Max-Cut is a fundamental problem that has been studied extensively in various settings. We design an algorithm for Euclidean Max-Cut, where the input is a set of points in $\mathbb{R}^d$, in the model of dynamic geometric streams, where the input $X\subseteq [\Delta]^d$ is presented as a sequence of point insertions and deletions. Previously, Frahling and Sohler [STOC 2005] designed a $(1+\epsilon)$-approximation algorithm for the low-dimensional regime, i.e., it uses space $\exp(d)$. To tackle this problem in the high-dimensional regime, which is of growing interest, one must improve the dependence on the dimension $d$, ideally to space complexity $\mathrm{poly}(\epsilon^{-1} d \log\Delta)$. Lammersen, Sidiropoulos, and Sohler [WADS 2009] proved that Euclidean Max-Cut admits dimension reduction with target dimension $d' = \mathrm{poly}(\epsilon^{-1})$. Combining this with the aforementioned algorithm that uses space $\exp(d')$, they obtain an algorithm whose overall space complexity is indeed polynomial in $d$, but unfortunately exponential in $\epsilon^{-1}$. We devise an alternative approach of \emph{data reduction}, based on importance sampling, and achieve space bound $\mathrm{poly}(\epsilon^{-1} d \log\Delta)$, which is exponentially better (in $\epsilon$) than the dimension-reduction approach. To implement this scheme in the streaming model, we employ a randomly-shifted quadtree to construct a tree embedding. While this is a well-known method, a key feature of our algorithm is that the embedding's distortion $O(d\log\Delta)$ affects only the space complexity, and the approximation ratio remains $1+\epsilon$.

翻译：最大割是一个基本问题，已在多种场景下得到广泛研究。我们针对动态几何流模型下的欧几里得最大割问题设计了一种算法，其中输入为$\mathbb{R}^d$中的点集$X\subseteq [\Delta]^d$，并以点的插入和删除序列形式呈现。此前，Frahling与Sohler [STOC 2005] 针对低维场景设计了一种$(1+\epsilon)$-近似算法，但其空间复杂度为$\exp(d)$。为应对日益受关注的高维场景，必须改进对维度$d$的依赖，理想情况下空间复杂度应为$\mathrm{poly}(\epsilon^{-1} d \log\Delta)$。Lammersen、Sidiropoulos与Sohler [WADS 2009] 证明欧几里得最大割可支持目标维度$d' = \mathrm{poly}(\epsilon^{-1})$的维度压缩。结合上述空间复杂度为$\exp(d')$的算法，他们得到的算法总体空间复杂度确实关于$d$呈多项式级，但不幸关于$\epsilon^{-1}$呈指数级。我们提出一种基于重要性抽样的替代性\emph{数据归约}方法，实现了空间复杂度$\mathrm{poly}(\epsilon^{-1} d \log\Delta)$，在$\epsilon$上相比维度压缩方法呈指数级改进。为在流式模型中实现该方案，我们采用随机偏移四叉树构建树嵌入。尽管这是已知方法，但本算法的关键特性在于：嵌入的畸变$O(d\log\Delta)$仅影响空间复杂度，而近似比仍保持$1+\epsilon$。