Computing Dominating Sets in Disk Graphs with Centers in Convex Position

TL;DR

This paper presents the first polynomial-time algorithms for finding minimum dominating sets in disk graphs with points in convex position, a problem that is NP-hard in general, with efficient solutions for both unweighted and weighted cases.

Contribution

The paper introduces the first polynomial-time algorithms for the dominating set problem in disk graphs with points in convex position, improving understanding of special cases.

Findings

Polynomial-time algorithm with $O(k^2 n \,\log^2 n)$ complexity for unweighted case.

Polynomial-time algorithm with $O(n^5 \,\log^2 n)$ complexity for weighted case.

First known efficient algorithms for this special geometric dominating set problem.

Abstract

Given a set $P$ of $n$ points in the plane and a collection of disks centered at these points, the disk graph $G(P)$ has vertex set $P$, with an edge between two vertices if their corresponding disks intersect. We study the dominating set problem in $G(P)$ under the special case where the points of $P$ are in convex position. The problem is NP-hard in general disk graphs. Under the convex position assumption, however, we present the first polynomial-time algorithm for the problem. Specifically, we design an $O(k^2 n \log^2 n)$-time algorithm, where $k$ denotes the size of a minimum dominating set. For the weighted version, in which each disk has an associated weight and the goal is to compute a dominating set of minimum total weight, we obtain an $O(n^5 \log^2 n)$-time algorithm.