International Workshop on Approximation and Online Algorithms

Approximation and Online Algorithms pp 122-132 | Cite as

An \(O(\log \mathrm{OPT})\)-Approximation for Covering/Packing Minor Models of \(\theta _{r}\)

  • Dimitris Chatzidimitriou
  • Jean-Florent Raymond
  • Ignasi Sau
  • Dimitrios M. Thilikos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9499)


Let \(\mathcal{C}_{H}\) be the class of graphs containing some fixed graph H as a minor. We define \(\mathbf{c}^\mathsf{v}_{H}(G)\) (resp. \(\mathbf{c}^\mathsf{e}_{H}(G)\)) as the minimun number of vertices (resp. edges) whose removal from G produces a graph without any subgraph isomorphic to a graph in \(\mathcal{C}_{H}\). Also \(\mathbf{p}^\mathsf{v}_{H}(G)\) (resp. \(\mathbf{p}^\mathsf{e}_{H}(G)\)) is the the maximum number of vertex- (resp. edge-) disjoint subgraphs of G that are isomorphic to some graph in \(\mathcal{C}_{H}\). We denote by \(\theta _{r}\) the graph with two vertices and r parallel edges between them. When \(H=\theta _{r}\), the parameters \(\mathbf{c}^\mathsf{v/e}_{H}\) and \(\mathbf{p}^\mathsf{v/e}_{H}\) are NP-complete to compute (for sufficiently large r). In this paper we prove a series of combinatorial and algorithmic lemmata that imply that if \(\mathbf{p}^\mathsf{v/e}_{\theta _r}(G)\le k\), then \(\mathbf{c}^\mathsf{v/e}_{\theta _r}(G) = O(k\log k)\). This means that for every r, the class \(\mathcal{C}_{\theta _{r}}\) has the vertex/edge Erdős-Pósa property. Using the combinatorial ideas from our proofs we introduce a unified approach for the design of an \(O(\log \mathrm{OPT})\)-approximation algorithm for \(\mathbf{c}^\mathsf{v}_{\theta _{r}}\), \(\mathbf{p}^\mathsf{v}_{\theta _{r}}\), \(\mathbf{c}^\mathsf{e}_{\theta _{r}}\) and \(\mathbf{p}^\mathsf{e}_{\theta _{r}}\) that runs in \(O(n\cdot \log (n)\cdot m)\) steps.


Erdős-Pósa properties Minors Graph packing Covering 


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Dimitris Chatzidimitriou
    • 1
  • Jean-Florent Raymond
    • 2
    • 3
  • Ignasi Sau
    • 2
  • Dimitrios M. Thilikos
    • 1
    • 2
  1. 1.Department of MathematicsNational and Kapodistrian University of AthensAthensGreece
  2. 2.AlGCo Project-TeamCNRS, LIRMMMontpellierFrance
  3. 3.Computer Science InstituteUniversity of WarsawWarsawPoland

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