# How Good Are Sparse Cutting-Planes?

• Santanu S. Dey
• Marco Molinaro
• Qianyi Wang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8494)

## 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 ) = $$\max_{x \in{\bf P}^k} (\min_{y \in \bf P} \| x - y\|)$$ as a measure of the quality of sparse cuts.

In our first result, we present general upper bounds on d(P, P 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. 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.

## Keywords

Convex Hull Mixed Integer Linear Programming Extended Formulation Valid Inequality Full Version
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer International Publishing Switzerland 2014

## Authors and Affiliations

• Santanu S. Dey
• 1
• Marco Molinaro
• 1
• Qianyi Wang
• 1
1. 1.School of Industrial and Systems EngineeringGeorgia Institute of TechnologyUSA