(1-ε)-Approximate Maximum Weighted Matching in poly(1/ε, log n) Time in the Distributed and Parallel Settings
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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 poly(1/ϵ, log n)-round algorithm for obtaining a (1-ϵ)-approximate solution for the unweighted maximum matching, it had been an open problem whether a (1-ϵ)-approximate MWM can be obtained in poly(log n, 1/ϵ) 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/ϵ) rounds for obtaining a (1-ϵ)-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 poly(1/ϵ, log n)-round algorithm for computing a (1-ϵ)-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 poly(1/ϵ, log n) span using only O(m) processors.
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