Approximating polyhedra with sparse inequalities

In this paper, we study how well one can approximate arbitrary polytopes using sparse inequalities. Our motivation comes from the use of sparse cutting-planes in mixed-integer programing (MIP) solvers, since they help in solving the linear programs encountered during branch&-bound more efficiently. However, how well can we approximate the integer hull by just using sparse cutting-planes? In order to understand this question better, given a polyope P (e.g. the integer hull of a MIP), let P k be its best approximation using cuts with at most k non-zero coefficients. We consider d(P, P ) = maxx∈Pk (miny∈P ‖x− y‖) as a measure of the quality of sparse cuts. In our first result, we present general upper bounds on d(P, P ) which depend on the number of vertices in the polytope. Our bounds imply that if P has polynomially many vertices, using half sparsity already approximates it very well. Second, we present a lower bound on d(P, P ) for random polytopes that show that the upper bounds are quite tight. Third, we show that for a class of hard packing IPs, sparse cutting-planes do not approximate the integer hull well, that is d(P, P ) is large for such instances unless k is very close to n. Finally, we show that using sparse cutting-planes in polytope extensions is at least as good as using them in the original polyhedron, and give an example where the former is actually much better.