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The Parameterized Complexity of the Unique Coverage Problem

  • Hannes Moser
  • Venkatesh Raman
  • Somnath Sikdar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4835)

Abstract

We consider the parameterized complexity of the Unique Coverage problem: given a family of sets and a parameter k, we ask whether there exists a subfamily that covers at least k elements exactly once. This NP-complete problem has applications in wireless networks and radio broadcasting and is also a natural generalization of the well-known Max Cut problem. We show that this problem is fixed-parameter tractable with respect to the parameter k. That is, for every fixed k, there exists a polynomial-time algorithm for it. One way to prove a problem fixed-parameter tractable is to show that it is kernelizable. To this end, we show that if no two sets in the input family intersect in more than c elements there exists a problem kernel of size k c + 1. This yields a k k kernel for the Unique Coverage problem, proving fixed-parameter tractability. Subsequently, we show a 4 k kernel for this problem. However a more general weighted version, with costs associated with each set and profits with each element, turns out to be much harder. The question here is whether there exists a subfamily with total cost at most a prespecified budget B such that the total profit of uniquely covered elements is at least k, where B and k are part of the input. In the most general setting, assuming real costs and profits, the problem is not fixed-parameter tractable unless \(\mbox{\it P} = \mbox{\it NP}\). Assuming integer costs and profits we show the problem to be W[1]-hard with respect to B as parameter (that is, it is unlikely to be fixed-parameter tractable). However, under some reasonable restriction, the problem becomes fixed-parameter tractable with respect to both B and k as parameters.

Keywords

Parameterized Complexity Parameterized Problem Reduction Rule Mobile Client Unique Coverage 
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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Hannes Moser
    • 1
  • Venkatesh Raman
    • 2
  • Somnath Sikdar
    • 2
  1. 1.Institut für Informatik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, D-07743 JenaGermany
  2. 2.The Institute of Mathematical Sciences, C.I.T Campus, Taramani, Chennai 600113India

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