Complexity and Approximability of Edge-Vertex Domination in UDG

TL;DR

This paper studies the edge-vertex dominating set problem in graphs, proving its NP-hardness on unit disk graphs, and presents a polynomial-time approximation scheme along with a simple 5-factor approximation algorithm.

Contribution

It establishes NP-hardness of the problem on unit disk graphs and provides both a PTAS and a simple 5-approximation algorithm.

Findings

NP-hardness of ev-dominating set problem on unit disk graphs

Existence of a polynomial-time approximation scheme (PTAS) for the problem

A simple linear-time 5-factor approximation algorithm

Abstract

Given an undirected graph $G=(V,E)$, a vertex $v\in V$ is edge-vertex (ev) dominated by an edge $e\in E$ if $v$ is either incident to $e$ or incident to an adjacent edge of $e$. A set $S^{ev}\subseteq E$ is an edge-vertex dominating set (referred to as \textit{ev}-dominating set and in short as \textit{EVDS}) of $G$ if every vertex of $G$ is \textit{ev}-dominated by at least one edge of $S^{ev}$. The minimum cardinality of an \textit{ev}-dominating set is the \textit{ev}-domination number. The edge-vertex dominating set problem is to find a minimum \textit{ev}-domination number. In this paper, we prove that the \textit{ev}-dominating set problem is {\tt NP-hard} on unit disk graphs. We also prove that this problem admits a polynomial-time approximation scheme on unit disk graphs. Finally, we give a simple 5-factor linear-time approximation algorithm.