## Abstract

Fundamental local symmetry breaking problems such as Maximal Independent Set (MIS) and coloring have been recognized as important by the community, and studied extensively in (standard) graphs. In particular, fast (i.e., logarithmic run time) randomized algorithms are well-established for MIS and Δ + 1-coloring in both the LOCAL and CONGEST distributed computing models. On the other hand, comparatively much less is known on the complexity of distributed symmetry breaking in hypergraphs. In particular, a key question is whether a fast (randomized) algorithm for MIS exists for hypergraphs.

In this paper, we study the distributed complexity of symmetry breaking in hypergraphs by presenting distributed randomized algorithms for a variety of fundamental problems under a natural distributed computing model for hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can be solved in O(log^{2}n) rounds (n is the number of nodes of the hypergraph) in the LOCAL model. We then present a key result of this paper — an O(Δ^{∈}polylog n)-round hypergraph MIS algorithm in the CONGEST model where Δ is the maximum node degree of the hypergraph and ∈ >0 is any arbitrarily small constant. We also present distributed algorithms for coloring, maximal matching, and maximal clique in hypergraphs.

To demonstrate the usefulness of hypergraph MIS, we present applications of our hypergraph algorithm to solving problems in (standard) graphs. In particular, the hypergraph MIS yields fast distributed algorithms for the balanced minimal dominating set problem (left open in Harris et al. [ICALP 2013]) and the minimal connected dominating set problem.

Our work shows that while some local symmetry breaking problems such as coloring can be solved in polylogarithmic rounds in both the LOCAL and CONGEST models, for many other hypergraph problems such as MIS, hitting set, and maximal clique, it remains challenging to obtain polylogarithmic time algorithms in the CONGEST model. This work is a step towards understanding this dichotomy in the complexity of hypergraph problems as well as using hypergraphs to design fast distributed algorithms for problems in (standard) graphs.

Original language | English (US) |
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Title of host publication | Distributed Computing - 28th International Symposium, DISC 2014, Proceedings |

Editors | Fabian Kuhn |

Publisher | Springer Verlag |

Pages | 469-483 |

Number of pages | 15 |

ISBN (Electronic) | 9783662451731 |

DOIs | |

State | Published - 2014 |

Externally published | Yes |

Event | 28th International Symposium on Distributed Computing, DISC 2014 - Austin, United States Duration: Oct 12 2014 → Oct 15 2014 |

### 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 | 8784 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 28th International Symposium on Distributed Computing, DISC 2014 |
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Country/Territory | United States |

City | Austin |

Period | 10/12/14 → 10/15/14 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science