The Parameterized Complexity of the Unique Coverage Problem

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. We also 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. In the most general setting, assuming real costs and profits, the problem is not fixed-parameter tractable unless P = NP. Assuming integer costs and profits we show the problem to be W[1]-hard with respect to B as parameter. However, under some reasonable restriction, the problem becomes fixed-parameter tractable with respect to both B and k as parameters.