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Given a set D of n unit disks in the plane and an integer k ≤ n, the maximum area connected subset problem asks for a set D′ ⊆ D of size k that maximizes the area of the union of disks, under the constraint that this union is connected. This problem is motivated by wireless router deployment and is a special case of maximizing a submodular function under a connectivity constraint. We prove that the problem is NP-hard and analyze a greedy algorithm, proving that it is a 2 approximation. We then give a polynomial-time approximation scheme (PTAS) for this problem with resource augmentation, i.e., allowing an additional set of εk disks that are not drawn from the input. Additionally, for two special cases of the problem we design a PTAS without resource augmentation. 2012 ACM Subject Classification Theory of computation → Design and analysis of algorithms

Maximizing Covered Area in the Euclidean Plane with Connectivity Constraint.