Computing Dominating Sets in Disk Graphs with Centers in Convex Position

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.

💡 Research Summary

The paper addresses the Minimum Dominating Set (MDS) problem on disk graphs whose vertices are the centers of disks in the plane. An edge connects two vertices if the corresponding disks intersect. While the problem is NP‑hard for general disk graphs (and even for unit‑disk graphs), the authors focus on the special geometric setting where all centers lie in convex position—that is, every point appears as a vertex of the convex hull of the set. This configuration models scenarios such as sensors deployed along a fence or shoreline.

The main contributions are two exact algorithms that run in polynomial time under the convex‑position assumption. For the unweighted version, they present an O(k² n log² n) algorithm, where k is the size of a minimum dominating set. When k is constant the running time drops to O(n log² n). For the weighted version, where each disk has a positive weight and the goal is to minimize total weight subject to a cardinality bound k, they give an O(n⁵ log² n) algorithm that works for any k (setting k = n yields the unrestricted optimum).

The technical core is a structural property they call “line‑separable”. Given an optimal dominating set S, they construct a partition A of the vertex set into contiguous sublists (subsequences along the convex hull) and an assignment ϕ that maps each sublist to a dominating vertex that actually dominates all points in the sublist. By building an additively weighted Voronoi diagram of S (weights equal to the negative radii), each vertex p∈S owns a star‑shaped cell R(p) that contains p. All points whose centers lie inside R(p) are assigned to p, forming the groups Aₚ. Because Voronoi cells are interior‑disjoint, any line segment connecting representatives of two consecutive sublists belonging to the same group never crosses a segment belonging to another group. Consequently, the collection of such segments (called diagonals) is planar and there always exists a diagonal that separates the convex hull into two regions, one of which contains no other diagonal. This yields a recursive decomposition of the problem.

Lemma 1 proves the existence of a line‑separable partition and assignment for any optimal solution; Lemma 2 guarantees that at least one dominating vertex has a group consisting of a single sublist (its “main sublist”). These lemmas enable a divide‑and‑conquer strategy. For the unweighted case, the algorithm recursively selects a separating diagonal, solves the subproblems on each side, and combines the solutions while ensuring that the total number of dominating vertices does not exceed k. The dynamic programming table stores, for each interval of sublists, the minimum number of dominating vertices needed, leading to the O(k² n log² n) bound.

For the weighted case, the authors introduce the notion of a “rank‑t center”. A rank‑t center for an interval A