Making a Hard Problem Easier: 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, find 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 an O(n)-time algorithm for it, where d is a constant independent of k. 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. This yields a k kernel for the Unique Coverage problem, proving fixed-parameter tractability. Subsequently, we show a 4 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 a much harder problem. 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 P = 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.