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


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).


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ajtai, M., Komlós, J., Szemeredi, E.: A note on Ramsey numbers. J. Comb. Theory, Ser. A 29, 354–360 (1980)Google Scholar
  2. 2.
    Bar-Yehuda, R., Even, S.: On approximating a vertex cover for planar graphs. Proc. 14th Ann. ACM Symp. Th. Computing, pp. 303–309. San Francisco 1982Google Scholar
  3. 3.
    Bar-Yehuda, R, Even, S.: A 2−log logn/2logn performance ratio for the weighted vertex cover problem. Technion Haifa, Technical Report #260, January 1983Google Scholar
  4. 4.
    Erdös, P.: Graph theory and probability, Can. J. Math. 11, 34–38 (1959)Google Scholar
  5. 5.
    Erdös, P., Gallai, T.: On the minimal number of vertices representing the edges of a graph, pp. 181–203. Magyar Tudományos Akadémia Matematikai Kutato Intézetének Közleményei VI. Evfolyam 1961Google Scholar
  6. 6.
    Even, S.: Graph algorithms. Rockville, MD: Computer Science Press 1979Google Scholar
  7. 7.
    Fajtlowicz, S.: On the size of independent sets in graphs. In: Proc. 9th S.E. Conf. Combinatorics, Graph Theory and Computing, Boca Raton, Congressum Numerantium No. XXI, pp. 269–274. Utilitas Mathematical-Publ. Inc. 1978Google Scholar
  8. 8.
    Graver, J.E, Yackel, J.: Some graph theoretic results with Ramsey's theorem. J. Comb. Theory, Ser. A 4, 125–175 (1968)Google Scholar
  9. 9.
    Griggs, J.R.: Lower bounds on the independence number in terms of the degrees. J. Comb. Theory, Ser. B 34, 22–39 (1983)Google Scholar
  10. 10.
    Hochbaum, D.S.: Efficient bounds for the stable set, vertex cover and set packing problems. Discrete Appl. Math. 6, 243–254 (1983)Google Scholar
  11. 11.
    Hopkins, G., Staton, W.: Girth and independence ratio. Can. Math. Bull. 25, 179–186 (1982)Google Scholar
  12. 12.
    König, D.: Theorie der endlichen und unendlichen Graphen. Leipzig: Akademische Verlagsgesellschaft 1936Google Scholar
  13. 13.
    Monien, B.: The complexity of determining a shortest cycle of even length. Computing 31, 355–369 (1983)Google Scholar
  14. 14.
    Monien, B., Speckenmeyer, E.: Some further approximation algorithms for the vertex cover problem. In: Proc. 8th Coll. on Trees in Algebra and Programming (CAAP 83), L'Aquila. Lecture Notes in Computer Science 159, 341–349 (1983)Google Scholar
  15. 15.
    Nemhauser, G.L., Trotter, L.E.: Vertex packing: structural properties and algorithms. Math. Program. 8, 232–248 (1975)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

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

Personalised recommendations