## Abstract

A matching of a graph is a subset of edges no two of which share a common vertex, and a maximum matching is a matching of maximum cardinality. In a b-matching every vertex v has an associated bound b_{v}, and a maximum b-matching is a maximum set of edges, such that every vertex v appears in at most b_{v} of them. We study an extension of this problem, termed Hierarchical b-Matching. In this extension, the vertices are arranged in a hierarchical manner. At the first level the vertices are partitioned into disjoint subsets, with a given bound for each subset. At the second level the set of these subsets is again partitioned into disjoint subsets, with a given bound for each subset, and so on. We seek for a maximum set of edges, that obey all bounds (that is, no vertex v participates in more than b_{v} edges, then all the vertices in one subset do not participate in more that subset’s bound of edges, and so on hierarchically). This is a sub-problem of the matroid matching problem which is NP -hard in general. It corresponds to the special case where the matroid is restricted to be laminar and the weights are unity. A pseudo-polynomial algorithm for the weighted laminar matroid matching problem is presented in [8]. We propose a polynomial-time algorithm for Hierarchical b-matching, i.e. the unweighted laminar matroid matching problem, and discuss how our techniques can possibly be generalized to the weighted case.

Original language | English |
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Title of host publication | SOFSEM 2021 |

Subtitle of host publication | Theory and Practice of Computer Science - 47th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2021, Proceedings |

Editors | Tomáš Bureš, Riccardo Dondi, Johann Gamper, Giovanna Guerrini, Tomasz Jurdzinski, Claus Pahl, Florian Sikora, Prudence W. Wong |

Pages | 189-202 |

Number of pages | 14 |

DOIs | |

State | Published - 2021 |

Event | 47th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2021 - Bolzano-Bozen, Italy Duration: 25 Jan 2021 → 29 Jan 2021 |

### 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 | 12607 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 47th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2021 |
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Country/Territory | Italy |

City | Bolzano-Bozen |

Period | 25/01/21 → 29/01/21 |

## Keywords

- Matching
- Matroids
- b-matching

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science