Distributed CONGEST Approximation of Weighted Vertex Covers and Matchings

Abstract

We provide CONGEST model algorithms for approximating the minimum weighted vertex cover and the maximum weighted matching problem. For bipartite graphs, we show that a (1 + ε)-approximate weighted vertex cover can be computed deterministically in poly log n ε rounds. 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 λ • w(V ) (where w(V ) denotes the total weight of all nodes) in polylogarithmic time in the CONGEST model, one can compute a (2 − 2λ + ε)-approximate weighted vertex cover in poly log n ε rounds 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 problem in poly log n ε rounds in the CONGEST model. For maximum weighted matchings, we show that a (1 − ε)-approximate solution can be computed deterministically in time 2 O(1/ε) • poly log n in the CONGEST model. We also provide a randomized algorithm that with arbitrarily good constant probability succeeds in computing a (1 − ε)-approximate weighted matching in time 2 O(1/ε) • poly log(∆W ) • log * n, where W denotes the ratio between the largest and the smallest edge weight. Our algorithm generalizes results of [Lotker, Patt-Shamir, Pettie; SPAA '08] and [Bar-Yehuda, Hillel, Ghaffari, Schwartzman; PODC '17], who gave 2 O(1/ε) • log n and 2 O(1/ε) • log ∆ log log ∆ -round randomized approximations for the unweighted matching problem.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 a result of [Göös, Suomela; DISC '12], who showed that computing a (1 + ε 0 )-approximation in such graphs requires Ω(log n) rounds.