Stochastic Minimum Vertex Cover in General Graphs: a $3/2$-Approximation

TL;DR

This paper presents a new algorithm for stochastic minimum vertex cover in general graphs that achieves a better approximation ratio of 3/2, using fewer queries, and extends to correlated edge realizations.

Contribution

It introduces a 3/2-approximation algorithm for stochastic vertex cover with fewer queries and analyzes its performance under correlated edge realizations.

Findings

Achieves a 3/2-approximation ratio with $O(n/initepsilon p)$ queries.

Improves over the previous 2-approximation algorithm.

Provides a tight lower bound for correlated stochastic graphs.

Abstract

Our main result is designing an algorithm that returns a vertex cover of $\mathcal{G}^\star$ with size at most $(3/2+\epsilon)$ times the expected size of the minimum vertex cover, using only $O(n/\epsilon p)$ non-adaptive queries. This improves over the best-known 2-approximation algorithm by Behnezhad, Blum, and Derakhshan [SODA'22], who also show that $\Omega(n/p)$ queries are necessary to achieve any constant approximation. Our guarantees also extend to instances where edge realizations are not fully independent. We complement this upper bound with a tight $3/2$-approximation lower bound for stochastic graphs whose edges realizations demonstrate mild correlations.