## Abstract

A maximal matching can be maintained in fully dynamic (supporting both addition and deletion of edges) n-vertex graphs using a trivial deterministic algorithm with a worstcase update time of On). No deterministic algorithm that outperforms the näive O(n) one was reported up to this date. The only progress in this direction is due to Ivkovíc and Lloyd [14], who in 1993 devised a deterministic algorithm with an amortized update time of O((n+m) √ ^{2/2}), where m is the number of edges. In this paper we show the first deterministic fully dynamic algorithm that outperforms the trivial one. Specifically, we provide a deterministic worst-case update time of O( √ m). Moreover, our algorithm maintains a matching which is in fact a 3/2-approximate maximum cardinality matching (MCM). We remark that no fully dynamic algorithm for maintaining (2-ε)-approximate MCM improving upon the näive O(n) was known prior to this work, even allowing amortized time bounds and randomization. For low arboricity graphs (e.g., planar graphs and graphs excluding fixed minors), we devise another simple deterministic algorithm with sub-logarithmic update time. Specifically, it maintains a fully dynamic maximal matching with amortized update time of O(log n/ log log n). This result addresses an open question of Onak and Rubinfeld [19]. We also show a deterministic algorithm with optimal space usage of O(n + m), that for arbitrary graphs maintains a maximal matching with amortized update time of O( √ m).

Original language | English |
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Title of host publication | STOC 2013 - Proceedings of the 2013 ACM Symposium on Theory of Computing |

Pages | 745-753 |

Number of pages | 9 |

DOIs | |

State | Published - 11 Jul 2013 |

Event | 45th Annual ACM Symposium on Theory of Computing, STOC 2013 - Palo Alto, CA, United States Duration: 1 Jun 2013 → 4 Jun 2013 |

### Conference

Conference | 45th Annual ACM Symposium on Theory of Computing, STOC 2013 |
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Country/Territory | United States |

City | Palo Alto, CA |

Period | 1/06/13 → 4/06/13 |

## Keywords

- Dynamic algorithms
- Maximal matching
- Maximum matching

## ASJC Scopus subject areas

- Software