Computing maximally separated sets in the plane

Let S be a set of n points in ℝ². Given an integer 1 ≤ k ≤ n, we wish to find a maximally separated subset I ⊆ S of size k; this is a subset for which the minimum among the (₂^k) pairwise distances between its points is as large as possible. The decision problem associated with this problem is to determine whether there exists I ⊆ S, |I| = k, so that all (₂^k) pairwise distances in I are at least 2. This problem can also be formulated in terms of disk-intersection graphs: Let D be the set of unit disks centered at the points of S. The disk-intersection graph G of D has as edges all pairs of disks with nonempty intersection. Any set I with the above properties is then the set of centers of disks that form an independent set in the graph G. This problem is known to be NP-complete if k is part of the input. In this paper we first present a linear-time ε-approximation algorithm for any constant k. Next we give exact algorithms for the cases k = 3 and k = 4 that run in time O(n ^4/3polylog(n)). We also present a simpler n^O(√k)- time exact algorithm (as compared with the recent algorithm in [J. Alber and J. Fiala, J. Algorithms, 52 (2004), pp. 134-151]) for arbitrary values of k.