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Algorithmica

, Volume 40, Issue 4, pp 219–234 | Cite as

Approximating Min Sum Set Cover

  • Uriel Feige
  • László Lovász
  • Prasad Tetali
Article

Abstract

The input to the min sum set cover problem is a collection of n sets that jointly cover m elements. The output is a linear order on the sets, namely, in every time step from 1 to n exactly one set is chosen. For every element, this induces a first time step by which it is covered. The objective is to find a linear arrangement of the sets that minimizes the sum of these first time steps over all elements. We show that a greedy algorithm approximates min sum set cover within a ratio of 4. This result was implicit in work of Bar-Noy, Bellare, Halldorsson, Shachnai, and Tamir (1998) on chromatic sums, but we present a simpler proof. We also show that for every ε > 0, achieving an approximation ratio of 4 – ε is NP-hard. For the min sum vertex cover version of the problem (which comes up as a heuristic for speeding up solvers of semidefinite programs) we show that it can be approximated within a ratio of 2, and is NP-hard to approximate within some constant ρ > 1.

Greedy algorithm Randomized rounding NP-hardness Threshhold 

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Copyright information

© Springer 2004

Authors and Affiliations

  1. 1.Department of Computer Science and Applied Mathematics, The Weizmann Institute, Rehovot 7610Israel
  2. 2.Microsoft Research, One Microsoft Way, Redmond, WA 98052USA
  3. 3.School of Mathematics and College of Computing, Georgia Institute of Technology, Atlanta, GA 30332-0160USA

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