On Circuit Diameter and Straight Line Complexity
Daniel Dadush, Stefan Kober, Zhuan Khye Koh
TL;DR
This work establishes a quantitative bridge between circuit diameter and straight line complexity for linear programs, showing that the length of a shortest circuit walk from any feasible point to the optimal face is bounded by a polyhedral factor times the sum of straight line complexities of the coordinates along the max central path. It provides a concrete circuit augmentation algorithm whose iteration count matches this bound up to poly$(n)$ factors, thereby aligning the performance of boundary-based circuit methods with interior-point path-following methods. As a corollary, the circuit diameter of a polyhedron is upper bounded by the straight line complexity of its associated LPs up to poly$(n)$ factors, and for polyhedra with at most two non-zeros per column (or per inequality in the dual), this yields strongly polynomial diameter bounds like $O(mn^4\log^2 n)$. The approach relies on a polarized decomposition of the max central path, a lifting operator with singular subspaces, and a Ratio-Circuit oracle to generate augmenting directions, combining short “Wallacher-type” steps with longer, targeted adjustments. The results extend our understanding of how interior and boundary-following LP algorithms relate in practice and provide new tools for analyzing and implementing efficient circuit-based augmentation schemes.
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 Végh (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.