Tight results on minimum entropy set cover

  • Authors:

  • Jean Cardinal;Samuel Fiorini;Gwenaël Joret

  • Affiliations:

  • Computer Science Department, Université Libre de Bruxelles, Brussels, Belgium;Department of Mathematics, Université Libre de Bruxelles, Brussels, Belgium;Computer Science Department, Université Libre de Bruxelles, Brussels, Belgium

  • Venue:
  • APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
  • Year:
  • 2006

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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 = log2e 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.