On Circuit Diameter Bounds via Circuit Imbalances

TL;DR

This paper establishes new bounds on the circuit diameter of polyhedra using circuit imbalances and introduces efficient circuit augmentation algorithms for linear programming.

Contribution

It provides the first bounds on circuit diameter involving circuit imbalance and develops a polynomial-time circuit augmentation method for LPs.

Findings

Circuit diameter bounded by $O(m ext{min}\{m, n-mig) ext{log}(m+ \kappa_A)+n ext{log} n$.

Strongly polynomial bounds when matrix entries have polynomially bounded encoding length.

LPs can be solved in $O(mn^2 ext{log}(n+\kappa_A))$ augmentation steps.

Abstract

We study the circuit diameter of polyhedra, introduced by Borgwardt, Finhold, and Hemmecke (SIDMA 2015) as a relaxation of the combinatorial diameter. We show that the circuit diameter of a system $\{x \in \mathbb{R}^n: Ax=b, 0\leq x\leq u\}$ for $A \in \mathbb{R}^{m \times n}$ is bounded by $O(m \min\{m, n-m\} \log(m+ \kappa_A)+n \log n)$, where $\kappa_A$ is the circuit imbalance measure of the constraint matrix. This yields a strongly polynomial circuit diameter bound if e.g., all entries of $A$ have polynomially bounded encoding length in $n$. Further, we present circuit augmentation algorithms for LPs using the minimum-ratio circuit cancelling rule. Even though the standard minimum-ratio circuit cancelling algorithm is not finite in general, our variant can solve an LP in $O(mn^2\log(n+\kappa_A))$ augmentation steps.