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arXiv · 2208.08119

Fast Distributed Vertex Splitting with Applications

Magnús M. Halldórsson·Yannic Maus·Alexandre Nolin
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Abstract

We present ${\rm poly\log\log n}$-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 $\approx d(u)/k$ neighbors in each part. Our techniques can be seen as the first progress towards general ${\rm poly\log\log n}$-round algorithms for the Lov\'asz Local Lemma. As the main application of our result, we obtain a randomized ${\rm poly\log\log n}$-round CONGEST algorithm for $(1+\epsilon)\Delta$-edge coloring $n$-node graphs of sufficiently large constant maximum degree $\Delta$, for any $\epsilon>0$. 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.

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Magnús M. Halldórsson, Yannic Maus, Alexandre Nolin. 2022-08-17. Fast Distributed Vertex Splitting with Applications. https://arxiv.org/abs/2208.08119

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