In this paper we provide an $\tilde{O}(nd+d^{3})$ time randomized algorithm for solving linear programs with $d$ variables and $n$ constraints with high probability. To obtain this result we provide a robust, primal-dual $\tilde{O}(\sqrt{d})$-iteration interior point method inspired by the methods of Lee and Sidford (2014, 2019) and show how to efficiently implement this method using new data-structures based on heavy-hitters, the Johnson-Lindenstrauss lemma, and inverse maintenance. Interestingly, we obtain this running time without using fast matrix multiplication and consequently, barring a major advance in linear system solving, our running time is near optimal for solving dense linear programs among algorithms that do not use fast matrix multiplication.

翻译：在本文中, 我们提供了一种 $\ tilde{ O} (nd+d ⁇ 3}) 的时间随机算法, 用于用 $d$ 变量和 $n$ 限制以高概率解决线性程序。 为了获得这一结果, 我们提供了一种由 Lee 和 Sidford (2014, 2019) 方法启发的坚固的、 原始- 原始的 $ tilde{ O} (sqrt{d}) $- itel 内点法, 并展示了如何使用基于重光源、 Johnson- Lindenstruss Lemma 和反向维护的新数据结构来高效实施这种方法。 有趣的是, 我们获得这个运行时间时没有使用快速矩阵乘法, 因而, 如果不能在线性系统解答中取得重大进步, 我们运行时间几乎是最佳的解决不使用快速矩阵倍增法的计算法中密度线性线性程序的方法 。