  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.
