Following the breakthrough work of Tardos in the bit-complexity model, Vavasis and Ye gave the first exact algorithm for linear programming in the real model of computation with running time depending only on the constraint matrix. For solving a linear program (LP) $\max\, c^\top x,\: Ax = b,\: x \geq 0,\: A \in \mathbb{R}^{m \times n}$, Vavasis and Ye developed a primal-dual interior point method using a 'layered least squares' (LLS) step, and showed that $O(n^{3.5} \log (\bar{\chi}_A+n))$ iterations suffice to solve (LP) exactly, where $\bar{\chi}_A$ is a condition measure controlling the size of solutions to linear systems related to $A$. Monteiro and Tsuchiya, noting that the central path is invariant under rescalings of the columns of $A$ and $c$, asked whether there exists an LP algorithm depending instead on the measure $\bar{\chi}^*_A$, defined as the minimum $\bar{\chi}_{AD}$ value achievable by a column rescaling $AD$ of $A$, and gave strong evidence that this should be the case. We resolve this open question affirmatively. Our first main contribution is an $O(m^2 n^2 + n^3)$ time algorithm which works on the linear matroid of $A$ to compute a nearly optimal diagonal rescaling $D$ satisfying $\bar{\chi}_{AD} \leq n(\bar{\chi}^*)^3$. This algorithm also allows us to approximate the value of $\bar{\chi}_A$ up to a factor $n (\bar{\chi}^*)^2$. As our second main contribution, we develop a scaling invariant LLS algorithm, together with a refined potential function based analysis for LLS algorithms in general. With this analysis, we derive an improved $O(n^{2.5} \log n\log (\bar{\chi}^*_A+n))$ iteration bound for optimally solving (LP) using our algorithm. The same argument also yields a factor $n/\log n$ improvement on the iteration complexity bound of the original Vavasis-Ye algorithm.

翻译：继Tardos在比特复杂度模型中的突破性工作之后，Vavasis和Ye首次给出了实数计算模型下线性规划的精确算法，其运行时间仅依赖于约束矩阵。对于求解线性规划 $ \max\, c^\top x,\: Ax = b,\: x \geq 0,\: A \in \mathbb{R}^{m \times n} $，Vavasis和Ye开发了一种使用“分层最小二乘”（LLS）步长的原始-对偶内点法，并证明了 $O(n^{3.5} \log (\bar{\chi}_A+n))$ 次迭代足以精确求解该线性规划，其中 $ \bar{\chi}_A $ 是控制与$A$相关的线性系统解大小的条件测度。Monteiro和Tsuchiya注意到中心路径在$A$和$c$的列缩放下保持不变，因此提出是否存在一种依赖于测度 $ \bar{\chi}^*_A $（定义为通过对$A$进行列缩放$AD$所能达到的最小 $\bar{\chi}_{AD}$ 值）的LP算法，并给出了强有力的证据表明这种情况应该成立。我们肯定地回答了这个开放性问题。我们的第一个主要贡献是一个 $O(m^2 n^2 + n^3)$ 时间算法，该算法作用于$A$的线性拟阵，以计算一个接近最优的对角缩放$D$，使得 $ \bar{\chi}_{AD} \leq n(\bar{\chi}^*)^3 $。该算法还允许我们将 $ \bar{\chi}_A $ 的值近似到 $ n (\bar{\chi}^*)^2 $ 的因子内。作为我们的第二个主要贡献，我们开发了一种缩放不变的LLS算法，并针对一般LLS算法提出了一种基于势能函数的改进分析。通过这种分析，我们推导出使用我们的算法最优求解（LP）的改进迭代界 $O(n^{2.5} \log n\log (\bar{\chi}^*_A+n))$。同样的论证还使得原始Vavasis-Ye算法的迭代复杂度界提高了 $n/\log n$ 因子。