$(1-\epsilon)$-Approximate Maximum Weighted Matching in $\text{poly}(1/\epsilon, \log n)$ Time in the Distributed and Parallel Settings

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 the unweighted maximum matching, it had been an open problem whether a $(1-\epsilon)$-approximate MWM can be obtained in $\text{poly}(\log n, 1/\epsilon)$ 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.