A Distributed $(2+\epsilon)$-Approximation for Vertex Cover in $O(\log{\Delta}/\epsilon\log\log{\Delta})$ Rounds

TL;DR

This paper introduces a simple deterministic distributed algorithm that achieves a near-optimal approximation for the minimum weight vertex cover problem within a logarithmic number of rounds, challenging previous lower bounds.

Contribution

The paper presents a novel distributed $(2+ ext{epsilon})$-approximation algorithm for vertex cover with improved round complexity, valid for any epsilon > 0.

Findings

Achieves $(2+ ext{epsilon})$-approximation in $O(rac{ ext{log}\Delta}{ ext{epsilon} ext{log} ext{log}\Delta})$ rounds.

Constant approximation in $O(rac{ ext{log}\Delta}{ ext{log} ext{log}\Delta})$ rounds for fixed epsilon.

Contradicts existing lower bounds for certain parameter ranges.

Abstract

We present a simple deterministic distributed $(2+\epsilon)$-approximation algorithm for minimum weight vertex cover, which completes in $O(\log{\Delta}/\epsilon\log\log{\Delta})$ rounds, where $\Delta$ is the maximum degree in the graph, for any $\epsilon>0$ which is at most $O(1)$. For a constant $\epsilon$, this implies a constant approximation in $O(\log{\Delta}/\log\log{\Delta})$ rounds, which contradicts the lower bound of [KMW10].