How Good Are Sparse Cutting-Planes?

TL;DR

This paper investigates the effectiveness of sparse cutting-planes in approximating the integer hull of polytopes, providing bounds, tightness results, and insights into their performance in different classes of problems.

Contribution

It offers the first comprehensive analysis of sparse cutting-plane approximation quality, including upper and lower bounds, and compares their effectiveness in extended formulations.

Findings

Sparse cuts with half sparsity approximate polytopes well when vertices are polynomially bounded.

Lower bounds show the upper bounds are tight for random polytopes.

Sparse cuts are less effective for certain hard packing IPs unless sparsity is very high.

Abstract

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.