The b-Matching problem in distance-hereditary graphs and beyond

Guillaume Ducoffe (1, 2) , Alexandru Popa (1, 2)

1 ICI - National Institute for Research and Development in Informatics [Bucharest]
2 UniBuc - University of Bucharest

We make progress on the fine-grained complexity of Maximum-Cardinality Matching within graphs of bounded clique-width. Quasi linear-time algorithms for this problem have been 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((k log 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 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.

Mots clés: maximum-cardinality matching; b-matching; FPT in P; split decomposition; distance-hereditary graphs

Domaines: Complexité [cs.CC]; Algorithme et structure de données [cs.DS]

Guillaume Ducoffe, Alexandru Popa. The b-Matching problem in distance-hereditary graphs and beyond. Discrete Applied Mathematics, 2021, 305, pp.233-246. ⟨10.1016/j.dam.2021.09.012⟩. ⟨hal-03359175⟩