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 \(D\subseteq V\) such that every vertex \(u\in V\setminus D\) is adjacent to at least one vertex \(v\in D\). That is, the set D satisfies the condition that \(|N[v]\cap D|\ge 1\) for each \(v\in V\), where N[v] is the closed neighborhood of v. In this paper, we study two variants of the classical dominating set problem: \(\varvec{k}\)-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 (\(\frac{11}{2}\))-approximation algorithm for the Liar’s dominating set problem on unit disk graphs. Then, we design a polynomial time approximation scheme (PTAS) for the \(\varvec{k}\)-tuple dominating set problem on unit disk graphs. On the hardness side, we show a \(\varOmega (n^2)\) bits lower bound for the space complexity of any (randomized) streaming algorithm for Liar’s dominating set problem as well as for the \(\varvec{k}\)-tuple dominating set problem. Furthermore, we prove that the Liar’s dominating set problem on bipartite graphs is W[2]-hard.