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Abstract

In the minimum entropy set cover problem, one is given a collection of k sets which collectively cover an n-element ground set. A feasible solution of the problem is a partition of the ground set into parts such that each part is included in some of the k given sets. The goal is to find a partition minimizing the (binary) entropy of the corresponding probability distribution, i.e., the one found by dividing each part size by n. Halperin and Karp have recently proved that the greedy algorithm always returns a solution whose cost is at most the optimum plus a constant. We improve their result by showing that the greedy algorithm approximates the minimum entropy set cover problem within an additive error of 1 nat = log2 e bits ≃ 1.4427 bits. Moreover, inspired by recent work by Feige, Lovász and Tetali on the minimum sum set cover problem, we prove that no polynomial-time algorithm can achieve a better constant, unless P = NP. We also discuss some consequences for the related minimum entropy coloring problem.

Keywords

Greedy Algorithm Side Information Minimum Entropy Polynomial Time Approximation Scheme Approximation Guarantee 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Jean Cardinal
    • 1
  • Samuel Fiorini
    • 2
  • Gwenaël Joret
    • 1
  1. 1.Computer Science DepartmentUniversité Libre de BruxellesBrusselsBelgium
  2. 2.Department of MathematicsUniversité Libre de BruxellesBrusselsBelgium

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