We study the maximum cardinality matching problem in a standard distributed setting, where the nodes V of a given n-node network graph G = (V,E) communicate over the edges E in synchronous rounds. More specifically, we consider the distributed CONGEST model, where in each round, each node of G can send an O(log n)-bit message to each of its neighbors. We show that for every graph G and a matching M of G, there is a randomized CONGEST algorithm to verify M being a maximum matching of G in time O(|M|) and disprove it in time O(D + 𝓁), where D is the diameter of G and 𝓁 is the length of a shortest augmenting path. We hope that our algorithm constitutes a significant step towards developing a CONGEST algorithm to compute a maximum matching in time Õ(s^*), where s^* is the size of a maximum matching.
@InProceedings{ahmadi_et_al:LIPIcs.DISC.2020.37, author = {Ahmadi, Mohamad and Kuhn, Fabian}, title = {{Distributed Maximum Matching Verification in CONGEST}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {37:1--37:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.37}, URN = {urn:nbn:de:0030-drops-131151}, doi = {10.4230/LIPIcs.DISC.2020.37}, annote = {Keywords: distributed matching, distributed graph algorithms, augmenting paths} }
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