An algebraic approach to symmetric extended formulations

TL;DR

This paper reinterprets Yannakakis's technique for symmetric extended formulations using group theory, simplifying lower bound proofs and advancing understanding of when small symmetric formulations are possible.

Contribution

It introduces a group-theoretic framework for Yannakakis's method, providing a new perspective and simplifying the derivation of lower bounds for symmetric extended formulations.

Findings

Rephrases Yannakakis's technique in a group-theoretic context

Simplifies several lower bound constructions

Enhances understanding of symmetric extended formulations

Abstract

Extended formulations are an important tool to obtain small (even compact) formulations of polytopes by representing them as projections of higher dimensional ones. It is an important question whether a polytope admits a small extended formulation, i.e., one involving only a polynomial number of inequalities in its dimension. For the case of symmetric extended formulations (i.e., preserving the symmetries of the polytope) Yannakakis established a powerful technique to derive lower bounds and rule out small formulations. We rephrase the technique of Yannakakis in a group-theoretic framework. This provides a different perspective on symmetric extensions and considerably simplifies several lower bound constructions.