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A Threshold of ln n for Approximating Set Cover (1998)
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| Venue: | JOURNAL OF THE ACM |
| Citations: | 776 - 5 self |
BibTeX
@ARTICLE{Feige98athreshold,
author = {Uriel Feige},
title = {A Threshold of ln n for Approximating Set Cover},
journal = {JOURNAL OF THE ACM},
year = {1998},
volume = {45},
pages = {314--318}
}
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Abstract
Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max k-cover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NP-hard. We prove that (1 \Gamma o(1)) ln n is a threshold below which set cover cannot be approximated efficiently, unless NP has slightly superpolynomial time algorithms. This closes the gap (up to low order terms) between the ratio of approximation achievable by the greedy algorithm (which is (1 \Gamma o(1)) ln n), and previous results of Lund and Yannakakis, that showed hardness of approximation within a ratio of (log 2 n)=2 ' 0:72 lnn. For max k-cover we show an approximation threshold of (1 \Gamma 1=e) (up to low order terms), under the assumption that P != NP .
