## A Threshold of ln n for Approximating Set Cover (1998)

Venue: | JOURNAL OF THE ACM |

Citations: | 653 - 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}

}

### Years of Citing Articles

### OpenURL

### 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 .

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Citation Context ...s.) The first hardness of approximation results for set cover followed from work on probabilistically checkable proof systems (PCPs). The notion of PCPs grew out of the theory 2 of interactive proofs =-=[14, 4, 6, 13]-=- (parts of which we will review shortly) and from major breakthroughs in understanding their power [25, 32, 3]. The relevance of interactive proofs for proving hardness of approximation results was de... |

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Citation Context ...over can be approximated within a ratio of ln n, where ln denotes the natural logarithm, and that max k-cover can be approximated within a ratio of 1 \Gamma 1=e ' 0:632. The results and techniques in =-=[1, 28]-=- imply that there is a constant ffi ! 1 such that it is NPhard to approximate max k-cover within a ration better than ffi . Lund and Yannakakis [26] showed (under a complexity assumption that will be ... |

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Citation Context ...Gamma 1=e ' 0:632. The results and techniques in [1, 28] imply that there is a constant ffi ! 1 such that it is NPhard to approximate max k-cover within a ration better than ffi . Lund and Yannakakis =-=[26]-=- showed (under a complexity assumption that will be presented in Section 1.1) that it is hard to approximate set cover within a ratio of (log n)=2, where log denotes logarithms in base 2. We extend th... |

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Citation Context ... be approximated within any constant ratio, and unless NP ae TIME(n O(loglog n) ) then set cover cannot be approximated within a ratio of log n=8. Improved analysis of two prover proof systems by Raz =-=[30]-=- implies that unless NP ae TIME(n O(loglogn) ) then set cover cannot be approximated within a ratio of log n=4, and that unless NP ae ZTIME(n O(loglogn) ) then set cover cannot be approximated within ... |

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Citation Context ...for chromatic number [11]. Tight constant factor hardness of approximation results were obtain for several problems in [18], including a threshold of 7=8 for MAX 3SAT. As for set cover, Raz and Safra =-=[31]-=- constructed new low error constant prover proof systems and used them to show that for some constant c ? 0, it is NP-hard to approximate set cover within a ratio of c log n. It is not known whether h... |

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Citation Context ...l other thresholds for approximation were discovered. Essentially tight O(n 1\Gammaffl ) hardness of approximation results where obtained for clique and independent set [17], and for chromatic number =-=[11]-=-. Tight constant factor hardness of approximation results were obtain for several problems in [18], including a threshold of 7=8 for MAX 3SAT. As for set cover, Raz and Safra [31] constructed new low ... |

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Citation Context ... cover was among the first problems for which approximation algorithms were analysed. Johnson [23] showed that the greedy algorithm gives an approximation ratio of ln n. (This was extended by Chvatal =-=[7]-=- to the weighted version of set cover.) Lovasz [24] showed that a linear programming relaxation approximates set cover within a ratio of ln n. In both cases, the authors were interested mainly in the ... |

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Citation Context ...d mainly in the leading term of the approximation ratio. Analysis of the low order terms of the approximation ratio was provided by Srinivasan [34] (for the linear programming approach) and by Slavik =-=[33]-=- (for the greedy algorithm). For max k-cover, the greedy algorithm gives an approximation ratio of 1 \Gamma 1=e (up to low order terms). See [20] and references therein, and also Proposition 11. (A si... |

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Citation Context ...ure. This can be extended to other problems as well, as the same threshold of ln n holds for all problems that are equivalent to set cover in terms of approximation ratio, such as dominating set (see =-=[29]-=- and [26] for more details). A survey of hardness of approximation results and the techniques involved is provided by Arora and Lund in [20]. Remark: A preliminary version of this paper, without the r... |

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