For any $Δ$, let $k_Δ$ be the maximum integer $k$ such that $(k+1)(k+2)\le Δ$. We give a distributed \LOCAL algorithm that, given an integer $k < k_Δ$, computes a valid $Δ-k$-coloring if one exists. The algorithm runs in $\tilde{O}(\log^4 \log n)$ rounds, which is within a polynomial factor of the $Ω(\log\log n)$ lower bound, which already applies to the case $k=0$. It is also best possible in the sense that if $k \ge k_Δ$, the problem requires $Ω(n/Δ)$ distributed rounds [Molloy, Reed, '14, Bamas, Esperet '19]. For $Δ$ at most polylogarithmic, the algorithm is an exponential improvement over the current state of the art of $O(\log^{49/12} n)$ rounds. When $Δ\ge (\log n)^{50}$, our algorithm achieves an even faster runtime of $O(\log^* n)$ rounds.

翻译：设 $Δ$ 为任意正整数，记 $k_Δ$ 为满足 $(k+1)(k+2)\le Δ$ 的最大整数 $k$。我们给出一个分布式 \LOCAL 算法：对于任意整数 $k < k_Δ$，若存在合法的 $Δ-k$-染色方案，该算法可在 $\tilde{O}(\log^4 \log n)$ 轮内计算得到。该运行时间与已适用于 $k=0$ 情形的下界 $Ω(\log\log n)$ 仅相差多项式因子。同时，该结果在如下意义下是最优的：当 $k \ge k_Δ$ 时，问题需要 $Ω(n/Δ)$ 轮分布式计算 [Molloy, Reed, '14; Bamas, Esperet '19]。对于至多呈多对数规模的 $Δ$，该算法将现有最优结果 $O(\log^{49/12} n)$ 轮实现了指数级改进。当 $Δ\ge (\log n)^{50}$ 时，我们的算法可实现更快的运行时间 $O(\log^* n)$ 轮。