How Good Are Sparse Cutting-Planes?

Sparse cutting-planes are often the ones used 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 $d(P, P^k) = \max_{x \in P^k} \left(min_{y \in P} \| x - y\|\right)$ as a measure of the quality of sparse cuts. In our first result, we present general upper bounds on $d(P, P^k)$ which depend on the number of vertices in the polytope and exhibits three phases as $k$ increases. 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 $d(P, P^k)$ 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 $d(P, P^k)$ is large for such instances unless $k$ is very close to $n$. Finally, we show that using sparse cutting-planes in extended formulations is at least as good as using them in the original polyhedron, and give an example where the former is actually much better.