We present poly log log n-round randomized distributed algorithms to compute vertex splittings, a partition of the vertices of a graph into k parts such that a node of degree d(u) has ≈ d(u)/k neighbors in each part. Our techniques can be seen as the first progress towards general poly log log n-round algorithms for the Lovász Local Lemma. As the main application of our result, we obtain a randomized poly log log n-round CONGEST algorithm for (1+ε)Δ-edge coloring n-node graphs of sufficiently large constant maximum degree Δ, for any ε > 0. Further, our results improve the computation of defective colorings and certain tight list coloring problems. All the results improve the state-of-the-art round complexity exponentially, even in the LOCAL model.
@InProceedings{halldorsson_et_al:LIPIcs.DISC.2022.26, author = {Halld\'{o}rsson, Magn\'{u}s M. and Maus, Yannic and Nolin, Alexandre}, title = {{Fast Distributed Vertex Splitting with Applications}}, booktitle = {36th International Symposium on Distributed Computing (DISC 2022)}, pages = {26:1--26:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-255-6}, ISSN = {1868-8969}, year = {2022}, volume = {246}, editor = {Scheideler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2022.26}, URN = {urn:nbn:de:0030-drops-172176}, doi = {10.4230/LIPIcs.DISC.2022.26}, annote = {Keywords: Graph problems, Edge coloring, List coloring, Lov\'{a}sz local lemma, LOCAL model, CONGEST model, Distributed computing} }
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