Approximating Min Sum Set Cover

  • Authors:
  • Uriel Feige;Prasad Tetali

  • Affiliations:
  • Department of Computer Science and Applied Mathematics, The Weizmann Institute, Rehovot 7610, Israel;School of Mathematics and College of Computing, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA

  • Venue:
  • Algorithmica
  • Year:
  • 2004

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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.