Simple Extensions of Polytopes

TL;DR

This paper introduces the concept of simple extension complexity for polytopes, providing combinatorial methods to establish lower bounds and demonstrating large complexities for various important polytopes.

Contribution

It defines simple extension complexity, develops a combinatorial lower bound technique, and applies it to several key polytopes, expanding understanding of their extension properties.

Findings

Several polytopes have large simple extension complexities.

A combinatorial method for lower bounds is devised.

Generalization of adjacency structure results for perfect matching polytopes.

Abstract

We introduce the simple extension complexity of a polytope P as the smallest number of facets of any simple (i.e., non-degenerate in the sense of linear programming) polytope which can be projected onto P. We devise a combinatorial method to establish lower bounds on the simple extension complexity and show for several polytopes that they have large simple extension complexities. These examples include both the spanning tree and the perfect matching polytopes of complete graphs, uncapacitated flow polytopes for non-trivially decomposable directed acyclic graphs, hypersimplices, and random 0/1-polytopes with vertex numbers within a certain range. On our way to obtain the result on perfect matching polytopes we generalize a result of Padberg and Rao's on the adjacency structures of those polytopes. To complement the lower bounding techniques we characterize in which cases known construction techniques yield simple extensions.