Geometric Random Edge

TL;DR

This paper introduces a geometric property-based variant of the random-edge pivot rule that ensures a strongly polynomial-time simplex algorithm for certain linear programs, generalizing previous integral matrix results.

Contribution

It presents a new geometric condition on the constraint matrix that guarantees a strongly polynomial simplex algorithm, extending prior work to broader classes of matrices.

Findings

The algorithm's number of pivots is polynomial in dimension and 1/δ.

It applies to matrices with bounded sub-determinants, including totally unimodular matrices.

The approach connects geometric properties with algorithmic complexity.

Abstract

We show that a variant of the random-edge pivoting rule results in a strongly polynomial time simplex algorithm for linear programs $\max\{c^Tx \colon Ax\leq b\}$, whose constraint matrix $A$ satisfies a geometric property introduced by Brunsch and R\"oglin: The sine of the angle of a row of $A$ to a hyperplane spanned by $n-1$ other rows of $A$ is at least $\delta$. This property is a geometric generalization of $A$ being integral and all sub-determinants of $A$ being bounded by $\Delta$ in absolute value (since $\delta \geq 1/(\Delta^2 n)$). In particular, linear programs defined by totally unimodular matrices are captured in this famework ($\delta \geq 1/ n$) for which Dyer and Frieze previously described a strongly polynomial-time randomized algorithm. The number of pivots of the simplex algorithm is polynomial in the dimension and $1/\delta$ and independent of the number of constraints of the linear program. Our main result can be viewed as an algorithmic realization of the proof of small diameter for such polytopes by Bonifas et al., using the ideas of Dyer and Frieze.