Vizing's Theorem in Near-Linear Time (2410.05240v2)

Abstract: Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $\Delta$ can be edge colored using at most $\Delta + 1$ 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 $\tilde O(m\sqrt{n})$ 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 $\tilde{O}(n^2)$ by [Assadi, 2024] and $\tilde O(mn^{1/3})$ by Bhattacharya, Carmon, Costa, Solomon and Zhang, 2024. In this paper, we present a randomized algorithm that computes a $(\Delta+1)$-edge coloring in near-linear time -- in fact, only $O(m\log{\Delta})$ time -- with high probability, giving a near-optimal algorithm for this fundamental problem.

• The paper introduces a near-linear time algorithm that computes (Δ+1)-edge coloring, significantly reducing runtime compared to previous methods.
• It employs efficient color extension and innovative u-fan handling to systematically reduce uncolored edges in graphs.
• The breakthrough algorithm enhances practical network management and paves the way for advanced research in distributed and complex graph coloring.

Overview of "Vizing's Theorem in Near-Linear Time"

The paper, titled "Vizing's Theorem in Near-Linear Time" by Assadi et al., presents a significant advance in the domain of graph theory and algorithm design, specifically addressing the implementation of Vizing's theorem within a computationally efficient timeframe. Vizing's theorem, a fundamental concept in discrete mathematics, states that any graph can be edge-colored using at most Δ + 1 colors, where Δ is the maximum degree of the graph. Although this theorem has been established for decades, the challenge lies in finding an efficient algorithm to achieve this coloring, particularly for large graphs.

Previous Work

The traditional algorithmic approaches to Vizing's theorem required considerable computational resources. Vizing's original proof offered an operational runtime of $O(mn)$, where m is the number of edges and n is the number of vertices. Subsequent work by Arjomandi and Gabow managed to improve this to $\tilde{O}(m\sqrt{n})$. Recent efforts have pushed the boundary further to $\tilde{O}(n^2)$ using randomization as shown by Assadi, and $\tilde{O}(mn^{1/3})$ by Bhattacharya et al.

Contribution

Assadi et al. propose a new algorithm that computes a ($Δ+1$)-edge coloring in $\tilde{O}(m)$ time, thereby achieving near-optimal efficiency for this problem. This is a notable reduction compared to prior algorithms, making it practically viable for large-scale applications.

Technical Insights

The authors introduce two main components to effectively reduce the computational complexity:

1. Efficient Color Extension: This component is crucial in reducing the number of uncolored edges in a graph. The authors focus on batching the edge colorings and utilizing the connectivity properties of alternating paths efficiently, drawing inspiration from previous dynamic and sub-linear algorithms.
2. Handling U-Fans: A novel focus on manipulating uncolored edges using data structures that manage u-fans and u-edges, which are abstract representations catering to efficient edge manipulations. These structures enable more effective handling and rearrangement of uncolored edges to facilitate fast and conflict-free coloring.

Numerical Significance

The algorithm operates by achieving a balanced but systematic reduction of uncolored edges across graph sections, maintaining computational bounds consistently below $\tilde{O}(m)$ for the entire edge coloring process. The thorough analyses and algorithmic designs ensure that high probability bounds on algorithm runtime are maintained.

Implications and Future Directions

Practically, this near-linear time approach to edge coloring can significantly impact how large networks (communication, transportation, etc.) are managed and optimized. Additionally, theoretical implications extend to further understanding the interplay between graph structure properties and computational complexity, potentially influencing new graph-based algorithm designs.

The paper opens up potential expansions in distributed computing where large-scale graph problems must be solved efficiently in decentralized systems. Furthermore, the consideration of different kinds of constraints and adaptations of this theorem to hypergraphs or directed graphs presents intriguing avenues for future research.

In conclusion, this paper not only reinforces the computational efficiency of Vizing's theorem but also revolutionizes the approach to solving graph coloring problems by rethinking how such algorithmic processes can be optimized, even on a large scale.