A quasi linear-time b-Matching algorithm on distance-hereditary graphs and bounded split-width graphs

We present a quasi linear-time algorithm for Maximum Matching on distance-hereditary graphs and some of their generalizations. This improves on [Dragan, WG'97], who proposed such an algorithm for the subclass of (tent,hexahedron)-free distance-hereditary graphs. Furthermore, our result is derived from a more general one that is obtained for b-Matching. In the (unit cost) b-Matching problem, we are given a graph G = (V, E) together with a nonnegative integer capacity b v for every vertex v $\in$ V. The objective is to assign nonnegative integer weights (x e) e$\in$E so that: for every v $\in$ V the sum of the weights of its incident edges does not exceed b v , and e$\in$E x e is maximized. We present the first algorithm for solving b-Matching on cographs, distance-hereditary graphs and some of their generalizations in quasi linear time. For that, we use a decomposition algorithm that outputs for any graph G a collection of subgraphs of G with no edge-cutsets inducing a complete bipartite subgraph (a.k.a., splits). The latter collection is sometimes called a split decomposition of G. Furthermore, there exists a generic method in order to design graph algorithms based on split decomposition [Rao, DAM'08]. However, this technique only applies to "localized" problems: for which a "best" partial solution for any given subgraph in a split decomposition can be computed almost independently from the remaining of the graph. Such framework does not apply to matching problems since an augmenting path may cross the subgraphs arbitrarily. We introduce a new technique that somehow captures all the partial solutions for a given union of subgraphs in a split decomposition, in a compact and amenable way for algorithms - assuming some piecewise linear assumption holds on the value of such solutions. The latter assumption is shown to hold for b-Matching. Doing so, we prove that solving b-Matching on any pair G, b can be reduced in quasi linear-time to solving this problem on a collection of smaller graphs: that are obtained from the subgraphs in any split decomposition of G by replacing every vertex with a constant-size module. In particular, if G has a split decomposition where all subgraphs have order at most a fixed k, then we can solve b-Matching for G, b in O((k log 2 k)$\times$(m+n)$\times$log ||b|| 1)-time. This answers an open question of [Coudert et al., SODA'18].