Dominating Set, Independent Set, Discrete $k$-Center, Dispersion, and Related Problems for Planar Points in Convex Position

Abstract

Given a set $P$ of $n$ points in the plane, its unit-disk graph $G(P)$ is a graph with $P$ as its vertex set such that two points of $P$ are connected by an edge if their (Euclidean) distance is at most $1$. We consider several classical problems on $G(P)$ in a special setting when points of $P$ are in convex position. These problems are all NP-hard in the general case. We present efficient algorithms for these problems under the convex position assumption. The considered problems include the following: finding a minimum weight dominating set in $G(P)$, the discrete $k$-center problem for $P$, finding a maximum weight independent set in $G(P)$, the dispersion problem for $P$, and several of their variations. For some of these problems, our algorithms improve the previously best results, while for others, our results provide first-known solutions.