Even Faster $(Δ+ 1)$-Edge Coloring via Shorter Multi-Step Vizing Chains
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913548872974336 |
|---|---|
| author | Bhattacharya, Sayan Costa, Martín Solomon, Shay Zhang, Tianyi |
| author_facet | Bhattacharya, Sayan Costa, Martín Solomon, Shay Zhang, Tianyi |
| contents | Vizing's Theorem from 1964 states that any $n$-vertex $m$-edge graph with maximum degree $Δ$ can be {\em edge colored} using at most $Δ+ 1$ colors. For over 40 years, the state-of-the-art running time for computing such a coloring, obtained independently by Arjomandi [1982] and by Gabow, Nishizeki, Kariv, Leven and Terada~[1985], was $\tilde O(m\sqrt{n})$. Very recently, this time bound was improved in two independent works, by Bhattacharya, Carmon, Costa, Solomon and Zhang to $\tilde O(mn^{1/3})$, and by Assadi to $\tilde O(n^2)$.
In this paper we present an algorithm that computes such a coloring in $\tilde O(mn^{1/4})$ time. Our key technical contribution is a subroutine for extending the coloring to one more edge within time $\tilde O(Δ^2 + \sqrt{Δn})$. The best previous time bound of any color extension subroutine is either the trivial $O(n)$, dominated by the length of a Vizing chain, or the bound $\tilde{O}(Δ^6)$ by Bernshteyn [2022], dominated by the length of {\em multi-step Vizing chains}, which is basically a concatenation of multiple (carefully chosen) Vizing chains. Our color extension subroutine produces significantly shorter multi-step Vizing chains than in previous works, for sufficiently large $Δ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_12479 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Even Faster $(Δ+ 1)$-Edge Coloring via Shorter Multi-Step Vizing Chains Bhattacharya, Sayan Costa, Martín Solomon, Shay Zhang, Tianyi Data Structures and Algorithms Vizing's Theorem from 1964 states that any $n$-vertex $m$-edge graph with maximum degree $Δ$ can be {\em edge colored} using at most $Δ+ 1$ colors. For over 40 years, the state-of-the-art running time for computing such a coloring, obtained independently by Arjomandi [1982] and by Gabow, Nishizeki, Kariv, Leven and Terada~[1985], was $\tilde O(m\sqrt{n})$. Very recently, this time bound was improved in two independent works, by Bhattacharya, Carmon, Costa, Solomon and Zhang to $\tilde O(mn^{1/3})$, and by Assadi to $\tilde O(n^2)$. In this paper we present an algorithm that computes such a coloring in $\tilde O(mn^{1/4})$ time. Our key technical contribution is a subroutine for extending the coloring to one more edge within time $\tilde O(Δ^2 + \sqrt{Δn})$. The best previous time bound of any color extension subroutine is either the trivial $O(n)$, dominated by the length of a Vizing chain, or the bound $\tilde{O}(Δ^6)$ by Bernshteyn [2022], dominated by the length of {\em multi-step Vizing chains}, which is basically a concatenation of multiple (carefully chosen) Vizing chains. Our color extension subroutine produces significantly shorter multi-step Vizing chains than in previous works, for sufficiently large $Δ$. |
| title | Even Faster $(Δ+ 1)$-Edge Coloring via Shorter Multi-Step Vizing Chains |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2410.12479 |