Fast Distributed Vertex Splitting with Applications

TL;DR

This paper introduces fast randomized distributed algorithms with polyloglog n rounds for vertex splitting, leading to improved solutions for edge coloring, defective coloring, and list coloring problems in large graphs.

Contribution

It presents the first polyloglog n-round algorithms for vertex splitting and applies them to significantly improve distributed graph coloring methods.

Findings

Polyloglog n-round randomized algorithms for vertex splitting.

Exponential improvement in round complexity for graph coloring tasks.

Enhanced algorithms for defective and list coloring problems.

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.