Faster Vizing and Near-Vizing Edge Coloring Algorithms

TL;DR

This paper introduces a fast randomized algorithm for near-Vizing edge coloring, achieving near-linear expected time for coloring graphs with a number of colors close to the maximum degree, improving over decades-old bounds.

Contribution

The paper presents the first near-linear time randomized algorithm for near-Vizing edge coloring, significantly improving the runtime for coloring graphs close to the optimal number of colors.

Findings

Achieves $ ext{O}(m ext{log} ext{Delta})$ expected time for $ ext{Delta}+O( ext{log} n)$ coloring.

Provides a $ ext{O}(n^2 ext{log} n)$ expected time algorithm for $( ext{Delta}+1)$ coloring.

Offers an $ ext{O}(m ext{log}(1/ ext{epsilon}))$ expected time algorithm for $(1+ ext{epsilon}) ext{Delta}$ coloring.

Abstract

Vizing's celebrated theorem states that every simple graph with maximum degree $\Delta$ admits a $(\Delta+1)$ edge coloring which can be found in $O(m \cdot n)$ time on $n$-vertex $m$-edge graphs. This is just one color more than the trivial lower bound of $\Delta$ colors needed in any proper edge coloring. After a series of simplifications and variations, this running time was eventually improved by Gabow, Nishizeki, Kariv, Leven, and Terada in 1985 to $O(m\sqrt{n\log{n}})$ time. This has effectively remained the state-of-the-art modulo an $O(\sqrt{\log{n}})$-factor improvement by Sinnamon in 2019. As our main result, we present a novel randomized algorithm that computes a $\Delta+O(\log{n})$ coloring of any given simple graph in $O(m\log{\Delta})$ expected time; in other words, a near-linear time randomized algorithm for a ``near''-Vizing's coloring. As a corollary of this algorithm, we also obtain the following results: * A randomized algorithm for $(\Delta+1)$ edge coloring in $O(n^2\log{n})$ expected time. This is near-linear in the input size for dense graphs and presents the first polynomial time improvement over the longstanding bounds of Gabow et.al. for Vizing's theorem in almost four decades. * A randomized algorithm for $(1+\varepsilon) \Delta$ edge coloring in $O(m\log{(1/\varepsilon)})$ expected time for any $\varepsilon = \omega(\log{n}/\Delta)$. The dependence on $\varepsilon$ exponentially improves upon a series of recent results that obtain algorithms with runtime of $\Omega(m/\varepsilon)$ for this problem.