Understanding the Strength of General-Purpose Cutting Planes

Cutting planes for a mixed-integer program are linear inequalities which are satisfied by all feasible solutions of the latter. These are fundamental objects in mixed-integer programming that are critical for solving large-scale problems in practice. One of the main challenge in employing them is that there are limitless possibilities for generating cutting planes; the selection of the strongest ones is crucial for their effective use. In this thesis, we provide a principled study of the strength of generalpurpose cutting planes, giving a better understanding of the relationship between the different families of cuts available and analyzing the properties and limitations of our current methods for deriving cuts. We start by analyzing the strength of disjunctive cuts that generalize the ubiquitous split cuts. We first provide a complete picture of the containment relationship of the split closure, second split closure, cross closure, crooked cross closure and tbranch split closure. In particular, we show that rank-2 split cuts and crooked cross cuts are neither implied by cross cuts, which points out the limitations of the latter; these results answer questions left open in [56, 65]. Moreover, given the prominent role of relaxations and their computational advantages, we explore how strong are cross cuts obtained from basic and 2-row relaxations. Unfortunately we show that not all cross cuts can be obtained as cuts based on these relaxation, answering a question left open in [56]. One positive message from this result, though, is that cross cuts do not suffer from the limitations of these relaxations. Our second contribution is the introduction of a probabilistic model for comparing the strength of families of cuts for the continuous relaxation. We employ this model to compare the important split and triangle cuts, obtaining results that provide improved information about their behavior. More precisely, while previous works indicated that triangle cuts should be much stronger than split cuts, we provide the first theoretical support for the effect that is observed in practice: for most instances,