Distributed local approximation of the minimum k-tuple dominating set in planar graphs

Andrzej Czygrinow, Michal Hanćkowiak, Edyta Szymańska, Wojciech Wawrzyniak, Marcin Witkowski

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Scopus citations

Abstract

In this paper we consider a generalization of the classical dominating set problem to the k-tuple dominating set problem (kMDS). For any positive integer k, we look for a smallest subset of vertices D ⊆ V with the property that every vertex in V\D is adjacent to at least k vertices of D. We are interested in the distributed complexity of this problem in the model, where the nodes have no identifiers. The most challenging case is when k = 2, and for this case we propose a distributed local algorithm, which runs in a constant number of rounds, yielding a 7-approximation in the class of planar graphs. On the other hand, in the class of algorithms in which every vertex uses only its degree and the degree of its neighbors to make decisions, there is no algorithm providing a (5 -ϵ)-approximation of the 2MDS problem. In addition, we show a lower bound of (4 - ϵ) for the 2MDS problem even if unique identifiers are allowed.

For k ≥ 3, we show that for the problem kMDS in planar graphs, a trivial algorithm yields a k/(k - 2)-approximation. In the model with unique identifiers this, surprisingly, is optimal for k = 3, 4, 5, and 6, as we provide a matching lower bound.

Original languageEnglish (US)
Title of host publicationPrinciples of Distributed Systems - 18th International Conference, OPODIS 2014, Proceedings
EditorsMarcos K. Aguilera, Leonardo Querzoni, Marc Shapiro
PublisherSpringer Verlag
Pages49-59
Number of pages11
ISBN (Electronic)9783319144719
DOIs
StatePublished - Jan 1 2014
Event18th International Conference on Principles of Distributed Systems, OPODIS 2014 - Cortina d’Ampezzo, Italy
Duration: Dec 16 2014Dec 19 2014

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume8878
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other18th International Conference on Principles of Distributed Systems, OPODIS 2014
CountryItaly
CityCortina d’Ampezzo
Period12/16/1412/19/14

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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  • Cite this

    Czygrinow, A., Hanćkowiak, M., Szymańska, E., Wawrzyniak, W., & Witkowski, M. (2014). Distributed local approximation of the minimum k-tuple dominating set in planar graphs. In M. K. Aguilera, L. Querzoni, & M. Shapiro (Eds.), Principles of Distributed Systems - 18th International Conference, OPODIS 2014, Proceedings (pp. 49-59). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8878). Springer Verlag. https://doi.org/10.1007/978-3-319-14472-6_4