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- W4298162536 abstract "This paper 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 $F subseteq E$ with the objective of minimizing the cardinality (or cost in the weighted case) of $F$. We consider a parameterized relaxation of this problem, where given some $0 leq epsilon < 1$, the goal is to construct an edge $(1 - epsilon)$-cover, namely, a subset of edges incident to all but an $epsilon$-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 trade-off between $epsilon$ and the approximation ratio: We design a semi-streaming algorithm that on input graph $G$, constructs a succinct data structure $mathcal{D}$ such that for every $0 leq epsilon < 1$, an edge $(1 - epsilon)$-cover that approximates the optimal edge mbox{($1$-)cover} within a factor of $f(epsilon, n)$ can be extracted from $mathcal{D}$ (efficiently and with no additional space requirements), where [ f(epsilon, n) = left{ begin{array}{ll} O (1 / epsilon), & text{if } epsilon > 1 / sqrt{n} O (sqrt{n}), & text{otherwise} end{array} right. , . ] In particular for the traditional set cover problem we obtain an $O(sqrt{n})$-approximation. This algorithm is proved to be best possible by establishing a family (parameterized by $epsilon$) of matching lower bounds." @default.
- W4298162536 created "2022-10-01" @default.
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- W4298162536 date "2014-04-27" @default.
- W4298162536 modified "2023-10-17" @default.
- W4298162536 title "Semi-Streaming Set Cover" @default.
- W4298162536 doi "https://doi.org/10.48550/arxiv.1404.6763" @default.
- W4298162536 hasPublicationYear "2014" @default.
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