Circuit Diameter of Polyhedra is Strongly Polynomial

TL;DR

This paper proves a strongly polynomial upper bound on the circuit diameter of polyhedra, advancing the understanding of polyhedral geometry and potentially leading to a strongly polynomial linear programming algorithm.

Contribution

It establishes the first strongly polynomial bound on the circuit diameter of polyhedra, resolving the circuit analogue of the polynomial Hirsch conjecture.

Findings

Circuit diameter is bounded by O(m^2 log m) for polyhedra defined by A x = b, x ≥ 0.

Monotone circuit walks also satisfy the same polynomial bound.

This result links circuit diameter bounds to the complexity of linear programming algorithms.

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.