Streaming Euclidean Max-Cut: Dimension vs Data Reduction

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.