On efficient fixed-parameter algorithms for weighted vertex cover

  • Authors:

  • Rolf Niedermeier;Peter Rossmanith

  • Affiliations:

  • Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Sand 13, D-72076 Tübingen, Federal Republic of Germany;Institut für Informatik, Technische Universität München, Arcisstr. 21, D-80290 München, Federal Republic of Germany

  • Venue:
  • Journal of Algorithms
  • Year:
  • 2003

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Abstract

We investigate the fixed-parameter complexity of 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 vertex subset C ⊆ V of total weight at most k such that every edge of G has at least one endpoint in C. WVC and its natural variants are NP-complete. We observe that, when restricting the range of ω 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 ω is restricted to positive reals ≥ 1, then so-called REAL-WVC can be solved in time O(1.3954k + k|V|). By way of contrast, unless P = NP, the problem is not fixed-parameter tractable if arbitrary weights 0 are allowed. Using dynamic programming, at the expense of exponential memory use, we can improve the running time of REALWVC to O(1.3788k + k|V|). The same technique applied to a known algorithm yields the so far fastest algorithm for unweighted VERTEX COVER, running in time O(1.2832kk + k|V|).