The distributed coloring problem is at the core of the area of distributed graph algorithms and it is a problem that has seen tremendous progress over the last few years. Much of the remarkable recent progress on deterministic distributed coloring algorithms is based on two main tools: a) defective colorings in which every node of a given color can have a limited number of neighbors of the same color and b) list coloring, a natural generalization of the standard coloring problem that naturally appears when colorings are computed in different stages and one has to extend a previously computed partial coloring to a full coloring. In this paper, we introduce \emph{list defective colorings}, which can be seen as a generalization of these two coloring variants. Essentially, in a list defective coloring instance, each node $v$ is given a list of colors $x_{v,1},\dots,x_{v,p}$ together with a list of defects $d_{v,1},\dots,d_{v,p}$ such that if $v$ is colored with color $x_{v, i}$, it is allowed to have at most $d_{v, i}$ neighbors with color $x_{v, i}$. We highlight the important role of list defective colorings by showing that faster list defective coloring algorithms would directly lead to faster deterministic $(\Delta+1)$-coloring algorithms in the LOCAL model. Further, we extend a recent distributed list coloring algorithm by Maus and Tonoyan [DISC '20]. Slightly simplified, we show that if for each node $v$ it holds that $\sum_{i=1}^p \big(d_{v,i}+1)^2 > \mathrm{deg}_G^2(v)\cdot polylog\Delta$ then this list defective coloring instance can be solved in a communication-efficient way in only $O(\log\Delta)$ communication rounds. This leads to the first deterministic $(\Delta+1)$-coloring algorithm in the standard CONGEST model with a time complexity of $O(\sqrt{\Delta}\cdot polylog \Delta+\log^* n)$, matching the best time complexity in the LOCAL model up to a $polylog\Delta$ factor.

翻译：分布式染色问题是分布式图算法领域的核心问题，近年来取得了显著进展。近期确定性分布式染色算法的许多重要突破主要基于两个工具：a）缺陷染色，其中每个给定颜色的节点允许有有限数量的同色邻居；b）列表染色，这是标准染色问题的自然推广，通常出现在染色分阶段计算时，需要将先前计算的部分染色扩展为完整染色。本文引入了*列表缺陷染色*，可视为这两种染色变体的推广。本质上，在列表缺陷染色实例中，每个节点$v$被赋予一组颜色列表$x_{v,1},\dots,x_{v,p}$及一组缺陷列表$d_{v,1},\dots,d_{v,p}$，使得若$v$被染为颜色$x_{v,i}$，则最多允许有$d_{v,i}$个邻居也染为$x_{v,i}$。我们通过证明更快的列表缺陷染色算法将直接导致LOCAL模型中更快的确定性$(\Delta+1)$-染色算法，强调了列表缺陷染色的重要作用。此外，我们扩展了Maus和Tonoyan [DISC '20]的最新分布式列表染色算法。简化而言，我们证明：若对于每个节点$v$满足$\sum_{i=1}^p \big(d_{v,i}+1)^2 > \mathrm{deg}_G^2(v)\cdot polylog\Delta$，则该列表缺陷染色实例可在仅$O(\log\Delta)$通信轮次内以通信高效的方式解决。这带来了标准CONGEST模型中首个确定性$(\Delta+1)$-染色算法，其时间复杂度为$O(\sqrt{\Delta}\cdot polylog \Delta+\log^* n)$，与LOCAL模型中的最优时间复杂度仅相差一个$polylog\Delta$因子。