Bottleneck Convex Subsets: Finding k Large Convex Sets in a Point Set.

Chvatal and Klincsek (1980) gave an \(O(n^3)\)-time algorithm for the problem of finding a maximum-cardinality convex subset of an arbitrary given set P of n points in the plane. This paper examines a generalization of the problem, the Bottleneck Convex Subsets problem: given a set P of n points in the plane and a positive integer k, select k pairwise disjoint convex subsets of P such that the cardinality of the smallest subset is maximized. Equivalently, a solution maximizes the cardinality of k mutually disjoint convex subsets of P of equal cardinality. We show the problem is NP-hard when k is an arbitrary input parameter, we give an algorithm that solves the problem exactly, with running time polynomial in n when k is fixed, and we give a fixed-parameter tractable algorithm parameterized in terms of the number of points strictly interior to the convex hull.