## Abstract

Approximate matchings in fully dynamic graphs have been intensively studied in recent years. Gupta and Peng [FOCS'13J presented a deterministic algorithm for maintaining fully dynamic (1 +∈)-approximate maximum cardinality matching (MCM) in general graphs with worst-case update time 0(√m· ∈^{-2}), for any ∈ > 0, where m denotes the current number of edges in the graph. Despite significant research efforts, this √m update time barrier remains the state-of-the-art even if amortized time bounds and randomization are allowed or the approximation factor is allowed to increase from 1 + ∈ to 2-∈, and even in basic graph families such as planar graphs. This paper presents a simple deterministic algorithm whose performance depends on the density of the graph. Specifically, we maintain fully dynamic (1 + ∈)-approximate MCM with worst-case update time O(α· ∈^{-2}) for graphs with arboricity1 bounded by α. The update time bound holds even if the arboricity bound a changes dynamically. Since the arboricity ranges between 1 and √m, our density-sensitive bound O(α· ∈^{-2}) naturally generalizes the O(√m· ∈^{-2}) bound of Gupta and Peng. For the family of bounded arboricity graphs (which includes forests, planar graphs, and graphs excluding a fixed minor), in the regime ∈ = O(1) our update time reduces to a constant. This should be contrasted with the previous best 2-approximation results for bounded arboricity graphs, which achieve either an O(logn) worst-case bound (Kopelowitz et al., ICALP'14) or an O(√logn) amortized bound (He et al., ISAAC'14), where n stands for the number of vertices in the graph. En route to this result, we provide local algorithms of independent interest for maintaining fully dynamic approximate matching and vertex cover.

Original language | English |
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Title of host publication | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 |

Editors | Robert Krauthgamer |

Publisher | Association for Computing Machinery |

Pages | 712-729 |

Number of pages | 18 |

ISBN (Electronic) | 9781510819672 |

DOIs | |

State | Published - 2016 |

Externally published | Yes |

Event | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 - Arlington, United States Duration: 10 Jan 2016 → 12 Jan 2016 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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Volume | 2 |

### Conference

Conference | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 |
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Country/Territory | United States |

City | Arlington |

Period | 10/01/16 → 12/01/16 |

### Bibliographical note

Funding Information:Supported in part by the Israel Science Foundation (grant 1549/13), the I-CORE program of the Israel PBC and ISF (grant 4/11), and the United States-Israel Binational Science Foundation (grant 2008348).%blankline%

### Funding

Supported in part by the Israel Science Foundation (grant 1549/13), the I-CORE program of the Israel PBC and ISF (grant 4/11), and the United States-Israel Binational Science Foundation (grant 2008348).%blankline%

Funders | Funder number |
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Israel PBC | |

United States-Israel Binational Science Foundation | 2008348 |

Israel Science Foundation | 1549/13, 4/11 |

Israeli Centers for Research Excellence |