Circuit Diameter of Polyhedra is Strongly Polynomial
Bento Natura
TL;DR
This work resolves the circuit diameter question for polyhedra by proving the first strongly polynomial bound: the circuit diameter of $P = \{x : \mathbf{A}x = b, x \ge 0\}$ is $O(m^2 \log m)$. The authors introduce a two-phase circuit-augmentation algorithm: Phase 1 reduces the non-basic support to at most $m$, and Phase 2 maintains a trapped set of basic indices while leveraging norm-reduction and elimination steps to force progress in a combinatorial, condition-number-free manner. Central to the proof is a conformal circuit decomposition of $x^* - x$ and carefully designed extrapolation steps that guarantee either a non-basic coordinate becomes zero or the trapped set expands, leading to a total of $O(m^2 \log m)$ augmentations. They also show the monotone circuit diameter enjoys the same bound, and discuss implications for Smale’s 9th problem and the computational-existence gap, noting that while a short circuit walk can be found efficiently between two given vertices, solving LP optimally remains algorithmically distinct.
Abstract
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.