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International Symposium on Algorithms and Computation

ISAAC 2000: Algorithms and Computation pp 180–191Cite as

On Efficient Fixed Parameter Algorithms for Weighted Vertex Cover

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1969))

Abstract

We investigate the fixed parameter complexity of one of the most popular problems in combinatorial optimization, Weighted Vertex Cover. Given a graph G = (V,E), a weight function ω : V → R+, and k ∈ R+, Weighted Vertex Cover (WVC for short) asks for a subset C of vertices in V of weight at most k such that every edge of G has at least one endpoint in C. WVC and its variants have all been shown to be NP-complete. We show that, when restricting the range of ! to positive integers, the so-called Integer-WVC can be solved as fast as unweighted Vertex Cover. Our main result is that if the range of ! is restricted to positive reals ≥ 1, then so-called Real-WVC can be solved in time O(1.3954k +k∣V∣). If we modify the problem in such a way that k is not the weight of the vertex cover we are looking for, but the number of vertices in a minimum weight vertex cover, then the same running time can be obtained. If the weights are arbitrary (referred to by General-WVC), however, the problem is not fixed parameter tractable unless P = NP.

Work performed within the “PEAL” project (Parameterized complexity and Exact ALgorithms), supported by the Deutsche Forschungsgemeinschaft (NI-369/1-1).

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Author information

Authors and Affiliations

  1. Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Sand 13, D-72076, Tübingen, Fed. Rep. of Germany

    Rolf Niedermeier

  2. Institut für Informatik, Technische Universität München, Arcisstr. 21, D-80290, München, Fed. Rep. of Germany

    Peter Rossmanith

Authors
  1. Rolf Niedermeier
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  2. Peter Rossmanith
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Editor information

Editors and Affiliations

  1. Karlsruhe University, Germany

    Gerhard Goos

  2. Cornell University, NY, USA

    Juris Hartmanis

  3. Utrecht University, The Netherlands

    Jan van Leeuwen

  4. Academia Sinica, Institute of Information Science, 128 Academia Road, Section 2, Nankang, 115, Taipei, Taiwan, R.O.C.

    D. T. Lee

  5. Department of Computer Science and Akamai Technologies, University of Illinois at Urbana Champaign, 500 Technology Square, 02139, Cambridge, MA, USA

    Shang-Hua Teng

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© 2000 Springer-Verlag Berlin Heidelberg

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Niedermeier, R., Rossmanith, P. (2000). On Efficient Fixed Parameter Algorithms for Weighted Vertex Cover. In: Goos, G., Hartmanis, J., van Leeuwen, J., Lee, D.T., Teng, SH. (eds) Algorithms and Computation. ISAAC 2000. Lecture Notes in Computer Science, vol 1969. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-40996-3_16

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  • DOI: https://doi.org/10.1007/3-540-40996-3_16

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41255-7

  • Online ISBN: 978-3-540-40996-0

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