2022/11/10 by Yu Chen, Chen, Xiaoyu, Shaofeng H. -C. Jiang +3 · 2 citations
Computer Science · #Complexity and Algorithms in Graphs #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #Privacy-Preserving Technologies in Data #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2211.05293
openalex publication_date 2022/11/10 · openalex created_date 2022/11/16 · openalex updated_date 2026/07/28
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+ε)-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(ε-1 d logΔ). Lammersen, Sidiropoulos, and Sohler [WADS 2009] proved that Euclidean Max-Cut admits dimension reduction with target dimension d' = poly(ε-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 ε-1. We devise an alternative approach of data reduction, based on importance sampling, and achieve space bound poly(ε-1 d logΔ), which is exponentially better (in ε) 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+ε.