Max-Cut is a fundamental problem that has been studied extensively in various settings. We study Euclidean Max-Cut, where the input is a set of points in $\mathbb{R}^d$, in the model of dynamic geometric streams, that is, the input is $X\subseteq [\Delta]^d$ presented as a sequence of point insertions and deletions. Previous results by Frahling and Sohler [STOC'05] only address the low-dimensional regime, as their $(1+\epsilon)$-approximation algorithm uses space $\exp(d)$. We design the first streaming algorithms that use space $\mathrm{poly}(d)$, and are thus suitable for a high dimension $d$. We tackle this challenge of high dimension using two well-known approaches. The first one is via \emph{dimension reduction}, where we show that target dimension $\mathrm{poly}(\epsilon^{-1})$ suffices for the Johnson-Lindenstrauss transform to preserve Max-Cut within factor $(1 \pm \epsilon)$. This result extends the applicability of the prior work (algorithm with $\exp(d)$-space) also to high dimension. The second approach is \emph{data reduction}, based on importance sampling. We implement this scheme in streaming by employing a randomly-shifted quadtree. While this is a well-known method to construct a tree embedding, a key feature of our algorithm is that the distortion $O(d\log\Delta)$ affects only the space requirement $\mathrm{poly}(\epsilon^{-1} d\log\Delta)$, and not the approximation ratio $1+\epsilon$. These results are in line with the growing interest and recent results on streaming (and other) algorithms for high dimension.

翻译：Max- Cut 是一个在各种设置中广泛研究过的基本问题。 我们研究 Euclidean Max- Cut, 输入是用 $\ mathb{R ⁇ d$ 的一组点数, 在动态几何流模型中, 也就是说, 输入是 $X\ subseteq [\ Delta]\ d$ 作为点插入和删除的序列来显示。 Frahling 和 Sohler [STOC'05] 的先前结果只针对低维机制, 因为它们的 $( \\\ lipslon) $- approcolm 运算法只使用 $\ exp( d) $。 我们设计了第一个流算算法, 以 $( little) max- max maxal- max max mail lax max mail max max max max max max max max maxil max mail mail max max max max max max max max max max max max max max maxx maxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx