Maximum Matching in Almost Linear Time on Graphs of Bounded Clique-Width

Abstract

Recently, independent groups of researchers have presented algorithms to compute a maximum matching in \(\tilde{\mathcal{O}}(f(k) \cdot (n+m))\) 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 \(P_4\)-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 \(MSO_1\) 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. We also present how to compute the Edmonds-Gallai decomposition in almost linear time by using the same framework. For that, we do apply Courcelle’s theorem but to the classic Tutte-Berge formula, that can easily be expressed as a \(CMSO_1\) optimization problem. Doing so, we can compute the cardinality of a maximum matching, but not the matching itself. To obtain with this approach a maximum matching, we need to combine it with a recursive dissection scheme for bounded clique-width graphs 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. Finally, for the bipartite graphs of clique-width at most k, we present an alternative \(\tilde{\mathcal{O}}(k^2\cdot (n+m))\)-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.