Distributed CONGEST Approximation of Weighted Vertex Covers and Matchings

Abstract: We provide CONGEST model algorithms for approximating minimum weighted vertex cover and the maximum weighted matching. For bipartite graphs, we show that a $(1+\varepsilon)$-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 $\lambda$-fraction of the total weight, one can compute a $(2-2\lambda +\varepsilon)$-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 $(2-1/a+\varepsilon)$-approximate weighted vertex cover. For maximum weighted matchings, we show that a $(1-\varepsilon)$-approximate solution can be computed deterministically in polylogarithmic CONGEST rounds (for constant $\varepsilon$). 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 $\leq 3$, if $\varepsilon<\varepsilon_0$ for some constant $\varepsilon_0>0$, then computing a $(1+\varepsilon)$-approximation for the unweighted minimum vertex cover problem requires $\Omega\big(\frac{\log n}{\varepsilon}\big)$ rounds. This generalizes aresult of [G\"o\"os, Suomela; DISC '12], who showed that computing a $(1+\varepsilon_0)$-approximation in such graphs requires $\Omega(\log n)$ rounds.