This tutorial presents a theory of valid inequalities for mixed integer linear sets. It introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as lift-and-project cuts, Gomory mixed integer cuts, mixed integer rounding cuts, split cuts and intersection cuts, and it reveals the relationships between these...

A market basket survey for beef retail cut composition at the retail level (four stores each from two chains in each city) was conducted in 11 US cities from January to March 2006. Beef cuts (n=17,495) were measured for external fat thickness with cuts from the chuck (0.05cm), round (0.05cm), and miscellaneous (0.04cm) having less (P<0.05) fat than cuts from the loin (0.11cm) and rib (0.11cm). ...

In technology mapping, enumeration of subcircuits or cuts to be replaced by a standard cell is an important step that decides both the quality of the solution and execution speed. In this work, we view cuts as set of edges instead of as set of nodes and based on it, provide a classification of cuts. It is shown that if enumeration is restricted to a subclass of cuts called unidirectional cuts, ...

We analyze split cuts from the perspective of cut generating functions via geometric lifting. We show that α-cuts, a natural higher-dimensional generalization of the k-cuts of Cornuéjols et al., gives all the split cuts for the mixed-integer corner relaxation. As an immediate consequence we obtain that the k-cuts are equivalent to split cuts for the 1-row mixed-integer relaxation. Further, we s...

Gomory mixed-integer (GMI) cuts are among the most effective cutting planes for general mixed-integer programs (MIP). They are traditionally generated from an optimal basis of a linear programming (LP) relaxation of a MIP. In this paper we propose a heuristic to generate useful GMI cuts from additional bases of the initial LP relaxation. The cuts we generate have rank one, i.e., they do not use...

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 stronges...

This tutorial presents a theory of valid inequalities for mixed integer linear sets. It introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as lift-and-project cuts, Gomory mixed integer cuts, mixed integer rounding cuts, split cuts and intersection cuts, and it reveals the relationships between these...

We explore the idea of obtaining valid inequalities from a finite-domain formulation of a problem, rather than a 0-1 formulation. A finite-domain model represents discrete choices with variables that have several possible values, as is frequently done in constraint programming. We apply the idea to graph coloring and identify facet-defining cuts that, when converted to cuts in a 0-1 model of th...

Gomory mixed-integer cuts are an important ingredient in state-ofthe-art software for solving mixed-integer linear programs. In particular, much attention has been paid to the strengthening of these cuts. In this paper, we give an overview of existing approaches for improving the performance of Gomory mixed-integer cuts. More precisely, we consider k-cuts, combined Gomory mixed-integer cuts, re...