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Set Cover, Set Packing and Hitting Set for Tree Convex and Tree-Like Set Systems

  • Min Lu
  • Tian Liu
  • Weitian Tong
  • Guohui Lin
  • Ke Xu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8402)

Abstract

A set system is a collection of subsets of a given finite universe. A tree convex set system has a tree defined on the universe, such that each subset in the system induces a subtree. A circular convex set system has a circular ordering defined on the universe, such that each subset in the system induces a circular arc. A tree-like set system has a tree defined on the system, such that for each element in the universe, all subsets in the system containing this element induce a subtree. A circular-like set system has a circular ordering defined on the system, such that for each element in the universe, all subsets in the system containing this element induce a circular arc. In this paper, we restrict the trees to be stars, combs, triads, respectively, and restrict the set system to be unweighted. We show tractability of Triad Convex Set Cover, Circular-like Set Packing, and Triad-like Hitting Set, intractability of Comb Convex Set Cover and Comb-like Hitting Set. Our results not only complement the known results in literatures, but also rise interesting questions such as which other kind of trees will lead to tractability or intractability results of Set Cover, Set Packing and Hitting Set for tree convex and tree-like set systems.

Keywords

Tree convex set systems tree-like set systems set cover set packing hitting set polynomial time \(\mathcal{NP}\)-complete 

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Min Lu
    • 1
  • Tian Liu
    • 1
  • Weitian Tong
    • 2
  • Guohui Lin
    • 2
  • Ke Xu
    • 3
  1. 1.Key Laboratory of High Confidence Software Technologies, Ministry of Education, Institute of Software, School of Electronic Engineering and Computer SciencePeking UniversityBeijingChina
  2. 2.Department of Computing ScienceUniversity of Alberta EdmontonAlbertaCanada
  3. 3.National Lab of Software Development EnvironmentBeihang UniversityBeijingChina

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