Faster Vizing and Near-Vizing Edge Coloring Algorithms

TL;DR

The paper tackles the problem of faster edge coloring, aiming beyond Vizing’s classical Δ+1 coloring. It introduces a randomized near-Vizing coloring that runs in $O(m\log\Delta)$ time and yields a coloring of size $\Delta + O(\log n)$ by repeatedly constructing and peeling off so-called fair matchings via a novel Matching Random Walk framework. The key ideas avoid heavy reliance on Vizing fans and chains, instead using local reductions to obtain a small additive gap and enabling near-linear-time performance, with corollaries that beat longstanding bounds for Δ+1 coloring on many graphs and deliver $(1+\varepsilon)\Delta$-colorings in $O(m\log(1/\varepsilon))$ time for suitable parameters. The results substantially improve the state of the art for arbitrary graphs, particularly in dense regimes, and offer a practical randomized toolkit for near-Vizing colorings with broad implications in graph algorithms. Overall, the work provides a conceptual shift by leveraging fair matchings and a sophisticated probabilistic analysis to achieve near-linear performance for a problem long dominated by slower combinatorial methods.

Abstract

Vizing's celebrated theorem states that every simple graph with maximum degree $Δ$ admits a $(Δ+1)$ 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 $Δ$ 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 $Δ+O(\log{n})$ coloring of any given simple graph in $O(m\logΔ)$ 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 $(Δ+1)$ 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 $(1+\varepsilon) Δ$ edge coloring in $O(m\log{(1/\varepsilon)})$ expected time for any $\varepsilon = ω(\log{n}/Δ)$. The dependence on $\varepsilon$ exponentially improves upon a series of recent results that obtain algorithms with runtime of $Ω(m/\varepsilon)$ for this problem.

Figures (2)

• Figure 1: An illustration of the $M$-digraph $\mathcal{G}$. The vertices of $\mathcal{G}$ are shown via (gray) circles encompassing a vertex (white circle) or a matching edge (thick black) from the original graph. The vertices $s$ and $t$ are designated vertices unique to $\mathcal{G}$. The (black) thin edges in the middle are connected according to the original graph, the edges going out of $s$ (blue) are only going to $\mathcal{V}_{\textnormal{s}}$, the long-dashed edges (green) are going to $t$, and short-dashed edges (red) are coming from $t$. Finally, the middle edges (black) are simple edges and the number next to them shows the different number of those edges leaving or arriving at a vertex, while all other edges are parallel edges and their number shows their multiplicity.
• Figure 2: An illustration of $\textnormal{AltPath}\xspace$ when $v_{i+1}$ is labeled previously and the algorithm calls $\textnormal{FixEven}\xspace$ or $\textnormal{FixOdd}\xspace$. Double-circles mark matched vertices and single circle mark unmatched ones. Thick (red) edges are matching edges.

Theorems & Definitions (54)

• Proposition 2.1: Chernoff bound; c.f. AlonS16
• Proposition 2.2
• proof
• Theorem 1
• Definition 3.1
• Lemma 3.2
• Definition 3.4
• proof
• Lemma 3.6
• proof