Distributed CONGEST Approximation of Weighted Vertex Covers and Matchings
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We provide CONGEST model algorithms for approximating minimum weighted vertex cover and the maximum weighted matching. For bipartite graphs, we show that a (1+ε)-approximate weighted vertex cover can be computed deterministically in polylogarithmic time. This generalizes a corresponding result for the unweighted vertex cover problem shown in [Faour, Kuhn; OPODIS '20]. Moreover, we show that in general weighted graph families that are closed under taking subgraphs and in which we can compute an independent set of weight at least a λ-fraction of the total weight, one can compute a (2-2λ +ε)-approximate weighted vertex cover in polylogarithmic time in the CONGEST model. Our result in particular implies that in graphs of arboricity a, one can compute a (2-1/a+ε)-approximate weighted vertex cover. For maximum weighted matchings, we show that a (1-ε)-approximate solution can be computed deterministically in polylogarithmic CONGEST rounds (for constant ε). We also provide a more efficient randomized algorithm. Our algorithm generalizes results of [Lotker, Patt-Shamir, Pettie; SPAA '08] and [Bar-Yehuda, Hillel, Ghaffari, Schwartzman; PODC '17] for the unweighted case. Finally, we show that even in the LOCAL model and in bipartite graphs of degree ≤ 3, if ε<ε_0 for some constant ε_0>0, then computing a (1+ε)-approximation for the unweighted minimum vertex cover problem requires Ω(log n/ε) rounds. This generalizes aresult of [Göös, Suomela; DISC '12], who showed that computing a (1+ε_0)-approximation in such graphs requires Ω(log n) rounds.
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