In this paper we present a deterministic CONGEST algorithm to compute an $O(k\Delta)$-vertex coloring in $O(\Delta/k)+\log^* n$ rounds, where $\Delta$ is the maximum degree of the network graph and $1\leq k\leq O(\Delta)$ can be freely chosen. The algorithm is extremely simple: Each node locally computes a sequence of colors and then it "tries colors" from the sequence in batches of size $k$. Our algorithm subsumes many important results in the history of distributed graph coloring as special cases, including Linial's color reduction [Linial, FOCS'87], the celebrated locally iterative algorithm from [Barenboim, Elkin, Goldenberg, PODC'18], and various algorithms to compute defective and arbdefective colorings. Our algorithm can smoothly scale between these and also simplifies the state of the art $(\Delta+1)$-coloring algorithm. At the cost of losing the full algorithm's simplicity we also provide a $O(k\Delta)$-coloring algorithm in $O(\sqrt{\Delta/k})+\log^* n$ rounds. We also provide improved deterministic algorithms for ruling sets, and, additionally, we provide a tight characterization for one-round color reduction algorithms.

翻译：在本文中，我们提出一种确定性的CONGEST算法，可在$O(\Delta/k)+\log^* n$轮内计算出$O(k\Delta)$顶点着色，其中$\Delta$为网络图的最大度数，且$1\leq k\leq O(\Delta)$可自由选择。该算法极其简单：每个节点本地计算一个颜色序列，然后以大小为$k$的批次从该序列中“尝试颜色”。我们的算法将分布式图着色历史上的许多重要结果作为特例囊括其中，包括Linial的颜色缩减算法[Linial, FOCS'87]、著名的[Barenboim, Elkin, Goldenberg, PODC'18]局部迭代算法，以及多种缺陷着色与拱缺陷着色算法。该算法可平滑地在这些方法间缩放，并简化了当前最先进的$(\Delta+1)$着色算法。在牺牲完全简洁性的代价下，我们还提供一种可在$O(\sqrt{\Delta/k})+\log^* n$轮内完成的$O(k\Delta)$着色算法。此外，我们针对统治集问题给出了改进的确定性算法，并为一轮颜色缩减算法提供了紧致刻画。