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

Sandip Banerjee; Sujoy Bhore

doi:10.1007/978-3-030-25005-8_5

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

Sandip Banerjee, Sujoy Bhore

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

3 Scopus citations

Abstract

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.

Original language	English
Title of host publication	Combinatorial Algorithms - 30th International Workshop, IWOCA 2019, Proceedings
Editors	Charles J. Colbourn, Roberto Grossi, Nadia Pisanti
Publisher	Springer Verlag
Pages	48-60
Number of pages	13
ISBN (Print)	9783030250041
DOIs	https://doi.org/10.1007/978-3-030-25005-8_5
State	Published - 1 Jan 2019
Externally published	Yes
Event	30th International Workshop on Combinatorial Algorithms, IWOCA 2019 - Pisa, Italy Duration: 23 Jul 2019 → 25 Jul 2019

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	11638 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	30th International Workshop on Combinatorial Algorithms, IWOCA 2019
Country/Territory	Italy
City	Pisa
Period	23/07/19 → 25/07/19

ASJC Scopus subject areas

Theoretical Computer Science
General Computer Science

Access to Document

10.1007/978-3-030-25005-8_5

Cite this

Banerjee, S., & Bhore, S. (2019). Algorithm and Hardness Results on Liar’s Dominating Set and k-tuple Dominating Set. In C. J. Colbourn, R. Grossi, & N. Pisanti (Eds.), Combinatorial Algorithms - 30th International Workshop, IWOCA 2019, Proceedings (pp. 48-60). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11638 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-030-25005-8_5

Banerjee, Sandip ; Bhore, Sujoy. / Algorithm and Hardness Results on Liar’s Dominating Set and k-tuple Dominating Set. Combinatorial Algorithms - 30th International Workshop, IWOCA 2019, Proceedings. editor / Charles J. Colbourn ; Roberto Grossi ; Nadia Pisanti. Springer Verlag, 2019. pp. 48-60 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

@inproceedings{1b7428f579784e3d9e79c9c76016c8bb,

title = "Algorithm and Hardness Results on Liar{\textquoteright}s Dominating Set and k-tuple Dominating Set",

abstract = "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{\textquoteright}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{\textquoteright}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{\textquoteright}s dominating set problem as well as for the k-tuple dominating set problem. Furthermore, we prove that the Liar{\textquoteright}s dominating set problem on bipartite graphs is W[2]-hard.",

author = "Sandip Banerjee and Sujoy Bhore",

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Banerjee, S & Bhore, S 2019, Algorithm and Hardness Results on Liar’s Dominating Set and k-tuple Dominating Set. in CJ Colbourn, R Grossi & N Pisanti (eds), Combinatorial Algorithms - 30th International Workshop, IWOCA 2019, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 11638 LNCS, Springer Verlag, pp. 48-60, 30th International Workshop on Combinatorial Algorithms, IWOCA 2019, Pisa, Italy, 23/07/19. https://doi.org/10.1007/978-3-030-25005-8_5

Algorithm and Hardness Results on Liar’s Dominating Set and k-tuple Dominating Set. / Banerjee, Sandip; Bhore, Sujoy.
Combinatorial Algorithms - 30th International Workshop, IWOCA 2019, Proceedings. ed. / Charles J. Colbourn; Roberto Grossi; Nadia Pisanti. Springer Verlag, 2019. p. 48-60 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11638 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

TY - GEN

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AU - Bhore, Sujoy

PY - 2019/1/1

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N2 - 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.

AB - 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.

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SN - 9783030250041

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

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BT - Combinatorial Algorithms - 30th International Workshop, IWOCA 2019, Proceedings

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Banerjee S, Bhore S. Algorithm and Hardness Results on Liar’s Dominating Set and k-tuple Dominating Set. In Colbourn CJ, Grossi R, Pisanti N, editors, Combinatorial Algorithms - 30th International Workshop, IWOCA 2019, Proceedings. Springer Verlag. 2019. p. 48-60. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-030-25005-8_5

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

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