### 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 language | English (US) |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Publisher | Springer Verlag |

Pages | 49-59 |

Number of pages | 11 |

Volume | 8878 |

ISBN (Print) | 9783319144719 |

State | Published - 2014 |

Event | 18th International Conference on Principles of Distributed Systems, OPODIS 2014 - Cortina d’Ampezzo, Italy Duration: Dec 16 2014 → Dec 19 2014 |

### Publication series

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

Volume | 8878 |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 18th International Conference on Principles of Distributed Systems, OPODIS 2014 |
---|---|

Country | Italy |

City | Cortina d’Ampezzo |

Period | 12/16/14 → 12/19/14 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 8878, pp. 49-59). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8878). Springer Verlag.

**Distributed local approximation of the minimum k-tuple dominating set in planar graphs.** / Czygrinow, Andrzej; Hanćkowiak, Michal; Szymańska, Edyta; Wawrzyniak, Wojciech; Witkowski, Marcin.

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

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 8878, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 8878, Springer Verlag, pp. 49-59, 18th International Conference on Principles of Distributed Systems, OPODIS 2014, Cortina d’Ampezzo, Italy, 12/16/14.

}

TY - GEN

T1 - Distributed local approximation of the minimum k-tuple dominating set in planar graphs

AU - Czygrinow, Andrzej

AU - Hanćkowiak, Michal

AU - Szymańska, Edyta

AU - Wawrzyniak, Wojciech

AU - Witkowski, Marcin

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84917690957&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84917690957&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84917690957

SN - 9783319144719

VL - 8878

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

SP - 49

EP - 59

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

PB - Springer Verlag

ER -