### Abstract

We make progress on the fine-grained complexity of Maximum-Cardinality Matching on graphs of bounded clique-width. Quasi linear-time algorithms for this problem have been recently proposed for the important subclasses of bounded-treewidth graphs (Fomin et al., SODA'17) and graphs of bounded modular-width (Coudert et al., SODA'18). We present such algorithm for bounded split-width graphs - a broad generalization of graphs of bounded modular-width, of which an interesting subclass are the distance-hereditary graphs. Specifically, we solve Maximum-Cardinality Matching in O((klog
^{2}
k)·(m+n)·log n)-time on graphs with split-width at most k. We stress that the existence of such algorithm was not even known for distance-hereditary graphs until our work. Doing so, we improve the state of the art (Dragan, WG'97) and we answer an open question of (Coudert et al., SODA'18). Our work brings more insights on the relationships between matchings and splits, a.k.a., join operations between two vertex-subsets in different connected components. Furthermore, our analysis can be extended to the more general (unit cost) b-Matching problem. On the way, we introduce new tools for b-Matching and dynamic programming over split decompositions, that can be of independent interest.

Original language | English |
---|---|

Title of host publication | 29th International Symposium on Algorithms and Computation, ISAAC 2018 |

Editors | Der-Tsai Lee, Chung-Shou Liao, Wen-Lian Hsu |

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

ISBN (Electronic) | 9783959770941 |

DOIs | |

Publication status | Published - Dec 1 2018 |

Event | 29th International Symposium on Algorithms and Computation, ISAAC 2018 - Jiaoxi, Yilan, Taiwan Duration: Dec 16 2018 → Dec 19 2018 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
---|---|

Volume | 123 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 29th International Symposium on Algorithms and Computation, ISAAC 2018 |
---|---|

Country | Taiwan |

City | Jiaoxi, Yilan |

Period | 12/16/18 → 12/19/18 |

### Fingerprint

### Keywords

- B-matching
- Distance-hereditary graphs
- FPT in P
- Maximum-cardinality matching
- Split decomposition

### ASJC Scopus subject areas

- Software

### Cite this

*29th International Symposium on Algorithms and Computation, ISAAC 2018*(Leibniz International Proceedings in Informatics, LIPIcs; Vol. 123). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ISAAC.2018.30

**The b-matching problem in distance-hereditary graphs and beyond.** / Ducoffe, Guillaume; Popa, Alexandru.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*29th International Symposium on Algorithms and Computation, ISAAC 2018.*Leibniz International Proceedings in Informatics, LIPIcs, vol. 123, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 29th International Symposium on Algorithms and Computation, ISAAC 2018, Jiaoxi, Yilan, Taiwan, 12/16/18. https://doi.org/10.4230/LIPIcs.ISAAC.2018.30

}

TY - GEN

T1 - The b-matching problem in distance-hereditary graphs and beyond

AU - Ducoffe, Guillaume

AU - Popa, Alexandru

PY - 2018/12/1

Y1 - 2018/12/1

N2 - We make progress on the fine-grained complexity of Maximum-Cardinality Matching on graphs of bounded clique-width. Quasi linear-time algorithms for this problem have been recently proposed for the important subclasses of bounded-treewidth graphs (Fomin et al., SODA'17) and graphs of bounded modular-width (Coudert et al., SODA'18). We present such algorithm for bounded split-width graphs - a broad generalization of graphs of bounded modular-width, of which an interesting subclass are the distance-hereditary graphs. Specifically, we solve Maximum-Cardinality Matching in O((klog 2 k)·(m+n)·log n)-time on graphs with split-width at most k. We stress that the existence of such algorithm was not even known for distance-hereditary graphs until our work. Doing so, we improve the state of the art (Dragan, WG'97) and we answer an open question of (Coudert et al., SODA'18). Our work brings more insights on the relationships between matchings and splits, a.k.a., join operations between two vertex-subsets in different connected components. Furthermore, our analysis can be extended to the more general (unit cost) b-Matching problem. On the way, we introduce new tools for b-Matching and dynamic programming over split decompositions, that can be of independent interest.

AB - We make progress on the fine-grained complexity of Maximum-Cardinality Matching on graphs of bounded clique-width. Quasi linear-time algorithms for this problem have been recently proposed for the important subclasses of bounded-treewidth graphs (Fomin et al., SODA'17) and graphs of bounded modular-width (Coudert et al., SODA'18). We present such algorithm for bounded split-width graphs - a broad generalization of graphs of bounded modular-width, of which an interesting subclass are the distance-hereditary graphs. Specifically, we solve Maximum-Cardinality Matching in O((klog 2 k)·(m+n)·log n)-time on graphs with split-width at most k. We stress that the existence of such algorithm was not even known for distance-hereditary graphs until our work. Doing so, we improve the state of the art (Dragan, WG'97) and we answer an open question of (Coudert et al., SODA'18). Our work brings more insights on the relationships between matchings and splits, a.k.a., join operations between two vertex-subsets in different connected components. Furthermore, our analysis can be extended to the more general (unit cost) b-Matching problem. On the way, we introduce new tools for b-Matching and dynamic programming over split decompositions, that can be of independent interest.

KW - B-matching

KW - Distance-hereditary graphs

KW - FPT in P

KW - Maximum-cardinality matching

KW - Split decomposition

UR - http://www.scopus.com/inward/record.url?scp=85063666747&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85063666747&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ISAAC.2018.30

DO - 10.4230/LIPIcs.ISAAC.2018.30

M3 - Conference contribution

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 29th International Symposium on Algorithms and Computation, ISAAC 2018

A2 - Lee, Der-Tsai

A2 - Liao, Chung-Shou

A2 - Hsu, Wen-Lian

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -