Semi-Streaming Set Cover

Abstract

This article studies the set cover problem under the semi-streaming model. The underlying set system is formalized in terms of a hypergraph G = ( V , E ) whose edges arrive one by one, and the goal is to construct an edge cover F ⊆ E with the objective of minimizing the cardinality (or cost in the weighted case) of F . We further consider a parameterized relaxation of this problem, where, given some 0 ⩽ ϵ < 1, the goal is to construct an edge (1 − ϵ)-cover, namely, a subset of edges incident to all but an ϵ-fraction of the vertices (or their benefit in the weighted case). The key limitation imposed on the algorithm is that its space is limited to (poly)logarithmically many bits per vertex. Our main result is an asymptotically tight tradeoff between ϵ and the approximation ratio: We design a semi-streaming algorithm that on input hypergraph G constructs a succinct data structure D such that for every 0 ⩽ ϵ < 1, an edge (1 − ϵ)-cover that approximates the optimal edge (1-)cover within a factor of f (ϵ, n ) can be extracted from D (efficiently and with no additional space requirements), where f (ϵ, n ) = { O (1/ ϵ ), if ϵ > 1/√ n O (√ n ), otherwise . In particular, for the traditional set cover problem, we obtain an O (√ n -approximation. This algorithm is proved to be best possible by establishing a family (parameterized by ϵ) of matching lower bounds.