We prove a strongly polynomial bound on the circuit diameter of polyhedra, resolving the circuit analogue of the polynomial Hirsch conjecture. Specifically, we show that the circuit diameter of a polyhedron $P = \{x\in \mathbb{R}^n:\, A x = b, \, x \ge 0\}$ with $A\in\mathbb{R}^{m\times n}$ is $O(m^2 \log m)$. Our construction yields monotone circuit walks, giving the same bound for the monotone circuit diameter. The circuit diameter, introduced by Borgwardt, Finhold, and Hemmecke (SIDMA 2015), is a natural relaxation of the combinatorial diameter that allows steps along circuit directions rather than only along edges. All prior upper bounds on the circuit diameter were only weakly polynomial. Finding a circuit augmentation algorithm that matches this bound would yield a strongly polynomial time algorithm for linear programming, resolving Smale's 9th problem.

翻译：我们证明了多面体电路直径的强多项式上界，从而解决了多项式赫希猜想的电路模拟问题。具体而言，我们证明了由 $A\in\mathbb{R}^{m\times n}$ 定义的多面体 $P = \{x\in \mathbb{R}^n:\, A x = b, \, x \ge 0\}$ 的电路直径为 $O(m^2 \log m)$。我们的构造产生了单调电路行走，从而为单调电路直径提供了相同的上界。电路直径由 Borgwardt、Finhold 和 Hemmecke（SIDMA 2015）引入，是组合直径的一种自然松弛形式，它允许沿电路方向移动而不仅仅是沿边移动。此前所有关于电路直径的上界都仅是弱多项式的。若能找到匹配该上界的电路增广算法，则将产生线性规划的强多项式时间算法，从而解决斯梅尔第九问题。