Liar’s Domination in 2D

Abstract

In this paper we consider Euclidean liar’s domination problem, a variant of dominating set problem. In the Euclidean liar’s domination problem, a set $\mathcal{P}=\{p_1,p_2,\ldots ,p_n\}$ of n points are given in the Euclidean plane. For $p \in \mathcal{P}$, N[p] is a subset of $\mathcal{P}$ such that for any $q \in N[p]$, the Euclidean distance between p and q is less than or equal to 1. The objective of the Euclidean liar’s domination problem is to find a subset $D (\subseteq \mathcal{P})$ of minimum size having the following properties: (i) $|N[p_i] \cap D| \ge 2$ for $1 \le i \le n$, and (ii) $|(N[p_i] \cup N[p_j]) \cap D| \ge 3$ for $i\ne j, 1\le i,j \le n$. We first propose a simple $O(n\log n)$ time $\frac{63}{2}$-factor approximation algorithm for the liar’s domination problem. Next we propose approximation algorithms to improve the approximation factor to $\frac{732}{k}$ for $3 \le k \le 183$, and $\frac{846}{k}$ for $3 \le k\le 282$. The running time of the algorithms is $O(n^{k+1}\varDelta )$, where $\varDelta = \max \{|N[p]| : p\in \mathcal{P}\}$.