Circuit Diameter of Polyhedra is Strongly Polynomial

2026-02-06

AI Summary

This work addresses the long-standing open question of whether the circuit diameter of polyhedra admits a strong polynomial upper bound—the circuit analogue of the polynomial Hirsch conjecture. By integrating circuit direction analysis, the geometric structure of linear programming, and combinatorial optimization techniques, the authors construct, for the first time, a monotone circuit walk and establish that the circuit diameter of any polyhedron in standard form is at most \(O(m^2 \log m)\). This result breaks through the previously known weakly polynomial bounds and provides the first strong polynomial upper bound on circuit diameters, offering crucial theoretical support for the existence of strongly polynomial-time algorithms for linear programming.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Applications of cryptography

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.

Problem

• circuit diameter
• polyhedra
• strongly polynomial
• Hirsch conjecture
• linear programming

Innovation

• circuit diameter
• strongly polynomial
• polyhedra
• monotone circuit walks
• linear programming