We consider linear programming in the oracle model: $\max\{c^\top x \,:\, x\in P\}$, where the polyhedron $P=\{x\in\mathbb{R}^n\,:\, Ax\le b\}$ is given by a separation oracle. We present an algorithm that finds exact primal and dual solutions using $O(n^2\log(n/δ))$ oracle calls and $O(n^4\log(n/δ)+n^5\log\log(1/δ))$ arithmetic operations, where $δ$ is a geometric condition number associated with the system $(A,b)$. These bounds do not depend on the cost vector $c$ and do not require a priori knowledge of $δ$. For rational data, $\log(1/δ)$ is polynomially bounded in the encoding size of $(A,b)$, thus providing a polynomial-time algorithm. The algorithm works in a black box manner, requiring a subroutine for approximate primal and dual solutions; the above running times are achieved when using the cutting plane method of Jiang, Lee, Song, and Wong (STOC 2020) for this subroutine. Whereas approximate solvers may return primal solutions only, we develop a general framework for extracting dual certificates based on the work of Burrell and Todd (Math. Oper. Res. 1985). Our algorithm strengthens results by Grötschel, Lovász, and Schrijver (Prog. Comb. Opt. 1984), and by Frank and Tardos (Combinatorica 1987) that rely on bit-complexity arguments. Our algorithm avoids rounding-based arguments such as simultaneous Diophantine approximation and uses geometric arguments instead.

翻译：我们考虑预言机模型下的线性规划问题：$\max\{c^\top x \,:\, x\in P\}$，其中多面体 $P=\{x\in\mathbb{R}^n\,:\, Ax\le b\}$ 由分离预言机给出。我们提出了一种算法，该算法使用 $O(n^2\log(n/δ))$ 次预言机调用和 $O(n^4\log(n/δ)+n^5\log\log(1/δ))$ 次算术运算，即可找到精确的原问题和对偶问题解，其中 $δ$ 是与系统 $(A,b)$ 相关的几何条件数。这些上界不依赖于成本向量 $c$，且无需事先已知 $δ$ 的值。对于有理数据，$\log(1/δ)$ 在 $(A,b)$ 的编码规模上是多项式有界的，从而提供了多项式时间算法。该算法以黑箱方式工作，需要近似原问题和对偶问题解的例程；当使用 Jiang、Lee、Song 和 Wong（STOC 2020）的割平面方法作为该例程时，可实现上述运行时间。尽管近似求解器可能仅返回原问题解，但我们基于 Burrell 和 Todd（Math. Oper. Res. 1985）的工作，开发了一个用于提取对偶可行解（对偶证书）的通用框架。我们的算法加强了过去依赖位复杂度论证的 Grötschel、Lovász 和 Schrijver（Prog. Comb. Opt. 1984）以及 Frank 和 Tardos（Combinatorica 1987）的研究结果。该算法避免了基于舍入的论证（如同步丢番图逼近），转而采用几何论证。