Algorithm and Hardness Results on Liar's Dominating Set and \varveck tuple Dominating Set
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.
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