## Abstract

Bipartite maximum matching and its variants are well-studied problems under various models of computation with the vast majority of approaches centering around various methods to find and eliminate augmenting paths. Beginning with the seminal papers of Demange, Gale and Sotomayor [DGS86] and Bertsekas [Ber81], bipartite maximum matching problems have also been studied in the context of auction algorithms. These algorithms model the maximum matching problem as an auction where one side of the bipartite graph consists of bidders and the other side consists of items; as such, these algorithms offer a very different approach to solving this problem that do not use classical methods. Dobzinski, Nisan and Oren [DNO14] demonstrated the utility of such algorithms in distributed, interactive settings by providing a simple and elegant O(log n/ε^{2}) round maximum cardinality bipartite matching (MCM) algorithm that has small round and communication complexity and gives a (1 − ε)-approximation for any (not necessarily constant) ε > 0. They leave as an open problem whether an auction algorithm, with similar guarantees, can be found for the maximum weighted bipartite matching (MWM) problem. Very recently, Assadi, Liu, and Tarjan [ALT21] extended the utility of auction algorithms for MCM into the semi-streaming and massively parallel computation (MPC) models, by cleverly using maximal matching as a subroutine, to give a new auction algorithm that uses O(1/ε^{2}) rounds and achieves the state-of-the-art bipartite MCM results in the streaming and MPC settings. In this paper, we give new auction algorithms for maximum weighted bipartite matching (MWM) and maximum cardinality bipartite b-matching (MCbM). Our algorithms run in O ^{(}log n/ε^{8)} and O ^{(}log n/ε^{2)} rounds, respectively, in the distributed setting. We show that our MWM algorithm can be implemented in the distributed, interactive setting using O(log^{2} n) and O(log n) bit messages, respectively, directly answering the open question posed by Demange, Gale and Sotomayor [DNO14]. Furthermore, we implement our algorithms in a variety of other models including the the semi-streaming model, the shared-memory work-depth model, and the massively parallel computation model. Our semi-streaming MWM algorithm uses O(1/ε^{8}) passes in O(n log n · log(1/ε)) space and our MCbM algorithm runs in O(1/ε^{2}) passes using O ^{((P}_{i}∈_{L} bi + |R|) log(1/ε)^{)} space (where parameters bi represent the degree constraints on the b-matching and L and R represent the left and right side of the bipartite graph, respectively). Both of these algorithms improves exponentially the dependence on ε in the space complexity in the semi-streaming model against the best-known algorithms for these problems, in addition to improvements in round complexity for MCbM. Finally, our algorithms eliminate the large polylogarithmic dependence on n in depth and number of rounds in the work-depth and massively parallel computation models, respectively, improving on previous results which have large polylogarithmic dependence on n (and exponential dependence on ε in the MPC model).

Original language | English (US) |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023 |

Editors | Nicole Megow, Adam Smith |

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

ISBN (Electronic) | 9783959772969 |

DOIs | |

State | Published - Sep 2023 |

Event | 26th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2023 and the 27th International Conference on Randomization and Computation, RANDOM 2023 - Atlanta, United States Duration: Sep 11 2023 → Sep 13 2023 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 275 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 26th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2023 and the 27th International Conference on Randomization and Computation, RANDOM 2023 |
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Country/Territory | United States |

City | Atlanta |

Period | 9/11/23 → 9/13/23 |

## Keywords

- auction algorithms
- distributed blackboard model
- massively parallel computation model
- maximum b-matching
- maximum weight bipartite matching
- parallel work-depth model
- streaming model

## ASJC Scopus subject areas

- Software