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Approximating the Unweighted k-Set Cover Problem: Greedy Meets Local Search

  • Asaf Levin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4368)

Abstract

In the unweighted set-cover problem we are given a set of elements E={ e 1,e 2, ...,e n } and a collection \(\cal F\) of subsets of E. The problem is to compute a sub-collection SOL ⊆\(\cal F\) such that \(\bigcup_{S_j\in SOL}S_j=E\) and its size |SOL| is minimized. When |S|≤k for all \(S\in\cal F\) we obtain the unweighted k-set cover problem. It is well known that the greedy algorithm is an H k -approximation algorithm for the unweighted k-set cover, where \(H_k=\sum_{i=1}^k {1 \over i}\) is the k-th harmonic number, and that this bound on the approximation ratio of the greedy algorithm, is tight for all constant values of k. Since the set cover problem is a fundamental problem, there is an ongoing research effort to improve this approximation ratio using modifications of the greedy algorithm. The previous best improvement of the greedy algorithm is an \(\left( H_k-{1\over 2}\right)\)-approximation algorithm. In this paper we present a new \(\left( H_k-{196\over 390}\right)\)-approximation algorithm for k ≥4 that improves the previous best approximation ratio for all values of k≥4 . Our algorithm is based on combining local search during various stages of the greedy algorithm.

Keywords

Approximation Algorithm Greedy Algorithm Approximation Ratio Cover Problem Performance 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 2007

Authors and Affiliations

  • Asaf Levin
    • 1
  1. 1.Department of StatisticsThe Hebrew UniversityJerusalemIsrael

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