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|\geq 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: 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 \Omega (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.
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|\geq 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: -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 -tuple dominating set problem on unit disk graphs. On the hardness side, we show a \Omega (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 -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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