The palette sparsification theorem (PST) of Assadi, Chen, and Khanna (SODA 2019) states that in every graph $G$ with maximum degree $Δ$, sampling a list of $O(\log{n})$ colors from $\{1,\ldots,Δ+1\}$ for every vertex independently and uniformly, with high probability, allows for finding a $(Δ+1)$ vertex coloring of $G$ by coloring each vertex only from its sampled list. PST naturally leads to a host of sublinear algorithms for $(Δ+1)$ 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 $(Δ+1)$ colored from the sampled lists using the standard greedy coloring algorithm. This way, we can recover nearly-optimal bounds for $(Δ+1)$ 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）指出：对于任意最大度为$Δ$的图$G$，若每个顶点独立且均匀地从$\{1,\ldots,Δ+1\}$中采样$O(\log{n})$种颜色构成列表，则高概率存在一种$(Δ+1)$顶点着色方案，使得每个顶点仅从其采样列表中选择颜色。PST自然导出了一系列$(Δ+1)$顶点着色的亚线性算法，涵盖半流式、亚线性时间及MPC计算模型，这些算法均被证明近乎最优，且前两类模型中的算法是该问题目前已知的唯一亚线性解法。尽管PST是一个表述简洁自然的定理，它仍存在两个缺陷：其一，所有已知证明均需依赖复杂的图分解与概率论证技术；其二，从采样列表高效构建图着色方案需要相当复杂的算法。我们证明，通过对PST进行自然弱化，可在保持算法质量（至多多对数因子差异）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），允许顶点列表具有不同尺寸，仅需约束列表的平均大小。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$(Δ+1)$着色。基于此，我们能够在所有前述计算模型中，以更易于实现和分析的算法，恢复近乎最优的$(Δ+1)$顶点着色边界。