Maximum Matching in almost linear time on graphs of bounded clique-width

Guillaume Ducoffe (1, 2)

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

Recently, independent groups of researchers have presented algorithms to compute a maximum matching in Õ(f (k) • (n + m)) time, for some computable function f, within the graphs where some clique-width upper bound is at most k (e.g., tree-width, modular-width and P4-sparseness). However, to the best of our knowledge, the existence of such algorithm within the graphs of bounded clique-width has remained open until this paper. Indeed, we cannot even apply Courcelle's theorem to this problem directly, because a matching cannot be expressed in MSO1 logic. Our first contribution is an almost linear-time algorithm to compute a maximum matching in any bounded clique-width graph, being given a corresponding clique-width expression. It can be used to also compute the Edmonds-Gallai decomposition. For that, we do apply Courcelle's theorem, but in order to compute the cardinality of a maximum matching rather than the matching itself, via the classic Tutte-Berge formula. To obtain with this approach a maximum matching, we need to combine it with a recursive dissection scheme for bounded clique-width graphs based on the existence of balanced edge-cuts with bounded neighbourhood diversity, and with a distributed version of Courcelle's theorem (Courcelle and Vanicat, DAM 2016) -- of which we present here a slightly stronger version than the standard one in the literature -- in order to evaluate the Tutte-Berge formula on various subgraphs of the input. Finally, for the bipartite graphs of clique-width at most k, we present an alternative Õ(k^2 •(n+m))-time algorithm for the problem. The algorithm is randomized and it is based on a completely different approach than above: combining various reductions to matching and flow problems on bounded tree-width graphs with a very recent result on the parameterized complexity of linear programming (Dong et. al., STOC'21). Our results for bounded clique-width graphs extend many prior works on the complexity of Maximum Matching within cographs, distance-hereditary graphs, series-parallel graphs and other subclasses.

Mots clés: Maximum Matching; Maximum b-matching; Clique-width; Tree-width; Courcelle's theorem; FPT in P

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

Guillaume Ducoffe. Maximum Matching in almost linear time on graphs of bounded clique-width. 16th International Symposium on Parameterized and Exact Computation (IPEC 2021), Sep 2021, Lisbon (virtual event), Portugal. ⟨10.4230/LIPIcs.IPEC.2021.15⟩. ⟨hal-03445479⟩