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 kc + 1. This yields a kk kernel for the Unique Coverage problem, proving fixed-parameter tractability. Subsequently, we show a 4k 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.

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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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