Graph coloring is a fundamental problem in computer science. In the semi-streaming model, an input graph $G$ on $n$ vertices and maximum degree $Δ$ is presented as a stream of edges, and the goal is to compute a vertex coloring using a small number of colors while storing only $\tilde{O}(n)$ bits of memory. Recent work has revealed an exponential separation between randomized and deterministic approaches in this setting: while randomized algorithms can achieve a $(Δ+1)$-coloring in a single pass [Assadi, Chen, and Khanna, 2019], any single-pass deterministic algorithm requires $\exp(Δ^{Ω(1)})$ colors [Assadi, Chen, and Sun, 2022]. Consequently, deterministic algorithms that use few colors must necessarily make multiple passes over the stream. Prior to this work, the best known deterministic trade-offs were: an $O(Δ^2)$-coloring in 2 passes, an $O(Δ)$-coloring in $O(\log Δ)$ passes [Assadi, Chen, and Sun, 2022], and a $(Δ+1)$-coloring in $O(\log Δ\cdot \log\log Δ)$ passes [Assadi, Chakrabarti, Ghosh, and Stoeckl, 2023]. It remained open whether better trade-offs -- particularly with sub-logarithmic pass complexity and linear-in-$Δ$ palette size -- were achievable. In this paper, we present a new deterministic semi-streaming algorithm that computes an $O(Δ)$-coloring in $O(\sqrt{\log Δ})$ passes. This is the first deterministic streaming algorithm to achieve a coloring with palette size linear-in-$Δ$ using sublogarithmic-in-$Δ$ passes.

翻译：图着色是计算机科学中的一个基本问题。在半流式模型下，输入图$G$有$n$个顶点和最大度$Δ$，其边以流形式呈现，目标是在仅存储$\tilde{O}(n)$比特内存的情况下，用少量颜色计算顶点着色。近期研究揭示，在此设定下，随机化方法与确定性方法之间存在指数级差距：随机化算法可在单次遍历中实现$(Δ+1)$-着色[Assadi, Chen, and Khanna, 2019]，而任何单次遍历确定性算法需要$\exp(Δ^{Ω(1)})$种颜色[Assadi, Chen, and Sun, 2022]。因此，使用少量颜色的确定性算法必须对数据流进行多次遍历。在本工作之前，已知最优的确定性权衡结果为：2次遍历中的$O(Δ^2)$-着色、$O(\log Δ)$次遍历中的$O(Δ)$-着色[Assadi, Chen, and Sun, 2022]，以及$O(\log Δ\cdot \log\log Δ)$次遍历中的$(Δ+1)$-着色[Assadi, Chakrabarti, Ghosh, and Stoeckl, 2023]。能否实现更优的权衡——特别是具有次对数遍历复杂度且调色板大小为$Δ$的线性阶——仍是一个悬而未决的问题。本文提出了一种新的确定性半流式算法，可在$O(\sqrt{\log Δ})$次遍历中计算$O(Δ)$-着色。这是首个使用$Δ$的次对数遍历次数实现调色板大小为$Δ$线性阶的确定性流式着色算法。