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

Chvátal and Klincsek (1980) gave an [Formula: see text]-time algorithm for the problem of finding a maximum-cardinality convex subset of an arbitrary given set [Formula: see text] of [Formula: see text] points in the plane. This paper examines a generalization of the problem, the Bottleneck Convex Subsets problem: given a set [Formula: see text] of [Formula: see text] points in the plane and a positive integer [Formula: see text], select [Formula: see text] pairwise disjoint convex subsets of [Formula: see text] such that the cardinality of the smallest subset is maximized. Equivalently, a solution maximizes the cardinality of [Formula: see text] mutually disjoint convex subsets of [Formula: see text] of equal cardinality. We give an algorithm that solves the problem exactly, with running time polynomial in [Formula: see text] when [Formula: see text] is fixed. We then show the problem to be NP-hard when [Formula: see text] is an arbitrary input parameter, even for points in general position. Finally, we give a fixed-parameter tractable algorithm parameterized in terms of the number of points strictly interior to the convex hull.