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

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