On Circuit Diameter and Straight Line Complexity

TL;DR

This paper explores the relationship between circuit diameter and straight line complexity in polyhedra, establishing bounds and algorithms that improve understanding of linear programming solution paths.

Contribution

It proves that circuit diameter is polynomially bounded by straight line complexity and provides a matching circuit augmentation algorithm.

Findings

Circuit diameter is polynomially bounded by straight line complexity.

Strongly polynomial bounds for polyhedra with at most 2 variables per inequality.

A circuit augmentation algorithm with matching iteration complexity.

Abstract

The circuit diameter of a polyhedron is the maximum length (number of steps) of a shortest circuit walk between any two vertices of the polyhedron. Introduced by Borgwardt, Finhold and Hemmecke (SIDMA 2015), it is a relaxation of the combinatorial diameter of a polyhedron. These two notions of diameter lower bound the number of iterations taken by circuit augmentation algorithms and the simplex method respectively for solving linear programs. Recently, an analogous lower bound for path-following interior point methods was introduced by Allamigeon, Dadush, Loho, Natura and V\'egh (SICOMP 2025). Termed straight line complexity, it refers to the minimum number of pieces of any piecewise linear curve that traverses a specified neighborhood of the central path. In this paper, we study the relationship between circuit diameter and straight line complexity. For a polyhedron $P:=\{x\in \mathbb{R}^n: Ax = b, x\geq \mathbf{0}\}$, we show that its circuit diameter is up to a $\mathrm{poly}(n)$ factor upper bounded by the straight line complexity of linear programs defined over $P$. This yields a strongly polynomial circuit diameter bound for polyhedra with at most 2 variables per inequality. We also give a circuit augmentation algorithm with matching iteration complexity.