The palette sparsification theorem (PST) of Assadi, Chen, and Khanna (SODA 2019) states that in every graph $G$ with maximum degree $\Delta$, sampling a list of $O(\log{n})$ colors from $\{1,\ldots,\Delta+1\}$ for every vertex independently and uniformly, with high probability, allows for finding a $(\Delta+1)$ vertex coloring of $G$ by coloring each vertex only from its sampled list. PST naturally leads to a host of sublinear algorithms for $(\Delta+1)$ vertex coloring, including in semi-streaming, sublinear time, and MPC models, which are all proven to be nearly optimal, and in the case of the former two are the only known sublinear algorithms for this problem. While being a quite natural and simple-to-state theorem, PST suffers from two drawbacks. Firstly, all its known proofs require technical arguments that rely on sophisticated graph decompositions and probabilistic arguments. Secondly, finding the coloring of the graph from the sampled lists in an efficient manner requires a considerably complicated algorithm. We show that a natural weakening of PST addresses both these drawbacks while still leading to sublinear algorithms of similar quality (up to polylog factors). In particular, we prove an asymmetric palette sparsification theorem (APST) that allows for list sizes of the vertices to have different sizes and only bounds the average size of these lists. The benefit of this weaker requirement is that we can now easily show the graph can be $(\Delta+1)$ colored from the sampled lists using the standard greedy coloring algorithm. This way, we can recover nearly-optimal bounds for $(\Delta+1)$ vertex coloring in all the aforementioned models using algorithms that are much simpler to implement and analyze.

翻译：Assadi、Chen和Khanna（SODA 2019）提出的调色板稀疏化定理（PST）表明：在任何最大度为Δ的图G中，以独立均匀的方式为每个顶点从{1,…,Δ+1}中采样O(log n)种颜色，则高概率地仅需从各顶点的采样颜色列表中着色，即可找到G的(Δ+1)顶点着色方案。PST自然导出了一系列(Δ+1)顶点着色的亚线性算法，包括半流式计算、亚线性时间计算和MPC模型中的算法，这些算法均被证明是近乎最优的，且前两类模型中的算法是该问题目前已知的唯一亚线性算法。尽管PST是一个表述简洁自然的定理，它仍存在两个缺陷：首先，所有已知证明都需要依赖复杂的图分解和概率论证的技术性论述；其次，从采样颜色列表中高效找到图的着色方案需要相当复杂的算法。我们证明，对PST进行自然弱化可在保持亚线性算法质量（至多相差多对数因子）的同时解决这两个缺陷。具体而言，我们提出了非对称调色板稀疏化定理（APST），该定理允许顶点的颜色列表具有不同尺寸，仅需约束这些列表的平均尺寸。这种弱化要求的优势在于，我们现在可以轻松证明使用标准贪心着色算法即可从采样列表实现图的(Δ+1)着色。通过这种方式，我们能够以更易于实现和分析的简单算法，在上述所有模型中恢复近乎最优的(Δ+1)顶点着色边界。