Algorithm and Hardness Results on Liar’s Dominating Set and $$\varvec{k}$$-tuple Dominating Set

Abstract

Given a graph G = (V, E), the dominating set problem asks for a minimum subset of vertices D ⊆ V such that every vertex u ∈ V \D is adjacent to at least one vertex v ∈ D. That is, the set D satisfies the condition that |NIn this paper, we study two variants of the classical dominating set problem: k-tuple dominating set (k-DS) problem and Liar's dominating set (LDS) problem, and obtain several algorithmic and hardness results.On the algorithmic side, we present a constant factor ( 11 2 )-approximation algorithm for the Liar's dominating set problem on unit disk graphs. Then, we obtain a PTAS for the k-tuple dominating set problem on unit disk graphs. On the hardness side, we show a Ω(n 2 ) bits lower bound for the space complexity of any (randomized) streaming algorithm for Liar's dominating set problem as well as for the k-tuple dominating set problem. Furthermore, we prove that the Liar's dominating set problem on bipartite graphs is W[2]-hard.

Liar's domination in unit disk graphs

“…Jallu and Das [7] first studied the geometric version of the MLDS problem, and presented constant factor approximation algorithms with high running time. Recently, Banerjee and Bhore [2] proposed a 11 2 -factor approximation algorithm in sub-quadratic time. However, unfortunately, their approximation analysis is erroneous and the approximation factor is at least 11 (refer Section 4).…”

“…Banerjee and Bhore [2] in their recent paper proposed an approximation algorithm and claim that their algorithm achieves a 5.5-factor approximation ratio for the MLDS problem in UDGs. However, their approximation analysis is erroneous.…”

“…For completeness here we give the idea of the algorithm proposed in [2] briefly. As a first step, the point set P (i.e., the set of disk centers) is sorted according to their x-coordinates.…”

“…We prove that the MLDS problem is NP-hard in unit disk graphs. Next, we show that the recent sub-quadratic time 11 2 -factor approximation algorithm [2] 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.…”

Balanced Independent and Dominating Sets on Colored Interval Graphs

“…The dominating set problem is another fundamental problem in theoretical computer science which also finds applications in various fields of science and engineering [6,12]. Several variants of the dominating set problem have been considered over the years: k-tuple dominating set [7], Liar's dominating set [3], independent dominating set [13], and more. The colored variant of the dominating set problem has been considered in parameterized complexity, namely, red-blue dominating set, where the objective is to choose a dominating set from one color class that dominates the other color class [9].…”