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

TL;DR

This paper studies the problem of selecting $k$ disjoint convex subsets from a point set to maximize the smallest subset size, providing complexity results and algorithms for fixed $k$ and interior points.

Contribution

It introduces the Bottleneck Convex Subsets problem, proves NP-hardness for arbitrary $k$, and offers polynomial-time and fixed-parameter algorithms for special cases.

Findings

NP-hardness when $k$ is arbitrary

Polynomial-time algorithm for fixed $k$

Fixed-parameter tractable algorithm based on interior points

Abstract

Chv\'{a}tal 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.