Discrepancy theory provides powerful tools for producing higher-quality objects which "beat the union bound" in fundamental settings throughout combinatorics and computer science. However, this quality has often come at the price of more expensive algorithms. We introduce a new framework for bridging this gap, by allowing for the efficient implementation of discrepancy-theoretic primitives. Our framework repeatedly solves regularized optimization problems to low accuracy to approximate the partial coloring method of [Rot17], and simplifies and generalizes recent work of [JSS23] on fast algorithms for Spencer's theorem. In particular, our framework only requires that the discrepancy body of interest has exponentially large Gaussian measure and is expressible as a sublevel set of a symmetric, convex function. We combine this framework with new tools for proving Gaussian measure lower bounds to give improved algorithms for a variety of sparsification and coloring problems. As a first application, we use our framework to obtain an $\widetilde{O}(m \cdot \epsilon^{-3.5})$ time algorithm for constructing an $\epsilon$-approximate spectral sparsifier of an $m$-edge graph, matching the sparsity of [BSS14] up to constant factors and improving upon the $\widetilde{O}(m \cdot \epsilon^{-6.5})$ runtime of [LeeS17]. We further give a state-of-the-art algorithm for constructing graph ultrasparsifiers and an almost-linear time algorithm for constructing linear-sized degree-preserving sparsifiers via discrepancy theory; in the latter case, such sparsifiers were not known to exist previously. We generalize these results to their analogs in sparsifying isotropic sums of positive semidefinite matrices. Finally, to demonstrate the versatility of our technique, we obtain a nearly-input-sparsity time constructive algorithm for Spencer's theorem (where we recover a recent result of [JSS23]).

翻译：差异理论为组合数学和计算机科学中的基本问题提供了强大工具，能够生成“打破联合界”的更高质量对象。然而，这种质量提升往往以更高成本的算法为代价。我们提出了一种新框架，通过高效实现差异理论原语来弥合这一差距。该框架通过反复求解低精度的正则化优化问题，近似实现[Rot17]的部分着色方法，并简化与推广了[JSS23]关于Spencer定理快速算法的近期工作。具体而言，我们的框架仅要求相关差异体的高斯测度呈指数级大且可表示为对称凸函数的子水平集。我们将此框架与证明高斯测度下界的新工具相结合，为多种稀疏化与着色问题提供了改进算法。作为首个应用，利用该框架获得了构建m边图ε-近似谱稀疏化器的$\widetilde{O}(m \cdot \epsilon^{-3.5})$时间算法，其稀疏度与[BSS14]在常数因子内匹配，并将[LeeS17]的$\widetilde{O}(m \cdot \epsilon^{-6.5})$运行时间优化。我们进一步提出了构建图超稀疏化器的先进算法，以及基于差异理论构建线性尺寸保持度稀疏化器的近线性时间算法（后者此前尚无已知的稀疏化器存在）。我们将这些结果推广至正半定矩阵各向同性和的稀疏化类似问题。最后，为展示技术的普适性，我们获得了Spencer定理的近输入稀疏度时间构造性算法（其中恢复了[JSS23]的近期成果）。