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Chv\'{a}tal and Klincsek (1980) gave an $O(n^3)$-time algorithm for theproblem of finding a maximum-cardinality convex subset of an arbitrary givenset $P$ of $n$ points in the plane. This paper examines a generalization of theproblem, the Bottleneck Convex Subsets problem: given a set $P$ of $n$ pointsin the plane and a positive integer $k$, select $k$ pairwise disjoint convexsubsets of $P$ such that the cardinality of the smallest subset is maximized.Equivalently, a solution maximizes the cardinality of $k$ mutually disjointconvex 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 theproblem exactly, with running time polynomial in $n$ when $k$ is fixed, and wegive a fixed-parameter tractable algorithm parameterized in terms of the numberof points strictly interior to the convex hull.

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