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

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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.

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Notes

  1. More generally, the goal is, for some problem solvable in \(\mathcal{O}(m^{q+o(1)})\) time on arbitrary m-edge graphs, to design an \(\mathcal{O}(f(k) \cdot m^{p+o(1)})\)-time algorithm, for some \(p < q\), within the class of graphs where some fixed parameter is at most k.

  2. The \(\tilde{\mathcal{O}}()\) notation suppresses poly-logarithmic factors.

  3. The result is stated in [26] for minimization problems. Since we only consider it here for Maximum b-Matching, we rather write it as a maximization problem.

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We thank the anonymous reviewers for their valuable feedback.

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  1. National Institute for Research and Development in Informatics, Bucharest, Romania

    Guillaume Ducoffe

  2. University of Bucharest, Bucharest, Romania

    Guillaume Ducoffe

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This work was supported by Grant TC ICUB-SSE 15109-26.07.2021, “The complexity landscape of Maximum Matching”. It was also supported by project PN-19-37-04-01 “New solutions for complex problems in current ICT research fields based on modelling and optimization”, funded by the Romanian Core Program of the Ministry of Research and Innovation (MCI) 2019-2022.

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Results of this paper were partially presented at the IPEC’21 conference [28].

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Ducoffe, G. Maximum Matching in Almost Linear Time on Graphs of Bounded Clique-Width. Algorithmica 84, 3489–3520 (2022). https://doi.org/10.1007/s00453-022-00999-9

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