Exponential lower bounds on the size of approximate formulations in the natural encoding for Capacitated Facility Location

TL;DR

This paper proves that any linear programming relaxation using the natural encoding for the metric capacitated facility location problem requires exponentially many constraints to achieve a constant integrality gap, highlighting fundamental limitations.

Contribution

It establishes exponential lower bounds on the size of approximate formulations in the natural encoding for this problem, a key open question in approximation algorithms.

Findings

Any constant-gap LP relaxation in the natural encoding must have exponential size.

The proof does not rely on properties like locality or symmetry.

This result indicates inherent complexity in approximating the problem via LP relaxations.

Abstract

The metric capacitated facility location is a well-studied problem for which, while constant factor approximations are known, no efficient relaxation with constant integrality gap is known. The question whether there is such a relaxation is among the most important open problems of approximation algorithms \cite{ShmoysWbook}. In this paper we show that, if one is restricted to linear programs that use the natural encoding for facility location, at least an exponential number of constraints is needed to achieve a constant gap. Our proof does not assume any special property of the relaxation such as locality or symmetry.