Streaming Euclidean Max-Cut: Dimension vs Data Reduction

Xiaoyu Chen, Shaofeng H.-C. Jiang, Robert Krauthgamer

Abstract: 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 ℝd, in the model of dynamic geometric streams, where the input X ⊆ [Δ]d is presented as a sequence of point insertions and deletions. Previously, Frahling and Sohler [STOC 2005] designed a (1+є)-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 poly(є−1 d logΔ). Lammersen, Sidiropoulos, and Sohler [WADS 2009] proved that Euclidean Max-Cut admits dimension reduction with target dimension d′ = poly(є−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 є−1.We devise an alternative approach of data reduction, based on importance sampling, and achieve space bound poly(є−1 d logΔ), which is exponentially better (in є) 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(dlogΔ) affects only the space complexity, and the approximation ratio remains 1+є.