In this paper, we study the problem of computing an edge-coloring in the (one-pass) W-streaming model. In this setting, the edges of an $n$-node graph arrive in an arbitrary order to a machine with a relatively small space, and the goal is to design an algorithm that outputs, as a stream, a proper coloring of the edges using the fewest possible number of colors. Behnezhad et al. [Behnezhad et al., 2019] devised the first non-trivial algorithm for this problem, which computes in $\tilde{O}(n)$ space a proper $O(\Delta^2)$-coloring w.h.p. (here $\Delta$ is the maximum degree of the graph). Subsequent papers improved upon this result, where latest of them [Ansari et al., 2022] shows that it is possible to deterministically compute an $O(\Delta^2/s)$-coloring in $O(ns)$ space. However, none of the improvements, succeeded in reducing the number of colors to $O(\Delta^{2-\epsilon})$ while keeping the same space bound of $\tilde{O}(n)$. In particular, no progress was made on the question of whether computing an $O(\Delta)$-coloring is possible with roughly $O(n)$ space, which was stated in [Behnezhad et al., 2019] to be a major open problem. In this paper we bypass the quadratic bound by presenting a new randomized $\tilde{O}(n)$-space algorithm that uses $\tilde{O}(\Delta^{1.5})$ colors.

翻译：本文研究了（单遍）W-流式模型中计算边着色的问题。在此设定下，一个包含$n$个节点的图的边以任意顺序到达具有相对较小空间的机器上，目标是设计一种算法，以流的形式输出使用尽可能少颜色的正确边着色。Behnezhad等人[Behnezhad et al., 2019]提出了该问题的首个非平凡算法，该算法在$\tilde{O}(n)$空间内以高概率计算出正确的$O(\Delta^2)$-着色（其中$\Delta$为图的最大度）。后续论文改进了这一结果，其中最新成果[Ansari et al., 2022]表明，可以在$O(ns)$空间内确定性地计算$O(\Delta^2/s)$-着色。然而，所有改进均未能在保持$\tilde{O}(n)$空间界的同时将颜色数减少至$O(\Delta^{2-\epsilon})$。特别地，关于能否用大约$O(n)$空间计算出$O(\Delta)$-着色的问题未取得进展，该问题在[Behnezhad et al., 2019]中被列为重要公开问题。本文通过提出一种新的随机化$\tilde{O}(n)$空间算法（使用$\tilde{O}(\Delta^{1.5})$种颜色）突破了二次界限。