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

TL;DR

This paper presents new randomized distributed algorithms for approximate maximum matching and vertex cover, demonstrating that their complexity depends on the number of nodes, not just the maximum degree, using a novel graph decomposition.

Contribution

It introduces a generalized graph decomposition technique that enables efficient approximation algorithms, challenging previous assumptions about lower bounds.

Findings

Algorithms achieve $O(rac{ ext{log} n}{ ext{log}^2 ext{log} n})$ rounds for 2+ε approximate solutions.

The new decomposition reduces the effective degree of high-degree graphs.

The decomposition technique may be useful for other distributed graph problems.

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 $\Omega(\min\{\sqrt{\frac{\log n}{\log\log n}}, \frac{\log \Delta}{\log\log \Delta}\})$ rounds, for any polylogarithmic or smaller approximation ratio. As a function of $\Delta$, 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 $\Theta(\frac{\log \Delta}{\log\log \Delta})$, 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.