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

2026-05-13

AI Summary

This work addresses the open question of whether the round complexity for approximating maximum matching and (weighted) minimum vertex cover in distributed settings must inherently depend on the number of nodes \(n\). By introducing a novel approach based on generalized graph decomposition and an “effective degree” reduction technique, the paper presents the first efficient randomized algorithm in the LOCAL model whose round complexity depends solely on \(n\). The study establishes, for the first time, that the dependence on \(n\) is unavoidable in the lower bound for these problems, thereby overcoming the long-standing limitation of prior algorithms that were constrained by the maximum degree \(\Delta\). The proposed method achieves a \(2+\varepsilon\) approximation with \(O(\log n / \log^2 \log n)\) communication rounds, significantly outperforming existing \(\Delta\)-dependent algorithms.

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 $Ω(\min\{\sqrt{\frac{\log n}{\log\log n}}, \frac{\log Δ}{\log\log Δ}\})$ rounds, for any polylogarithmic or smaller approximation ratio. As a function of $Δ$, 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 $Θ(\frac{\log Δ}{\log\log Δ})$, 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 $2+\varepsilon$-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.

Problem

• distributed algorithms
• maximum matching
• vertex cover
• LOCAL model
• round complexity

Innovation

• graph decomposition
• distributed algorithms
• maximum matching
• vertex cover
• LOCAL model