A threshold of ln n for approximating set cover

  • Authors:
  • Uriel Feige

  • Affiliations:
  • -

  • Venue:
  • Journal of the ACM (JACM)
  • Year:
  • 1998

Quantified Score

Hi-index 0.03

Visualization

Abstract

Given a collection F of subsets of S ={1,…,n}, setcover is the problem of selecting as few as possiblesubsets from F such that their union coversS,, and maxk-cover is the problem of selectingk subsets from F such that their union has maximum cardinality. Both these problems areNP-hard. We prove that (1 - o(1)) lnn is a threshold below which setcover cannot be approximated efficiently, unless NP has slightlysuperpolynomial time algorithms. This closes the gap (up to low-orderterms) between the ratio of approximation achievable by the greedyalogorithm (which is (1 - o(1)) lnn), and provious results of Lund and Yanakakis, that showed hardness ofapproximation within a ratio of log2n/2≃0.72 ln n. For maxk-cover, we show an approximationthreshold of (1 - 1/e)(up tolow-order terms), under assumption that P≠NP.