用平方（秩）线性系统解线性规划
Solving Linear Programs with Sqrt(rank) Linear System Solves
摘要：
我们提出了一个算法，给定一个线性规划的$n$变量，$m$约束，和约束矩阵$A$，计算一个$ε$ -近似解的$\tilde{O}(\sqrt{rank(A)}\log(1/ε))$迭代具有高概率。我们的方法的每次迭代包括求解$\tilde{O}(1)$线性系统和额外的近似线性时间计算，由于Renegar（1988）的迭代成本，与以前最快的方法相比，提高了$\tildeΩ((m/rank(A))^{1/2})$的因子。此外，我们提供了一个确定性的多项式时间可计算的$\tilde{O}(rank(A))$ -自协调的障碍函数的多面体，解决了一个悬而未决的问题Nesterov和Nemirovski（1994）的理论“通用障碍”的内点方法。应用我们的技术的线性规划制定的最大流产生一个$\tilde{O}(|E|\sqrt{|V|}\log(U))$时间算法解决最大流问题的有向图与$|E|$边，$|V|$顶点，和整数容量的大小最$U$。这改进了Goldberg和Rao（1998）以前实现的$O(|E|\min\{|E|^{1/2},|V|^{2/3}\}\log(|V|^{2}/|E|)\log(U))$的最快多项式运行时间。在求解稠密有向单位容量图的特殊情况下，我们的算法改进了Even和Tarjan（1975）和Karzanov（1973）之前实现的$O(|E|\min\{|E|^{1/2},|V|^{2/3}\})$的最快运行时间，以及最近由Mackdry（2013）实现的$\tilde{O}(|E|^{10/7})$的最快运行时间。
英文摘要：
We present an algorithm that given a linear program with $n$ variables, $m$ constraints, and constraint matrix $A$, computes an $ε$-approximate solution in $\tilde{O}(\sqrt{rank(A)}\log(1/ε))$ iterations with high probability. Each iteration of our method consists of solving $\tilde{O}(1)$ linear systems and additional nearly linear time computation, improving by a factor of $\tildeΩ((m/rank(A))^{1/2})$ over the previous fastest method with this iteration cost due to Renegar (1988). Further, we provide a deterministic polynomial time computable $\tilde{O}(rank(A))$-self-concordant barrier function for the polytope, resolving an open question of Nesterov and Nemirovski (1994) on the theory of "universal barriers" for interior point methods. Applying our techniques to the linear program formulation of maximum flow yields an $\tilde{O}(|E|\sqrt{|V|}\log(U))$ time algorithm for solving the maximum flow problem on directed graphs with $|E|$ edges, $|V|$ vertices, and integer capacities of size at most $U$. This improves upon the previous fastest polynomial running time of $O(|E|\min\{|E|^{1/2},|V|^{2/3}\}\log(|V|^{2}/|E|)\log(U))$ achieved by Goldberg and Rao (1998). In the special case of solving dense directed unit capacity graphs our algorithm improves upon the previous fastest running times of $O(|E|\min\{|E|^{1/2},|V|^{2/3}\})$ achieved by Even and Tarjan (1975) and Karzanov (1973) and of $\tilde{O}(|E|^{10/7})$ achieved more recently by Mądry (2013).
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