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On the Minimum Degree Hypergraph Problem with Subset Size Two and the Red-Blue Set Cover Problem with the Consecutive Ones Property

  • Biing-Feng Wang
  • Chih-Hsuan Li
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)

Abstract

Let S be a set and let C b (blue collection) and C r (red collection) be two collections of subsets of S. The MDH problem is to find a subset S′ ⊆ S such that S′ ∩ B ≠ ∅ for all B ∈ C b and |S′ ∩ R| ≤ k for all R ∈ C r , where k is a given non-negative integer. The RBSC problem is to find a subset S′ ⊆ S with S′ ∩ B ≠ ∅ for all B ∈ C b which minimizes |{R | R ∈ C r , S′ ∪ R ≠ ∅ }|. In this paper, improved algorithms are proposed for the MDH problem with k = 1 and all sets in C b having size two and the RBSC problem with C b  ∪ C r having the consecutive ones property. For the first problem, we give an optimal \(O(|S| + |C_{b}| + \sum_{R \in C_{r}} |R|)\)-time algorithm, improving the previous \(O(|S| + |C_{b}| + \sum_{R \in C_{r}} |R|^{2})\) bound by Dom et al. Our improvement is obtained by presenting a new representation of a dense directed graph, which may be of independent interest. For the second problem, we give an \(O(|C_{b}| + |C_{r}| \lg |S| + |S| \lg |S|)\)-time algorithm, improving the previous O(|C b ||S| + |C r ||S| + |S|2) bound by Chang et al.

Keywords

set cover minimum degree hypergraphs 2-satisfiability algorithms 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Biing-Feng Wang
    • 1
  • Chih-Hsuan Li
    • 1
  1. 1.Department of Computer ScienceNational Tsing Hua UniversityHsinchuTaiwan, Republic of China

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