We present a randomized algorithm that, given $\epsilon > 0$, outputs a proper $(1+\epsilon)\Delta$-edge-coloring of an $m$-edge simple graph $G$ of maximum degree $\Delta \geq 1/\epsilon$ in $O(m\,\log(1/\epsilon)/\epsilon^4)$ time. For constant $\epsilon$, this is the first linear-time algorithm for this problem without any restrictions on $\Delta$ other than the necessary bound $\Delta \geq 1/\epsilon$. The best previous result in this direction, very recently obtained by Assadi, gives a randomized algorithm with expected running time $O(m \, \log(1/\epsilon))$ under the assumption $\Delta \gg \log n/\epsilon$; removing the lower bound on $\Delta$ was explicitly mentioned as a challenging open problem by Bhattacharya, Costa, Panski, and Solomon. Indeed, even for edge-coloring with $2\Delta - 1$ colors (i.e., meeting the "greedy" bound), no linear-time algorithm covering the full range of $\Delta$ has been known until now. Additionally, when $\epsilon = 1/\Delta$, our result yields an $O(m\,\Delta^4\log \Delta)$-time algorithm for $(\Delta+1)$-edge-coloring, improving the bound $O(m\, \Delta^{17})$ from the authors' earlier work.

翻译：我们提出一种随机算法，给定$\epsilon > 0$，可在$O(m\,\log(1/\epsilon)/\epsilon^4)$时间内为最大度$\Delta \geq 1/\epsilon$的$m$条边简单图$G$输出正确的$(1+\epsilon)\Delta$边着色。对于常数$\epsilon$，这是首个线性时间算法，除必要约束$\Delta \geq 1/\epsilon$外不对$\Delta$施加任何限制。该方向的最佳先前结果由Assadi近期获得，在假设$\Delta \gg \log n/\epsilon$下给出了期望运行时间为$O(m \, \log(1/\epsilon))$的随机算法；消除$\Delta$的下界限制被Bhattacharya、Costa、Panski和Solomon明确列为具有挑战性的开放问题。事实上，即使对于满足"贪心"界的$2\Delta - 1$边着色问题，迄今为止也未曾出现覆盖完整$\Delta$范围的线性时间算法。此外，当$\epsilon = 1/\Delta$时，我们的结果产生了一个$O(m\,\Delta^4\log \Delta)$时间的$(\Delta+1)$边着色算法，改进了作者早期工作中$O(m\, \Delta^{17})$的界。