Capacitated domination and covering: a parameterized perspective

  • Authors:

  • Michael Dom;Daniel Lokshtanov;Saket Saurabh;Yngve Villanger

  • Affiliations:

  • Institut für Informatik, Friedrich-Schiller-Universität Jena, Jena, Germany;Department of Informatics, University of Bergen, Bergen, Norway;Department of Informatics, University of Bergen, Bergen, Norway;Department of Informatics, University of Bergen, Bergen, Norway

  • Venue:
  • IWPEC'08 Proceedings of the 3rd international conference on Parameterized and exact computation
  • Year:
  • 2008

Quantified Score

Hi-index 0.00

Visualization

Abstract

Capacitated versions of Vertex Cover and DominatingSet have been studied intensively in terms of polynomial timeapproximation algorithms. Although the problems Dominating Set andVertex Cover have been subjected to considerable scrutiny in theparameterized complexity world, this is not true for their capacitatedversions. Here we make an attempt to understand the behavior of theproblems Capacitated Dominating Set and Capacitated VertexCover from the perspective of parameterized complexity. The original, uncapacitated versions of these problems, VertexCover and Dominating Set, are known to be fixed parameter tractablewhen parameterized by a structure of the graph called the treewidth (tw).In this paper we show that the capacitated versions of these problemsbehave differently. Our results are: - Capacitated Dominating Set is W[1]-hard when parameterized bytreewidth. In fact, Capacitated Dominating Set is W[1]-hard whenparameterized by both treewidth and solution size k of the capacitateddominating set. - Capacitated Vertex Cover is W[1]-hard when parameterized bytreewidth. - Capacitated Vertex Cover can be solved in time 2O(tw log k)nO(1)where tw is the treewidth of the input graph and k is the solution size.As a corollary, we show that the weighted version of Capacitated VertexCover in general graphs can be solved in time 2O(k log k)nO(1).This improves the earlier algorithm of Guo et al. [15] running intime O(1.2k2 + n2). Capacitated Vertex Cover is, therefore, to ourknowledge the first known "subset problem" which has turned out tobe fixed parameter tractable when parameterized by solution size butW[1]-hard when parameterized by treewidth.