On the depth of cutting planes

TL;DR

This paper introduces a new notion of depth for cutting planes, providing theoretical properties and bounds that relate to their practical strength, with efficient computation methods for certain cases.

Contribution

It defines a novel depth measure for cutting planes, establishes bounds for split and intersection cuts, and offers polynomial-time computation methods.

Findings

Depth is a good proxy for practical strength of cutting planes.

Parametric bounds explain observed computational behaviors.

Depth can be computed efficiently for certain polyhedra.

Abstract

We introduce a natural notion of depth that applies to individual cutting planes as well as entire families. This depth has nice properties that make it easy to work with theoretically, and we argue that it is a good proxy for the practical strength of cutting planes. In particular, we show that its value lies in a well-defined interval, and we give parametric upper bounds on the depth of two prominent types of cutting planes: split cuts and intersection cuts from a simplex tableau. Interestingly, these parametric bounds can help explain behaviors that have been observed computationally. For polyhedra, the depth of an individual cutting plane can be computed in polynomial time through an LP formulation, and we show that it can be computed in closed form in the case of corner polyhedra.