# The Parameterized Complexity of the Unique Coverage Problem

## 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.

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