Distributed Approximate Maximum Matching and Minimum Vertex Cover via Generalized Graph Decomposition

Abstract

The classic lower bound of Kuhn, Moscibroda and Wattenhofer [JACM 2016] states that approximate maximum matching and approximate vertex cover (among other problems) in the LOCAL model require $\hat{I}\text{c◯}(\min \{\sqrt{\frac{\log n}{\log \log n}},\frac{\log \hat{I}”}{\log \log \hat{I}”}\})$ rounds, for any polylogarithmic or smaller approximation ratio. As a function of $\hat{I}”$, this complexity was subsequently matched for constant-approximate weighted vertex cover [Bar-Yehuda, Censor-Hillel and Schwartzman, JACM 2017] and constant-approximate maximum matching [Bar-Yehuda, Censor-Hillel, Ghaffari and Schwartzman, PODC 2017]. One might expect, therefore, that the true complexity should be $\hat{I}˜(\frac{\log \hat{I}”}{\log \log \hat{I}”})$, and the $n$-dependent term in the lower bound is just an artefact of the proof method. We show that this is not the case, and a term dependent on $n$ is in fact required. Specifically, we show randomized algorithms for $2+\epsilon$-approximate maximum matching and approximate (weighted) minimum vertex cover taking $O(\frac{\log n}{{\log }^2\log n})$ rounds. Our algorithms are based on a novel graph decomposition result generalizing the method of Miller, Peng and Xu [SPAA 2013], which we use to reduce the `effective' degree of high-degree graphs. We expect that this decomposition may be of further use for other problems.

To appear at PODC 2026