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S. Banerjee, S. Bhore:
"Algorithm and Hardness Results on Liar´s Dominating Set and k-tuple Dominating Set";
Talk: International Workshop on Combinatorial Algorithms (IWOCA), Pisa; 2019-07-23 - 2019-07-25; in: "IWOCA 2019: Combinatorial Algorithms", LNCS, 11638 (2019), ISBN: 978-3-030-25004-1; 48 - 60.

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 |N[v] ∩ D| ≥ 1 for each v ∈ 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 Ω(n2) 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.

http://dx.doi.org/10.1007/978-3-030-25005-8_4

https://publik.tuwien.ac.at/files/publik_284561.pdf