## Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs

Please always quote using this URN: urn:nbn:de:bvb:20-opus-267876

- A bipartite graph G=(U,V,E) is convex if the vertices in V can be linearly ordered such that for each vertex u∈U, the neighbors of u are consecutive in the ordering of V. An induced matching H of G is a matching for which no edge of E connects endpoints of two different edges of H. We show that in a convex bipartite graph with n vertices and m weighted edges, an induced matching of maximum total weight can be computed in O(n+m) time. An unweighted convex bipartite graph has a representation of size O(n) that records for each vertex u∈U theA bipartite graph G=(U,V,E) is convex if the vertices in V can be linearly ordered such that for each vertex u∈U, the neighbors of u are consecutive in the ordering of V. An induced matching H of G is a matching for which no edge of E connects endpoints of two different edges of H. We show that in a convex bipartite graph with n vertices and m weighted edges, an induced matching of maximum total weight can be computed in O(n+m) time. An unweighted convex bipartite graph has a representation of size O(n) that records for each vertex u∈U the first and last neighbor in the ordering of V. Given such a compact representation, we compute an induced matching of maximum cardinality in O(n) time. In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in O(n) time. If no compact representation is given, the cover can be computed in O(n+m) time. All of our algorithms achieve optimal linear running time for the respective problem and model, and they improve and generalize the previous results in several ways: The best algorithms for the unweighted problem versions had a running time of O(n\(^{2}\)) (Brandstädt et al. in Theor. Comput. Sci. 381(1–3):260–265, 2007. https://doi.org/10.1016/j.tcs.2007.04.006). The weighted case has not been considered before.…

Author: | Boris Klemz, Günter Rote |
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URN: | urn:nbn:de:bvb:20-opus-267876 |

Document Type: | Journal article |

Faculties: | Fakultät für Mathematik und Informatik / Institut für Informatik |

Language: | English |

Parent Title (English): | Algorithmica |

ISSN: | 1432-0541 |

Year of Completion: | 2022 |

Volume: | 84 |

Issue: | 4 |

Pagenumber: | 1064–1080 |

Source: | Algorithmica 2022, 84(4):1064–1080. DOI: 10.1007/s00453-021-00904-w |

DOI: | https://doi.org/10.1007/s00453-021-00904-w |

Dewey Decimal Classification: | 0 Informatik, Informationswissenschaft, allgemeine Werke / 00 Informatik, Wissen, Systeme / 004 Datenverarbeitung; Informatik |

Tag: | certifying algorithm; chain cover; convex bipartite graph; dynamic programming; graph algorithm; induced matching |

Release Date: | 2022/06/07 |

Licence (German): | CC BY: Creative-Commons-Lizenz: Namensnennung 4.0 International |