On the existence of compact ε-approximated formulations for knapsack in the original space

We show that there exists a family of Knapsack polytopes such that, for each polytope P from this family and each ε > 0, any ε-approximated formulation of P in the original space R^n requires a number of inequalities that is super-polynomial in n. This answers a question by Bienstock and McClosky (2012). We also prove that, for any down-monotone polytope, an ε-approximated formulation in the original space can be obtained with inequalities using at most O(min{log(n/ε),n}/ε) different coefficients.

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