Abstract
Chen and Kanj considered the Vertex Cover problem for graphs with perfect matchings (VC-PM). They showed that: (i) There is a reduction from general Vertex Cover to VC-PM, which guarantees that if one can achieve an approximation factor of less than two for VC-PM, then one can do so for general Vertex Cover as well. (ii) There is an algorithm for VC-PM whose approximation factor is given as \(1.069+0.069\overline{d}\) where \(\overline{d}\) is the average degree of the given graph. In this paper we improve (ii). Namely we give a new VC-PM algorithm which greatly outperforms the above one and its approximation factor is roughly \(2-\frac{6.74}{\overline{d} + 6.28}\). Our algorithm also works for graphs with “large” matchings although its approximation factor is degenerated.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Baker, B.S.: Approximation algorithms for np-complete problems on planar graphs. J. ACM 41(1), 153–180 (1994)
Bar-Yehuda, R., Even, S.: A local-ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics 25, 27–45 (1985)
Berman, P., Fujito, T.: On approximation properties of the independent set problem in degree 3 graphs. In: Sack, J.-R., Akl, S.G., Dehne, F., Santoro, N. (eds.) WADS 1995. LNCS, vol. 955, pp. 449–460. Springer, Heidelberg (1995)
Chen, J., Kanj, I.: On approximating minimum vertex cover for graphs with perfect matching. In: International Symposium on Algorithms and Computation, pp. 132–143 (2000)
Clementi, A.E.F., Trevisan, L.: Improved non-approximability results for minimum vertex cover with density constraints. TCS: Theoretical Computer Science 225(1-2), 113–128 (1999)
Dinur, I., Safra, S.: The importance of being biased. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC 2002), May 19–21, pp. 33–42. ACM Press, New York (2002)
Feige, U., Goemans, M.X.: Approximating the value of two prover proof systems, with applications to MAX 2SAT and MAX DICUT. In: Proceedings of the Third Israel Symposium on Theory of Computing and Systems (1995)
Garey, M., Johnson, D.: Computers and intractability: A guide to the theory of npcompleteness w (1979)
Halldórsson, M.M., Radhakrishnan, J.: Greed is good: Approximating independent sets in sparse and bounded-degree graphs. Algorithmica 18, 145–163 (1997)
Halperin, E.: Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs. SIAM Journal on Computing 31(5), 1608–1623 (2002)
Hochbaum: Efficient bounds for the stable set, vertex cover and set packing problems. DAMATH: Discrete Applied Mathematics and Combinatorial Operations Research and Computer Science 6 (1983)
Karpinski, M., Zelikovsky, A.: Approximating dense cases of covering problems. Electronic Colloquium on Computational Complexity (ECCC) 4(004) (1997)
Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2− ε. In: Proc. of 18th IEEE Annual Conference on Computational Complexity (CCC), pp. 379–386 (2003)
Monien, B., Speckenmeyer, E.: Ramsey numbers and an approximation algorithm for the vertex cover problem. Acta Informatica 22(1), 115–123 (1985), NCPY
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Imamura, T., Iwama, K., Tsukiji, T. (2004). Approximated Vertex Cover for Graphs with Perfect Matchings. In: Chwa, KY., Munro, J.I.J. (eds) Computing and Combinatorics. COCOON 2004. Lecture Notes in Computer Science, vol 3106. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27798-9_16
Download citation
DOI: https://doi.org/10.1007/978-3-540-27798-9_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22856-1
Online ISBN: 978-3-540-27798-9
eBook Packages: Springer Book Archive