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 by $O(m \min\{m, n-m\} \log(m+ \kappa_A)+n \log n)$, where $\kappa_A$ is the circuit imbalance measure of the constraint matrix. This yields a strongly polynomial circuit diameter bound if e.g., 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：我们研究了由Borgwardt、Finhold和Hemmecke（SIDMA 2015）引入的多面体电路直径，该概念是组合直径的一种松弛形式。我们证明，对于$A \in \mathbb{R}^{m \times n}$，系统$\{x \in \mathbb{R}^n: Ax=b, 0\leq x\leq u\}$的电路直径上界为$O(m \min\{m, n-m\} \log(m+ \kappa_A)+n \log n)$，其中$\kappa_A$是约束矩阵的电路不平衡度量。若$A$的所有元素在$n$中具有多项式有界编码长度，则该结果可导出强多项式电路直径界。此外，我们提出了使用最小比率电路消去规则的线性规划电路增广算法。尽管标准最小比率电路消去算法通常不具备有限性，我们的变体算法可在$O(mn^2\log(n+\kappa_A))$次增广步骤内求解线性规划问题。