Stochastic Vertex Cover with Few Queries (2112.05415v1)

Abstract: We study the minimum vertex cover problem in the following stochastic setting. Let $G$ be an arbitrary given graph, $p \in (0, 1]$ a parameter of the problem, and let $G_p$ be a random subgraph that includes each edge of $G$ independently with probability $p$. We are unaware of the realization $G_p$, but can learn if an edge $e$ exists in $G_p$ by querying it. The goal is to find an approximate minimum vertex cover (MVC) of $G_p$ by querying few edges of $G$ non-adaptively. This stochastic setting has been studied extensively for various problems such as minimum spanning trees, matroids, shortest paths, and matchings. To our knowledge, however, no non-trivial bound was known for MVC prior to our work. In this work, we present a: * $(2+\epsilon)$-approximation for general graphs which queries $O(\frac{1}{\epsilon^3 p})$ edges per vertex, and a * $1.367$-approximation for bipartite graphs which queries $poly(1/p)$ edges per vertex. Additionally, we show that at the expense of a triple-exponential dependence on $p^{-1}$ in the number of queries, the approximation ratio can be improved down to $(1+\epsilon)$ for bipartite graphs. Our techniques also lead to improved bounds for bipartite stochastic matching. We obtain a $0.731$-approximation with nearly-linear in $1/p$ per-vertex queries. This is the first result to break the prevalent $(2/3 \sim 0.66)$-approximation barrier in the $poly(1/p)$ query regime, improving algorithms of [Behnezhad et al; SODA'19] and [Assadi and Bernstein; SOSA'19].