On the existence of compact {\epsilon}-approximated formulations for knapsack in the original space

TL;DR

The paper demonstrates that certain knapsack polytopes lack compact {}-approximated formulations in the original space, but provides bounds for such formulations for any down-monotone polytope.

Contribution

It proves the non-existence of polynomial-sized {}-approximated formulations for some knapsack polytopes and offers bounds for all down-monotone polytopes.

Findings

Super-polynomial size needed for {}-approximations of some knapsack polytopes.

Bound of O(min{log(n/{}),n}/{}) inequalities for any down-monotone polytope.

Answers a question posed by Bienstock and McClosky (2012).

Abstract

We show that there exists a family of Knapsack polytopes such that, for each polytope P from this family and each {\epsilon} > 0, any {\epsilon}-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 {\epsilon}-approximated formulation in the original space can be obtained with inequalities using at most O(min{log(n/{\epsilon}),n}/{\epsilon}) different coefficients.