Total Domination in Unit Disk Graphs

TL;DR

This paper proves the NP-hardness of the total dominating set problem in unit disk graphs, introduces an 8-approximation algorithm, and demonstrates the existence of a PTAS for the problem.

Contribution

It establishes NP-hardness, provides an efficient approximation algorithm, and shows a PTAS for total domination in unit disk graphs, advancing theoretical understanding and algorithmic solutions.

Findings

TDS problem is NP-hard in unit disk graphs.

An 8-factor approximation algorithm with $O(n \, log \, k)$ runtime.

Existence of a PTAS for TDS in unit disk graphs.

Abstract

Let $G=(V,E)$ be an undirected graph. We call $D_t \subseteq V$ as a total dominating set (TDS) of $G$ if each vertex $v \in V$ has a dominator in $D$ other than itself. Here we consider the TDS problem in unit disk graphs, where the objective is to find a minimum cardinality total dominating set for an input graph. We prove that the TDS problem is NP-hard in unit disk graphs. Next, we propose an 8-factor approximation algorithm for the problem. The running time of the proposed approximation algorithm is $O(n \log k)$, where $n$ is the number of vertices of the input graph and $k$ is output size. We also show that TDS problem admits a PTAS in unit disk graphs.