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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.IPEC.2021.15
URN: urn:nbn:de:0030-drops-153987
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/15398/
Ducoffe, Guillaume
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 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(k²⋅(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.
BibTeX - Entry
@InProceedings{ducoffe:LIPIcs.IPEC.2021.15,
author = {Ducoffe, Guillaume},
title = {{Maximum Matching in Almost Linear Time on Graphs of Bounded Clique-Width}},
booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)},
pages = {15:1--15:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-216-7},
ISSN = {1868-8969},
year = {2021},
volume = {214},
editor = {Golovach, Petr A. and Zehavi, Meirav},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/15398},
URN = {urn:nbn:de:0030-drops-153987},
doi = {10.4230/LIPIcs.IPEC.2021.15},
annote = {Keywords: Maximum Matching, Maximum b-matching, Clique-width, Tree-width, Courcelle’s theorem, FPT in P}
}
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Keywords: |
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Maximum Matching, Maximum b-matching, Clique-width, Tree-width, Courcelle’s theorem, FPT in P |
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Collection: |
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16th International Symposium on Parameterized and Exact Computation (IPEC 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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22.11.2021 |
