Extended Formulation Lower Bounds via Hypergraph Coloring?

TL;DR

This paper introduces a new framework for proving lower bounds on the size of extended formulations in combinatorial optimization, using hypergraph coloring to establish exponential lower bounds for certain problems.

Contribution

It develops a novel methodology connecting hypergraph chromatic number to extended formulation size lower bounds, applicable to mixed integer sets and approximate relaxations.

Findings

Proves exponential lower bounds for approximate mixed product formulations of the metric capacitated facility location problem.

Introduces product relaxations and a hypergraph coloring approach to analyze extended formulation complexity.

Shows mixed product relaxations are as powerful as certain extended formulations.

Abstract

Exploring the power of linear programming for combinatorial optimization problems has been recently receiving renewed attention after a series of breakthrough impossibility results. From an algorithmic perspective, the related questions concern whether there are compact formulations even for problems that are known to admit polynomial-time algorithms. We propose a framework for proving lower bounds on the size of extended formulations. We do so by introducing a specific type of extended relaxations that we call product relaxations and is motivated by the study of the Sherali-Adams (SA) hierarchy. Then we show that for every approximate relaxation of a polytope P, there is a product relaxation that has the same size and is at least as strong. We provide a methodology for proving lower bounds on the size of approximate product relaxations by lower bounding the chromatic number of an underlying hypergraph, whose vertices correspond to gap-inducing vectors. We extend the definition of product relaxations and our methodology to mixed integer sets. However in this case we are able to show that mixed product relaxations are at least as powerful as a special family of extended formulations. As an application of our method we show an exponential lower bound on the size of approximate mixed product formulations for the metric capacitated facility location problem, a problem which seems to be intractable for linear programming as far as constant-gap compact formulations are concerned.