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 $O(m^2 \log(m+ \kappa_A)+n \log n)$, 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 optimization LP in $O(n^3\log(n+\kappa_A))$ augmentation steps.

翻译：我们用Borgwardt、Finhold和Hemmecke(SIDMA, 2015年)作为组合直径的放松。 我们显示一个系统的电路直径 $x\x\ in\ mathb{R ⁇ n: Ax=b, 0\leq x\leq u ⁇ $, $A\ in\mathbb{R ⁇ m\ times n}$, 由Borgwardt, Finhold, 和Hemmecke (SIDMA, 2015年)作为组合直径的放松。 我们显示一个系统的电路直径 $x\x\ in\ mathb{R+\kappa_ A) 。 如果$A值的所有条目都以美元以多元绑定的编码长度 。 此外, 我们使用最起码的拉皮电路取消规则为LPs提出电路增速算算法。 即使标准的最低限度取消算算算法不是一般的限定值, 我们的变式可以用 $A\\\\\\\\\\\\\\\\\\\\\\\\\\\ a