## Abstract

The Maximal Independent Set problem is a well-studied problem in the distributed community. We study Maximum and Maximal Independent Set problems on two geometric intersection graphs; interval graphs and axis-parallel segment intersection graphs, and present deterministic distributed algorithms in the local communication model. We compute the maximum independent set on interval graphs in O(k) rounds and O(n) messages, where k is the size of the maximum independent set and n is the number of nodes in the graph. We provide a matching lower bound of Ω(k) on the number of rounds whereas Ω(n) is a trivial lower bound on message complexity. Thus our algorithm is both time and message optimal. We also study the maximal independent set problem in bi-interval graphs, a special case of the interval graphs where the intervals have two different lengths. We prove that a maximal independent set can be computed in bi-interval graphs in constant rounds that is 16 -approximation. For axis-parallel segment intersection graphs, we design an algorithm that finds a maximal independent set in O(D) rounds, where D is the diameter of the graph. We further show that this independent set is a 12 -approximation. The results in this paper extend the results of Molla et al. [J. Parallel Distrib. Comput. 2019].

Original language | English |
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Title of host publication | SOFSEM 2021 |

Subtitle of host publication | Theory and Practice of Computer Science - 47th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2021, Proceedings |

Editors | Tomáš Bureš, Riccardo Dondi, Johann Gamper, Giovanna Guerrini, Tomasz Jurdzinski, Claus Pahl, Florian Sikora, Prudence W. Wong |

Publisher | Springer Science and Business Media Deutschland GmbH |

Pages | 175-188 |

Number of pages | 14 |

ISBN (Print) | 9783030677305 |

DOIs | |

State | Published - 1 Jan 2021 |

Externally published | Yes |

Event | 47th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2021 - Bolzano-Bozen, Italy Duration: 25 Jan 2021 → 29 Jan 2021 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 12607 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 47th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2021 |
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Country/Territory | Italy |

City | Bolzano-Bozen |

Period | 25/01/21 → 29/01/21 |

## Keywords

- Approximation algorithm
- Distributed algorithm
- Interval graph
- Maximal Independent Set
- Segment intersection graph

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science