Journal of Computer and System Sciences
Combination can be hard: approximability of the unique coverage problem
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Parameterized Complexity
Approximating geometric coverage problems
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Incompressibility through Colors and IDs
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
The Budgeted Unique Coverage Problem and Color-Coding
CSR '09 Proceedings of the Fourth International Computer Science Symposium in Russia on Computer Science - Theory and Applications
A polynomial-time approximation scheme for the geometric unique coverage problem on unit squares
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
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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 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.