From linear programming to colliding particles

TL;DR

This paper explores the complex behavior of max-slope pivot rules on simplices and related polytopes, revealing deep combinatorial structures linked to particle collision models and extending known geometric realizations.

Contribution

It establishes a novel connection between max-slope pivot rules and associahedra, multiplihedra, and constrainahedra, providing new geometric realizations and interpretations.

Findings

Pivot rule polytopes are isomorphic to associahedra for simplices.

Interpretation of pivot rules via particle collision combinatorics.

New realizations of constrainahedra for products of simplices.

Abstract

Although simplices are trivial from a linear optimization standpoint, the simplex algorithm can exhibit quite complex behavior. In this paper we study the behavior of max-slope pivot rules on (products of) simplices and describe the associated pivot rule polytopes. For simplices, the pivot rule polytopes are combinatorially isomorphic to associahedra. To prove this correspondence, we interpret max-slope pivot rules in terms of the combinatorics of colliding particles on a line. For prisms over simplices, we recover Stasheff's multiplihedra. For products of two simplices we get new realizations of constrainahedra, that capture the combinatorics of certain particle systems in the plane.