We develop a new `subspace layered least squares' interior point method (IPM) for solving linear programs. Applied to an $n$-variable linear program in standard form, the iteration complexity of our IPM is up to an $O(n^{1.5} \log n)$ factor upper bounded by the \emph{straight line complexity} (SLC) of the linear program. This term refers to the minimum number of segments of any piecewise linear curve that traverses the \emph{wide neighborhood} of the central path, a lower bound on the iteration complexity of any IPM that follows a piecewise linear trajectory along a path induced by a self-concordant barrier. In particular, our algorithm matches the number of iterations of any such IPM up to the same factor $O(n^{1.5}\log n)$. As our second contribution, we show that the SLC of any linear program is upper bounded by $2^{n + o(1)}$, which implies that our IPM's iteration complexity is at most exponential. This in contrast to existing iteration complexity bounds that depend on either bit-complexity or condition measures; these can be unbounded in the problem dimension. We achieve our upper bound by showing that the central path is well-approximated by a combinatorial proxy we call the \emph{max central path}, which consists of $2n$ shadow vertex simplex paths. Our upper bound complements the lower bounds of Allamigeon, Benchimol, Gaubert, and Joswig (SIAGA 2018), and Allamigeon, Gaubert, and Vandame (STOC 2022), who constructed linear programs with exponential SLC. Finally, we show that each iteration of our IPM can be implemented in strongly polynomial time. Along the way, we develop a deterministic algorithm that approximates the singular value decomposition of a matrix in strongly polynomial time to high accuracy, which may be of independent interest.

翻译：我们提出了一种新的"子空间分层最小二乘"内点法（IPM）用于求解线性规划。将该方法应用于标准形式的$n$变量线性规划时，其迭代复杂度与线性规划的\emph{直线复杂度}（SLC）相差至多$O(n^{1.5} \log n)$倍。直线复杂度指沿中心路径\emph{宽邻域}遍历任意分段线性曲线所需的最少线段数——这是任何沿自和谐障碍函数诱导路径的分段线性轨迹IPM迭代复杂度的下界。因此，我们的算法在相同倍数$O(n^{1.5}\log n)$范围内匹配了此类IPM的迭代次数。作为第二项贡献，我们证明任意线性规划的SLC上界为$2^{n + o(1)}$，这意味着我们IPM的迭代复杂度至多为指数级。这不同于现有依赖于比特复杂度或条件测度的迭代复杂度界——这些指标在问题维度中可能无界。我们通过证明中心路径可被称为\emph{最大中心路径}的组合代理精确逼近（该路径由$2n$条顶点单纯形路径组成）来得到上界。该上界补充了Allamigeon、Benchimol、Gaubert与Joswig（SIAGA 2018）以及Allamigeon、Gaubert与Vandame（STOC 2022）在构造指数SLC线性规划时建立的下界。最后，我们证明所提出IPM的每次迭代可在强多项式时间内实现。在此过程中，我们开发了一种确定性算法，可在强多项式时间内高精度逼近矩阵的奇异值分解，这一结果可能具有独立价值。