We study the circuit diameter of polyhedra, introduced by Borgwardt, Finhold, and Hemmecke (SIDMA, 2015) as a relaxation of the combinatorial diameter. We show that the circuit diameter of a system $\{x \in \mathbb{R}^n: Ax=b, 0\leq x\leq u\}$ for $A \in \mathbb{R}^{m \times n}$ is bounded as $O(m \min\{m, n-m\} \log(m+ \kappa_A)+n \log n)$, where $\kappa_A$ is the circuit imbalance measure of the constraint matrix. This yields a strongly polynomial circuit diameter bound e.g. if all entries of $A$ have polynomially bounded encoding length in $n$. Further, we present circuit augmentation algorithms for LPs using the minimum-ratio circuit cancelling rule. Even though the standard minimum-ratio circuit cancelling algorithm is not finite in general, our variant can solve an LP in $O(mn^2\log(n+\kappa_A))$ augmentation steps.

翻译：我们研究多面体的电路直径，该概念由Borgwardt、Finhold和Hemmecke（SIDMA, 2015）提出，作为组合直径的松弛形式。我们证明，对于系统$\{x \in \mathbb{R}^n: Ax=b, 0\leq x\leq u\}$（其中$A \in \mathbb{R}^{m \times n}$），其电路直径有界为$O(m \min\{m, n-m\} \log(m+ \kappa_A)+n \log n)$，这里$\kappa_A$是约束矩阵的电路不平衡度量。若$A$的所有条目在$n$的编码长度上呈多项式有界，该结果可给出强多项式电路直径界限。此外，我们提出基于最小比率电路消去规则的线性规划电路增广算法。尽管标准的最小比率电路消去算法通常不保证有限收敛，但我们的变体可在$O(mn^2\log(n+\kappa_A))$次增广步骤内求解线性规划。