A Deterministic Distributed $2$-Approximation for Weighted Vertex Cover in $O(\log n\log\Delta / \log^2\log\Delta)$ Rounds

TL;DR

This paper introduces a faster deterministic distributed algorithm that approximates the minimum weight vertex cover within a factor of 2, improving previous methods in round complexity and approximation dependency.

Contribution

It provides a deterministic distributed 2-approximation algorithm with improved round complexity and generalizes the $(2+ ext{epsilon})$-approximation with better epsilon dependency.

Findings

Achieves $O(rac{ ext{log} n ext{log} ext{Delta}}{ ext{log}^2 ext{log} ext{Delta}})$ round complexity.

Improves approximation dependency from linear to logarithmic in epsilon.

Provides asymptotically optimal $(2+ ext{epsilon})$-approximation in $O(rac{ ext{log} ext{Delta}}{ ext{log} ext{log} ext{Delta}})$ rounds.

Abstract

We present a deterministic distributed $2$-approximation algorithm for the Minimum Weight Vertex Cover problem in the CONGEST model whose round complexity is $O(\log n \log \Delta / \log^2 \log \Delta)$. This improves over the currently best known deterministic 2-approximation implied by [KVY94]. Our solution generalizes the $(2+\epsilon)$-approximation algorithm of [BCS17], improving the dependency on $\epsilon^{-1}$ from linear to logarithmic. In addition, for every $\epsilon=(\log \Delta)^{-c}$, where $c\geq 1$ is a constant, our algorithm computes a $(2+\epsilon)$-approximation in $O(\log \Delta / \log \log \Delta)$~rounds (which is asymptotically optimal).