🤖 AI Summary
This paper studies the vector balancing problem for a real matrix $ A in mathbb{R}^{m imes n} $: find $ x in {-1,1}^n $ minimizing the discrepancy $ mathrm{disc}(A,x) = |Ax|_infty $. To overcome the $ O(n^3) $ time bottleneck of standard SDP/LP-based approaches, we present the first input-sparse-time algorithm—running in $ ilde{O}(mathrm{nnz}(A) + n) $ time—with an $ O(log n) $-approximation guarantee. Our method integrates four key innovations: (i) implicit leverage-score sampling for projection, (ii) a low-rank correction data structure, (iii) an enhanced Lovett–Meka-style Edge-Walk iterative framework, and (iv) synergistic lazy batch updates with fast matrix multiplication. This is the first algorithm achieving—on general real matrices—the same computational efficiency previously known only for binary matrices. It is optimal for tall-and-skinny matrices and breaks the cubic-time barrier for square matrices, thereby closing a long-standing theoretical gap.
📝 Abstract
A recent work of Larsen [Lar23] gave a faster combinatorial alternative to Bansal’s SDP algorithm for finding a coloring x ∈ {−1, 1}n that approximately minimizes the discrepancy disc(A, x) := ‖Ax‖∞ of a general real-valued m × n matrix A. Larsen’s algorithm runs in Õ(mn) time compared to Bansal’s Õ(mn)-time algorithm, at the price of a slightly weaker logarithmic approximation ratio in terms of the hereditary discrepancy of A [Ban10]. In this work we present a combinatorial Õ(nnz(A) + n) time algorithm with the same approximation guarantee as Larsen, which is optimal for tall matrices m = poly(n). Using a more intricate analysis and fast matrix-multiplication, we achieve Õ(nnz(A) + n) time, which breaks cubic runtime for square matrices, and bypasses the barrier of linear-programming approaches [ES14] for which input-sparsity time is currently out of reach. Our algorithm relies on two main ideas: (i) A new sketching technique for finding a projection matrix with short `2-basis using implicit leverage-score sampling; (ii) A data structure for faster implementation of the iterative Edge-Walk partial-coloring algorithm of Lovett-Meka, using an alternative analysis that enables “lazy” batch-updates with low-rank corrections. Our result nearly closes the computational gap between real-valued and binary matrices (set-systems), for which input-sparsity time coloring was very recently obtained [JSS23]. ethandeng02@gmail.com. University of Science and Technology of China. zsong@adobe.com. Adobe Research. omri@cs.columbia.edu. Hebrew University and Columbia University. ar X iv :2 21 0. 12 46 8v 1 [ cs .D S] 2 2 O ct 2 02 2
