fischer、Mitrovic和Uitto [FMU22]的算法，融合了Duan与Pettie [DP14]的顺序算法以及Faour、Fuchs和Kuhn [FFK21]的工作。有趣的是，该方案还推导出一个仅需$O(m)$个处理器、跨度为$\text{poly}(1/\epsilon, \log n)$的CREW PRAM算法。此外，通过Gupta和Peng [GP13]的归约方法，我们进一步获得一个$\text{poly}(1/\epsilon)$轮次的半流式算法。当$\epsilon$小于常数$o(1)$但至少为$1/\log^{o(1)} n$时，我们的算法比Ahn与Guha的$\text{poly}(1/\epsilon, \log n)$轮次算法[AG13]和Gamlath、Kale、Mitrovic与Svensson的$(1/\epsilon)^{O(1/\epsilon^2)}$轮次算法[GKMS19]更高效。