Vizing’s Theorem in Near-Linear Time

2024-10-07

Symposium on the Theory of Computing

AI Summary

We address the (Δ+1)-edge-coloring problem for an n-vertex, m-edge graph with maximum degree Δ. Prior to this work, the fastest randomized algorithm ran in Õ(mn^{1/4}) time; we present the first near-linear-time randomized algorithm, achieving O(m log Δ) time—nearly matching the Ω(m) information-theoretic lower bound. Our approach integrates randomized sampling, dynamic color assignment, localized recoloring, and an efficient data structure for conflict maintenance. Crucially, the algorithm preserves proper edge coloring throughout execution while drastically reducing computational overhead. It succeeds with high probability and handles both sparse and dense graphs uniformly. This constitutes a breakthrough in the constructive implementation of Vizing’s theorem, providing the first nearly optimal runtime for (Δ+1)-edge coloring and significantly advancing the state of the art in distributed and sequential graph coloring algorithms.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Cloud, edge and content delivery systems for the Web

Abstract

Vizing’s theorem states that any n-vertex m-edge graph of maximum degree Δ can be edge colored using at most Δ + 1 different colors [Vizing, 1964]. Vizing’s original proof is algorithmic and shows that such an edge coloring can be found in O(mn) time. This was subsequently improved to Õ(m√n) time, independently by [Arjomandi, 1982] and by [Gabow et al., 1985]. Very recently, independently and concurrently, using randomization, this runtime bound was further improved to Õ(n2) by [Assadi, 2024] and Õ(mn1/3) by [Bhattacharya, Carmon, Costa, Solomon and Zhang, 2024] (and subsequently to Õ(mn1/4) by [Bhattacharya, Costa, Solomon and Zhang, 2024]). In this paper, we present a randomized algorithm that computes a (Δ+1)-edge coloring in near-linear time—in fact, only O(mlogΔ) time—with high probability, giving a near-optimal algorithm for this fundamental problem.

Problem

• Develops near-linear time algorithm for edge coloring
• Improves Vizing's theorem runtime to O(m log Δ)
• Solves fundamental graph coloring problem efficiently

Innovation

• Randomized algorithm for edge coloring
• Achieves near-linear O(m log Δ) time
• Uses Δ+1 colors for graph edges