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.

Figures (13)

• Figure 1: Illustrating the ordering property of $S$ (the centers of the disks).
• Figure 2: Illustrating $D_j^i$ and $D_i^j$, which are the blue and red regions, respectively, excluding $\overline{u_{ij}v_{ij}}$.
• Figure 3: Illustrating a schematic view of $P$ (i.e., the dotted circle) and the relative positions of $\alpha_i$, $q_1$, $\beta_1'$, $q_2$, $q_3$, $\beta_2'$, and $q_4$. The sublists $\alpha_i$, $\beta_1'$, and $\beta_2'$ are illustrated with solid arcs.
• Figure 4: Illustrating angles for Lemma \ref{['lem:balassign']}. Solid red segments have lengths at most $1$ while solid blue segments have lengths greater than $1$.
• Figure 5: $\beta'_{11}$ consists of the points in the blue part, and its portion left of $\ell$ is $\beta"_{11}$.

Theorems & Definitions (61)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof