## Abstract

We study distributed algorithms for the maximum matching problem in the CONGEST model, where each message must be bounded in size. We give new deterministic upper bounds, and a new lower bound on the problem. We begin by giving a distributed algorithm that computes an exact maximum (unweighted) matching in bipartite graphs, in O(n log n) rounds. Next, we give a distributed algorithm that approximates the fractional weighted maximum matching problem in general graphs. In a graph with maximum degree at most ∆, the algorithm computes a (1−ε)-approximation for the problem in time O log(∆W)/ε ^{2} , where W is a bound on the ratio between the largest and the smallest edge weight. Next, we show a slightly improved and generalized version of the deterministic rounding algorithm of Fischer [DISC’17]. Given a fractional weighted maximum matching solution of value f for a given graph G, we show that in time O((log ^{2} (∆) + log ^{∗} n)/ε), the fractional solution can be turned into an integer solution of value at least (1 − ε)f for bipartite graphs and (1 − ε) · ^{g−} _{g} ^{1} · f for general graphs, where g is the length of the shortest odd cycle of G. Together with the above fractional maximum matching algorithm, this implies a deterministic algorithm that computes a (1 − ε) · ^{g−} _{g} ^{1} -approximation for the weighted maximum matching problem in time O log(∆W)/ε ^{2} + (log ^{2} (∆) + log ^{∗} n)/ε. On the lower-bound front, we show that even for unweighted fractional maximum matching in bipartite graphs, computing an (1 − O(1/n))-approximate solution requires at least Ω( D + n) rounds in CONGEST. This lower bound requires the introduction of a new 2-party communication problem, for which we prove a tight lower bound.

Original language | English |
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Title of host publication | 32nd International Symposium on Distributed Computing, DISC 2018 |

Editors | Ulrich Schmid, Josef Widder |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770927 |

DOIs | |

State | Published - 1 Oct 2018 |

Event | 32nd International Symposium on Distributed Computing, DISC 2018 - New Orleans, United States Duration: 15 Oct 2018 → 19 Oct 2018 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 121 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 32nd International Symposium on Distributed Computing, DISC 2018 |
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Country/Territory | United States |

City | New Orleans |

Period | 15/10/18 → 19/10/18 |

## Keywords

- Communication complexity
- Deterministic rounding
- Distributed graph algorithms
- Maximum matching