Faster $(Δ+ 1)$-Edge Coloring: Breaking the $m \sqrt{n}$ Time Barrier (2405.15449v1)

Abstract: Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $\Delta$ can be {\em edge colored} using at most $\Delta + 1$ different colors [Diskret.~Analiz, '64]. Vizing's original proof is algorithmic and shows that such an edge coloring can be found in $\tilde{O}(mn)$ time. This was subsequently improved to $\tilde O(m\sqrt{n})$, independently by Arjomandi [1982] and by Gabow et al.~[1985]. In this paper we present an algorithm that computes such an edge coloring in $\tilde O(mn^{1/3})$ time, giving the first polynomial improvement for this fundamental problem in over 40 years.

• The paper introduces an innovative algorithm that computes a (Δ+1)-edge coloring in ō(mn^(1/3)) time, marking the first polynomial improvement in over 40 years.
• It employs random sampling, vertex partitioning, and path alternation techniques to surpass the previous m√n time complexity barrier.
• The breakthrough offers practical benefits for network design, scheduling, and resource management by enabling more efficient graph edge coloring.

Faster $(\Delta + 1)$-Edge Coloring: Breaking the $m \sqrt{n}$ Time Barrier

This paper presents a significant advancement in the theory of graph edge coloring by introducing a novel algorithm capable of computing a ($\Delta$+1)-edge coloring in time $\tilde{O}(mn^{1/3})$, thereby surpassing the longstanding $m \sqrt{n}$ time complexity barrier. The edge coloring problem, as defined by Vizing's theorem, is essential in many applications within computer science and mathematics. This theorem guarantees that any graph can be edge-colored with at most $\Delta + 1$ colors, where $\Delta$ is the graph's maximum degree.

State-of-the-Art and Historical Context

Initially, Vizing's proof provided an algorithmic solution with time complexity $\tilde{O}(mn)$. Over the decades, this was improved to $\tilde{O}(m\sqrt{n})$ by Arjomandi and independently by Gabow et al. in the 1980s. A minor improvement with simplifications was achieved recently by Sinnamon. However, no significant polynomial improvement had been achieved since these developments, highlighting the significance of this new algorithm.

Technical Contributions and Algorithmic Developments

The primary contribution of this paper is the development of an algorithm that can calculate a ($\Delta$+1)-edge coloring of a graph in $\tilde{O}(mn^{1/3})$ time—marking the first polynomial improvement for this foundational problem in over 40 years. The proposed algorithm strategically utilizes random sampling and path alterations combined with graph partitioning techniques to achieve enhanced performance.

The algorithm operates on a high-level strategy that partitions the vertex set based on a sampling method, which in turn leads to a more efficient edge coloring process. A key insight is the separation of vertices into high and low degree groups, allowing an efficient application of the edge coloring strategy by maintaining structural properties that enable shorter alternating paths necessary for the color extension process.

Implications and Further Research

This breakthrough has several theoretical and practical implications. Theoretically, it provides a new approach to tackle related graph coloring problems that were considered infeasible within certain time constraints. Practically, it could influence the development of software tools used in resource management, scheduling, and network design, where edge coloring is applicable.

Future research could focus on extending these findings to other coloring-related problems in graphs with more complex structures or constraints. Furthermore, improved deterministic algorithms that eliminate reliance on randomization could be explored. The exploration of parallel computing adaptations of this algorithm might also yield substantial benefits in real-world application scenarios where large-scale graphs are common.

Overall, this paper paves the way for more efficient algorithms in graph theory and its various applications, presenting an impactful milestone in the domain of combinatorial optimization.