We study the problem of solving linear program in the streaming model. Given a constraint matrix $A\in \mathbb{R}^{m\times n}$ and vectors $b\in \mathbb{R}^m, c\in \mathbb{R}^n$, we develop a space-efficient interior point method that optimizes solely on the dual program. To this end, we obtain efficient algorithms for various different problems: * For general linear programs, we can solve them in $\widetilde O(\sqrt n\log(1/\epsilon))$ passes and $\widetilde O(n^2)$ space for an $\epsilon$-approximate solution. To the best of our knowledge, this is the most efficient LP solver in streaming with no polynomial dependence on $m$ for both space and passes. * For bipartite graphs, we can solve the minimum vertex cover and maximum weight matching problem in $\widetilde O(\sqrt{m})$ passes and $\widetilde O(n)$ space. In addition to our space-efficient IPM, we also give algorithms for solving SDD systems and isolation lemma in $\widetilde O(n)$ spaces, which are the cornerstones for our graph results.

翻译：我们研究流式模型下求解线性规划问题。给定约束矩阵 $A\in \mathbb{R}^{m\times n}$ 以及向量 $b\in \mathbb{R}^m, c\in \mathbb{R}^n$，我们提出一种仅在对偶程序上进行优化的空间高效内点法。为此，我们针对多种不同问题获得了高效算法：* 对于一般线性规划，我们可以在 $\widetilde O(\sqrt n\log(1/\epsilon))$ 轮和 $\widetilde O(n^2)$ 空间内求解 $\epsilon$ 近似解。据我们所知，这是流式场景下空间和轮次复杂度均不依赖于 $m$ 的多项式项的最高效线性规划求解器。* 对于二分图，我们可以在 $\widetilde O(\sqrt{m})$ 轮和 $\widetilde O(n)$ 空间内求解最小顶点覆盖和最大权匹配问题。除空间高效内点法外，我们还给出了在 $\widetilde O(n)$ 空间内求解 SDD 系统和隔离引理的算法，这些是支撑我们图论结果的核心基石。