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 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.
Supported by a DAAD-DST exchange program, D/05/57666.
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Moser, H., Raman, V., Sikdar, S. (2007). The Parameterized Complexity of the Unique Coverage Problem. In: Tokuyama, T. (eds) Algorithms and Computation. ISAAC 2007. Lecture Notes in Computer Science, vol 4835. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77120-3_54
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DOI: https://doi.org/10.1007/978-3-540-77120-3_54
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-77118-0
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