Liar’s dominating set problem on unit disk graphs

In this paper, we consider Euclidean versions of the 2-tuple dominating set problem and the liar’s dominating set problem. For a given set P={p1,p2,…,pn} of n points in R2, the objective of the Euclidean 2-tuple dominating set problem is to find a minimum size set D⊆P such that |N[pi]∩D|≥2 for each pi∈P, where N[pi]={pj∈P∣δ(pi,pj)≤1} and δ(pi,pj) is the Euclidean distance between pi and pj. The objective of the Euclidean liar’s dominating set problem is to find a set D(⊆P) of minimum size satisfying the following two conditions: (i) D is a 2-tuple dominating set of P, and (ii) for every distinct pair of points pi and pj in P, |(N[pi]∪N[pj])∩D|≥3. We first propose a simple O(nlogn) time 632-factor approximation algorithm for the Euclidean liar’s dominating set problem. Next, we propose approximation algorithms to improve the approximation factor to 732α for 3≤α≤183, and 846α for 3≤α≤282. The running time of both the algorithms is O(nα+1Δ), where Δ=max{|N[p]|:p∈P}. Finally, we propose a PTAS for the Euclidean 2-tuple dominating set problem.