On Limits to the Scope of the Extended Formulations "Barriers"

TL;DR

This paper challenges key assumptions in extended formulations of polytopes, showing that some widely accepted implications are false and introducing the concept of augmentation to clarify the limitations of EF relations.

Contribution

It introduces the notion of augmentation for polytopes and demonstrates the invalidity of certain presumptions used in extended formulation arguments.

Findings

Counter-examples to EF presumptions are provided.

Augmentation can make EF relations degenerate or meaningless.

The projection relation does not always hold under augmentation.

Abstract

In this paper, we introduce the notion of augmentation for polytopes and use it to show the error in two presumptions that have been key in arriving at over-reaching/over-scoped claims of "impossibility" in recent extended formulations (EF) developments. One of these presumptions is that: "If Polytopes P and Q are described in the spaces of variables x and y respectively, and there exists a linear map x=Ay between the feasible sets of P and Q, then Q is an EF of P". The other is: "(An augmentation of Polytope A projects to Polytope B) ==> (The external descriptions of A and B are related)". We provide counter-examples to these presumptions, and show that in general: (1) If polytopes can always be arbitrarily augmented for the purpose of establishing EF relations, then the notion of EF becomes degenerate/meaningless in some cases, and that: (2) The statement: "(Polytope B is the projection of an augmentation of Polytope A) ==> (Polytope B is the projection of Polytope A)" is not true in general (although, as we show, the converse statement, "(B is the projection of A) ==> (B is the projection of every augmentation of A)", is true in general). We illustrate some of the ideas using the minimum spanning tree problem, as well as the "lower bounds" developments in Fiorini et al. (2011; 2012), in particular.