Simple Round Compression for Parallel Vertex Cover

TL;DR

This paper presents a simplified parallel algorithm for approximating the minimum vertex cover with a logarithmic approximation factor, achieving significantly fewer MPC rounds than previous methods by adapting and compressing existing distributed algorithms.

Contribution

It introduces a simpler MPC algorithm for vertex cover that compresses multiple rounds into fewer rounds, with a trade-off in approximation quality.

Findings

Achieves an O(log n) approximation in O(log log n) MPC rounds.

Simplifies analysis compared to previous algorithms for vertex cover.

Reduces complexity by avoiding intricate steps of prior methods.

Abstract

Recently, Czumaj et.al. (arXiv 2017) presented a parallel (almost) $2$-approximation algorithm for the maximum matching problem in only $O({(\log\log{n})^2})$ rounds of the massive parallel computation (MPC) framework, when the memory per machine is $O(n)$. The main approach in their work is a way of compressing $O(\log{n})$ rounds of a distributed algorithm for maximum matching into only $O({(\log\log{n})^2})$ MPC rounds. In this note, we present a similar algorithm for the closely related problem of approximating the minimum vertex cover in the MPC framework. We show that one can achieve an $O(\log{n})$ approximation to minimum vertex cover in only $O(\log\log{n})$ MPC rounds when the memory per machine is $O(n)$. Our algorithm for vertex cover is similar to the maximum matching algorithm of Czumaj et.al. but avoids many of the intricacies in their approach and as a result admits a considerably simpler analysis (at a cost of a worse approximation guarantee). We obtain this result by modifying a previous parallel algorithm by Khanna and the author (SPAA 2017) for vertex cover that allowed for compressing $O(\log{n})$ rounds of a distributed algorithm into constant MPC rounds when the memory allowed per machine is $O(n\sqrt{n})$.