Computing Dominating Sets in Disk Graphs with Centers in Convex Position

TL;DR

This work proves that the dominating set problem on disk graphs with centers in convex position is solvable in polynomial time, extending beyond the unit-disk setting. The authors introduce a line-separable structural property and a Voronoi-based partitioning to decompose the problem into tractable subproblems, which are then solved via dynamic programming using rank-$t$ centers. The weighted variant runs in $O(k^2n^3\log^2 n)$ time (yielding $O(n^5\log^2 n)$ when $k=n$), while the unweighted case achieves $O(k^2n\log^2 n)$ time (and $O(n^5\log^2 n)$ in the full-scale bound). A key contribution is the construction of a scalable, structure-driven DP that leverages the line-separable property to manage subproblem interactions, enabling efficient computation of minimum-weight and minimum-cardinality dominating sets in disk graphs under convex-position constraints.

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.

Figures (7)

• Figure 1: Illustrating an example in which $|\mathcal{A}_{p_i}|=\Omega(n)$.
• Figure 2: Illustrating the relative positions of points of $P$ (only their indices are shown): the circle represent $\mathcal{H}(P)$.
• Figure 3: Illustrating $L_x$, $L_y$, and $L$.
• Figure 4: Illustrating the sublists of $\mathcal{A}$, represented by the black solid arcs (the dotted circle represents $\mathcal{H}(P)$).
• Figure 5: Illustrating the proof of Lemma \ref{['lem:correct20']}. For points of $P$, only their indices are shown. The red (resp., blue) dashed arc represents $L_1'$ (resp., $L_2'$). The red (resp., blue, green) solid arc represents $L_1$ (resp., $L_2$, $L$). The black solid arcs on the dotted circle represents sublists of $\mathcal{A}$.

Theorems & Definitions (33)

• Lemma 1
• proof
• Lemma 2
• proof
• Definition 1
• Definition 2
• proof
• proof
• Lemma 3
• Lemma 4