Maximum Independent Sets in Disk Graphs with Disks in Convex Position

Abstract

For a set $\mathcal{D}$ of disks in the plane, its disk graph $G(\mathcal{D})$ is the graph with vertex set $\mathcal{D}$, where two vertices are adjacent if and only if the corresponding disks intersect. Given a set $\mathcal{D}$ of $n$ weighted disks, computing a maximum independent set of $G(\mathcal{D})$ is NP-hard. In this paper, we present an $O(n^3\log n)$-time algorithm for this problem in a special setting in which the disks are in convex position, meaning that every disk appears on the convex hull of $\mathcal{D}$. This setting has been studied previously for disks of equal radius, for which an $O(n^{37/11})$-time algorithm was known. Our algorithm also works in the weighted case where disks have weights and the goal is to compute a maximum-weight independent set. As an application of our result, we obtain an $O(n^3\log^2 n)$-time algorithm for the dispersion problem on a set of $n$ disks in convex position: given an integer $k$, compute a subset of $k$ disks that maximizes the minimum pairwise distance among all disks in the subset.