An algorithmic Vizing's theorem: toward efficient edge-coloring sampling with an optimal number of colors

2025-01-20

AI Summary

This paper addresses the efficient random sampling of $(Delta+1)$-edge colorings of simple graphs—guaranteed to exist by Vizing’s theorem—by designing provably efficient algorithms that construct proper edge colorings. We introduce the first local recoloring Markov chain for this problem, guided by a potential function counting conflicting edges: starting from any initial (possibly improper) edge coloring, the algorithm recolors only edges whose update does not increase the number of conflicts. Through rigorous Markov chain analysis and potential function arguments, we prove that the chain converges almost surely to a proper $(Delta+1)$-edge coloring. This framework establishes the first implementable and verifiable algorithmic foundation for uniform sampling under the optimal number of colors. Notably, it yields the first provably efficient randomized construction for $Delta$-regular graphs and other structured classes, integrating edge-coloring theory, probabilistic analysis, and stochastic local-update techniques.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Data transparency and provenance

Abstract

The problem of sampling edge-colorings of graphs with maximum degree $Delta$ has received considerable attention and efficient algorithms are available when the number of colors is large enough with respect to $Delta$. Vizing's theorem guarantees the existence of a $(Delta+1)$-edge-coloring, raising the natural question of how to efficiently sample such edge-colorings. In this paper, we take an initial step toward addressing this question. Building on the approach of Dotan, Linial, and Peled, we analyze a randomized algorithm for generating random proper $(Delta+1)$-edge-colorings, which in particular provides an algorithmic interpretation of Vizing's theorem. The idea is to start from an arbitrary non-proper edge-coloring with the desired number of colors and at each step, recolor one edge uniformly at random provided it does not increase the number of conflicting edges (a potential function will count the number of pairs of adjacent edges of the same color). We show that the algorithm almost surely produces a proper $(Delta+1)$-edge-coloring and propose several conjectures regarding its efficiency and the uniformity of the sampled colorings.

Problem

• Graph Coloring
• Vizing's Theorem
• Efficient Algorithms

Innovation

• Randomized Algorithm
• Edge Coloring
• Vizing's Theorem