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 '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：分布式着色问题是分布式图算法领域的核心问题，且近年来取得了巨大进展。近期确定性分布式着色算法诸多显著进展主要基于两种工具：a) 缺陷着色，其中给定颜色的每个节点可拥有有限数量的同色邻居；b) 列表着色，这是标准着色问题的自然泛化，当分阶段计算着色且需将先前计算的部分着色扩展为完整着色时自然出现。本文引入“列表缺陷着色”，可视为这两种着色变体的泛化。本质上，在列表缺陷着色实例中，每个节点$v$被赋予一个颜色列表$x_{v,1},\dots,x_{v,p}$以及一个缺陷列表$d_{v,1},\dots,d_{v,p}$，使得若$v$以颜色$x_{v,i}$着色，则其最多允许有$d_{v,i}$个邻居也着以颜色$x_{v,i}$。我们通过证明更快的列表缺陷着色算法将直接导向LOCAL模型中更快的确定性$(\Delta+1)$-着色算法，凸显了列表缺陷着色的重要作用。此外，我们扩展了Maus和Tonoyan [DISC '20]近期提出的分布式列表着色算法。简化而言，我们证明若对每个节点$v$有$\sum_{i=1}^p \big(d_{v,i}+1)^2 > \mathrm{deg}_G^2(v)\cdot polylog\Delta$，则该列表缺陷着色实例可在仅$O(\log\Delta)$通信轮次内以通信高效方式求解。这催生了标准CONGEST模型中首个时间复杂度为$O(\sqrt{\Delta}\cdot polylog \Delta+\log^* n)$的确定性$(\Delta+1)$-着色算法，与LOCAL模型中最优时间复杂度仅相差$polylog\Delta$因子。