Liar's Domination in Unit Disk Graphs

TL;DR

This paper investigates the NP-hardness of the minimum liar's dominating set problem in unit disk graphs, corrects previous approximation claims, proposes a new efficient approximation algorithm, and establishes the problem's polynomial-time approximability.

Contribution

It proves NP-hardness of MLDS in unit disk graphs, corrects a flawed approximation algorithm, introduces a simple efficient approximation, and shows MLDS admits a PTAS.

Findings

MLDS is NP-hard in unit disk graphs.

The previous $rac{11}{2}$-factor approximation algorithm is incorrect.

A new $O(n + m)$ time 7.31-factor approximation algorithm is proposed.

Abstract

In this article, we study a variant of the minimum dominating set problem known as the minimum liar's dominating set (MLDS) problem. We prove that the MLDS problem is NP-hard in unit disk graphs. Next, we show that the recent sub-quadratic time $\frac{11}{2}$-factor approximation algorithm \cite{bhore} for the MLDS problem is erroneous and propose a simple $O(n + m)$ time 7.31-factor approximation algorithm, where $n$ and $m$ are the number of vertices and edges in the input unit disk graph, respectively. Finally, we prove that the MLDS problem admits a polynomial-time approximation scheme.