An Optimal Algorithm for Stochastic Vertex Cover

TL;DR

This paper presents an optimal algorithm for the stochastic vertex cover problem, achieving a near-minimum approximation with a query complexity matching the known lower bound, and introduces a new concentration bound for random graphs.

Contribution

The authors develop a $(1+ ext{epsilon})$-approximate algorithm with $O_ ext{epsilon}(n/p)$ queries, resolving a major open question and improving upon prior results.

Findings

Achieved a $(1+ ext{epsilon})$-approximation with optimal $O_ ext{epsilon}(n/p)$ queries.

Matched the known lower bound for query complexity in this problem.

Introduced a new concentration bound for the size of minimum vertex cover in random graphs.

Abstract

The goal in the stochastic vertex cover problem is to obtain an approximately minimum vertex cover for a graph $G^\star$ that is realized by sampling each edge independently with some probability $p\in (0, 1]$ in a base graph $G = (V, E)$. The algorithm is given the base graph $G$ and the probability $p$ as inputs, but its only access to the realized graph $G^\star$ is through queries on individual edges in $G$ that reveal the existence (or not) of the queried edge in $G^\star$. In this paper, we resolve the central open question for this problem: to find a $(1+\varepsilon)$-approximate vertex cover using only $O_\varepsilon(n/p)$ edge queries. Prior to our work, there were two incomparable state-of-the-art results for this problem: a $(3/2+\varepsilon)$-approximation using $O_\varepsilon(n/p)$ queries (Derakhshan, Durvasula, and Haghtalab, 2023) and a $(1+\varepsilon)$-approximation using $O_\varepsilon((n/p)\cdot \mathrm{RS}(n))$ queries (Derakhshan, Saneian, and Xun, 2025), where $\mathrm{RS}(n)$ is known to be at least $2^{\Omega\left(\frac{\log n}{\log \log n}\right)}$ and could be as large as $\frac{n}{2^{\Theta(\log^* n)}}$. Our improved upper bound of $O_{\varepsilon}(n/p)$ matches the known lower bound of $\Omega(n/p)$ for any constant-factor approximation algorithm for this problem (Behnezhad, Blum, and Derakhshan, 2022). A key tool in our result is a new concentration bound for the size of minimum vertex cover on random graphs, which might be of independent interest.