Distributed Deterministic Edge Coloring using Bounded Neighborhood Independence

TL;DR

This paper introduces a significantly faster deterministic distributed algorithm for edge coloring in graphs, leveraging vertex coloring on graphs with bounded neighborhood independence, outperforming previous methods especially for large maximum degree Delta.

Contribution

The paper presents a novel deterministic algorithm for edge coloring that is faster than existing algorithms, utilizing a new vertex coloring approach for graphs with bounded neighborhood independence.

Findings

Achieves O(Delta)-edge-coloring in O(Delta^{epsilon}) + log-star n time.

Provides an O(Delta^{1 + epsilon})-edge-coloring in O(log Delta) + log-star n time.

Outperforms all existing randomized algorithms for small Delta values.

Abstract

We study the {edge-coloring} problem in the message-passing model of distributed computing. This is one of the most fundamental and well-studied problems in this area. Currently, the best-known deterministic algorithms for (2Delta -1)-edge-coloring requires O(Delta) + log-star n time \cite{PR01}, where Delta is the maximum degree of the input graph. Also, recent results of \cite{BE10} for vertex-coloring imply that one can get an O(Delta)-edge-coloring in O(Delta^{epsilon} \cdot \log n) time, and an O(Delta^{1 + epsilon})-edge-coloring in O(log Delta log n) time, for an arbitrarily small constant epsilon > 0. In this paper we devise a drastically faster deterministic edge-coloring algorithm. Specifically, our algorithm computes an O(Delta)-edge-coloring in O(Delta^{epsilon}) + log-star n time, and an O(Delta^{1 + epsilon})-edge-coloring in O(log Delta) + log-star n time. This result improves the previous state-of-the-art {exponentially} in a wide range of Delta, specifically, for 2^{Omega(\log-star n)} \leq Delta \leq polylog(n). In addition, for small values of Delta our deterministic algorithm outperforms all the existing {randomized} algorithms for this problem. On our way to these results we study the {vertex-coloring} problem on the family of graphs with bounded {neighborhood independence}. This is a large family, which strictly includes line graphs of r-hypergraphs for any r = O(1), and graphs of bounded growth. We devise a very fast deterministic algorithm for vertex-coloring graphs with bounded neighborhood independence. This algorithm directly gives rise to our edge-coloring algorithms, which apply to {general} graphs. Our main technical contribution is a subroutine that computes an O(Delta/p)-defective p-vertex coloring of graphs with bounded neighborhood independence in O(p^2) + \log-star n time, for a parameter p, 1 \leq p \leq Delta.