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Approximating the maximum clique minor and some subgraph homeomorphism problems.

Författare

Summary, in English

We consider the “minor” and “homeomorphic” analogues of the maximum clique problem, i.e., the problems of determining the largest h such that the input graph (on n vertices) has a minor isomorphic to Kh or a subgraph homeomorphic to Kh, respectively, as well as the problem of finding the corresponding subgraphs. We term them as the maximum clique minor problem and the maximum homeomorphic clique problem, respectively. We observe that a known result of Kostochka and Thomason supplies an O(sqrt n) source bound on the approximation factor for the maximum clique minor problem achievable in polynomial time. We also provide an independent proof of nearly the same approximation factor with explicit polynomial-time estimation, by exploiting the minor separator theorem of Plotkin et al.



Next, we show that another known result of Bollobás and Thomason and of Komlós and Szemerédi provides an O(sqrt n) source bound on the approximation factor for the maximum homeomorphic clique achievable in polynomial time. On the other hand, we show an Ω(n1/2−O(1/(logn)γ)) lower bound (for some constant γ, unless NP subset ZPTIME(2^(logn)^O(1)) on the best approximation factor achievable efficiently for the maximum homeomorphic clique problem, nearly matching our upper bound.



Finally, we derive an interesting trade-off between approximability and subexponential time for the problem of subgraph homeomorphism where the guest graph has maximum degree not exceeding three and low treewidth.

Avdelning/ar

  • Computer Science

Publiceringsår

2007

Språk

Engelska

Sidor

149-158

Publikation/Tidskrift/Serie

Theoretical Computer Science

Volym

374

Issue

1-3

Dokumenttyp

Artikel i tidskrift

Förlag

Elsevier

Ämne

  • Computer Science

Nyckelord

  • Graph homeomorphism
  • Approximation
  • Graph minors

Status

Published

Projekt

  • VR 2005-4085

ISBN/ISSN/Övrigt

  • ISSN: 0304-3975