The Polyhedral Geometry of Pivot Rules and Monotone Paths

TL;DR

This paper introduces a geometric framework for analyzing pivot rules in linear programming, using polytopes and arborescences to understand their behavior and complexity, with implications for optimization and geometric combinatorics.

Contribution

It develops a novel polyhedral approach to study normalized-weight pivot rules, including the max-slope rule, and connects these to classical polytopes like permutahedra and associahedra.

Findings

Defined pivot rule polytopes and neighbotopes capturing pivot rule behavior.

Established face structures in terms of multi-arborescences.

Provided bounds on the number of coherent arborescences.

Abstract

Motivated by the analysis of the performance of the simplex method we study the behavior of families of pivot rules of linear programs. We introduce normalized-weight pivot rules which are fundamental for the following reasons: First, they are memory-less, in the sense that the pivots are governed by local information encoded by an arborescence. Second, many of the most used pivot rules belong to that class, and we show this subclass is critical for understanding the complexity of all pivot rules. Finally, normalized-weight pivot rules can be parametrized in a natural continuous manner. We show the existence of two polytopes, the pivot rule polytopes and the neighbotopes, that capture the behavior of normalized-weight pivot rules on polytopes and linear programs. We explain their face structure in terms of multi-arborescences. We compute upper bounds on the number of coherent arborescences, that is, vertices of our polytopes. Beyond optimization, our constructions provide new perspectives on classical geometric combinatorics. We introduce a normalized-weight pivot rule, we call the max-slope pivot rule which generalizes the shadow-vertex pivot rule. The corresponding pivot rule polytopes and neighbotopes refine monotone path polytopes of Billera--Sturmfels. Moreover special cases of our polytopes yield permutahedra, associahedra, and multiplihedra. For the greatest improvement pivot rules we draw connections to sweep polytopes and polymatroids.