Algorithm and Hardness Results on Liar's Dominating Set and k-tuple Dominating Set

Given a graph G = (V, E), the dominating set problem asks for a minimum subset of vertices D subset of V such that every vertex u is an element of V \ D is adjacent to at least one vertex v is an element of D. That is, the set D satisfies the condition that vertical bar N[v] boolean AND D vertical bar >= 1 for each v is an element of 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 design a polynomial time approximation scheme (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.