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The goal in the stochastic vertex cover problem is to obtain an approximately minimum vertex cover for a graph G⋆ that is realized by sampling each edge independently with some probability p∈ (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⋆ is through queries on individual edges in G that reveal the existence (or not) of the queried edge in G⋆. In this paper, we resolve the central open question for this problem: to find a (1+ε)-approximate vertex cover using only Oε(n/p) edge queries. Prior to our work, there were two incomparable state-of-the-art results for this problem: a (3/2+ε)-approximation using Oε(n/p) queries (Derakhshan, Durvasula, and Haghtalab, 2023) and a (1+ε)-approximation using Oε((n/p)· RS(n)) queries (Derakhshan, Saneian, and Xun, 2025), where RS(n) is known to be at least 2Ω(logn/loglogn) and could be as large as n/2Θ(log* n). Our improved upper bound of Oε(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.
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Stochastic Minimum Vertex Cover in General Graphs: A 3/2-Approximation
We study the stochastic vertex cover problem. In this problem, G = (V, E) is an arbitrary known graph, and G⋆ is an unknown random subgraph of G where each edge e is realized independently with probability p. Edges of G⋆ can only be verified using edge queries. The goal in this problem is to find a minimum vertex cover of G⋆ using a small number of queries. Our main result is designing an algorithm that returns a vertex cover of G⋆ with size at most (3/2+є) times the expected size of the minimum vertex cover, using only O(n/є p) non-adaptive queries. This improves over the best-known 2-approximation algorithm by Behnezhad, Blum and Derakhshan [SODA’22] who also show that Ω(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 upperbound with a tight 3/2-approximation lower bound for stochastic graphs whose edges realizations demonstrate mild correlations.
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- Award ID(s):
- 2145898
- PAR ID:
- 10482192
- Publisher / Repository:
- ACM
- Date Published:
- ISBN:
- 9781450399135
- Page Range / eLocation ID:
- 242 to 253
- Format(s):
- Medium: X
- Location:
- Orlando FL USA
- Sponsoring Org:
- National Science Foundation
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