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Acta Informatica

, Volume 22, Issue 1, pp 115–123 | Cite as

Ramsey numbers and an approximation algorithm for the vertex cover problem

  • Burkhard Monien
  • Ewald Speckenmeyer
Article

Summary

We show two results. First we derive an upper bound for the special Ramsey numbers rk(q) where rk(q) is the largest number of nodes a graph without odd cycles of length bounded by 2k+1 and without an independent set of size q+1 can have. We prove \(r_k (q) \leqq \frac{k}{{k + {\text{1}}}}q^{\frac{{k + {\text{1}}}}{k}} + \frac{{k + {\text{2}}}}{{k + {\text{1}}}}q\) The proof is constructive and yields an algorithm for computing an independent set of that size. Using this algorithm we secondly describe an OV¦·¦E¦) time bounded approximation algorithm for the vertex cover problem, whose worst case ratio is \(\Delta \leqq {\text{2 - }}\frac{{\text{1}}}{{k + {\text{1}}}}\), for all graphs with at most (2k+3) k (2k+2) nodes (e.g. Δ≦1.8, if ¦V¦≦146000).

Keywords

Information System Operating System Data Structure Communication Network Information Theory 
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 1985

Authors and Affiliations

  • Burkhard Monien
    • 1
  • Ewald Speckenmeyer
    • 1
  1. 1.Fachbereich 17, Theoretische InformatikUniversität PaderbornPaderbornFederal Republic of Germany

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