An Optimal Algorithm for Stochastic Vertex Cover

2026-03-29

AI Summary

This work addresses the stochastic vertex cover problem, where the target graph is generated by independently sampling each edge of a base graph with probability $p$, and the algorithm can only access the graph via edge queries. The paper proposes an adaptive query strategy that achieves a $(1+\varepsilon)$-approximation to the minimum vertex cover using only $O_\varepsilon(n/p)$ queries. This result is the first to attain a $(1+\varepsilon)$-approximation with query complexity tightly matching the known lower bound of $\Omega(n/p)$, thereby overcoming the longstanding trade-off between approximation ratio and query complexity. A key technical contribution is a new concentration inequality for the size of the minimum vertex cover, which significantly improves upon existing approaches.

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 $2^{Ω\left(\frac{\log n}{\log \log n}\right)}$ and could be as large as $\frac{n}{2^{Θ(\log^* n)}}$. Our improved upper bound of $O_{\varepsilon}(n/p)$ matches the known lower bound of $Ω(n/p)$ 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.

Problem

• stochastic vertex cover
• random graph
• edge queries
• approximation algorithm
• vertex cover

Innovation

• stochastic vertex cover
• query complexity
• approximation algorithm
• concentration bound
• random graph