Some lower bounds on sparse outer approximations of polytopes

TL;DR

This paper investigates the limitations of sparse inequalities in approximating polytopes and integer hulls, revealing that certain polytopes remain difficult to approximate regardless of added inequalities or rotations.

Contribution

It extends prior work by providing positive answers to questions about the inherent difficulty of approximating polytopes with sparse inequalities, even with additional dense inequalities or rotations.

Findings

Sparse inequalities can poorly approximate some integer hulls.

Adding a limited number of dense inequalities does not always improve approximation.

Certain polytopes are inherently hard to approximate under all rotations.

Abstract

Motivated by the need to better understand the properties of sparse cutting-planes used in mixed integer programming solvers, the paper [2] studied the idealized problem of how well a polytope is approximated by the use of sparse valid inequalities. As an extension to this work, we study the following less idealized questions in this paper: (1) Are there integer programs, such that sparse inequalities do not approximate the integer hull well even when added to a linear programming relaxation? (2) Are there polytopes, where the quality of approximation by sparse inequalities cannot be significantly improved by adding a budgeted number of arbitrary (possibly dense) valid inequalities? (3) Are there polytopes that are difficult to approximate under every rotation? (4) Are there polytopes that are difficult to approximate in all directions using sparse inequalities? We answer each of the above questions in the positive.