An Optimal Algorithm for Stochastic Vertex Cover
Jan van den Brand, Inge Li Gørtz, Chirag Pabbaraju, Debmalya Panigrahi, Clifford Stein, Miltiadis Stouras, Ola Svensson, Ali Vakilian
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.