Getting Started

Abstract

AbstractWhat are integer programs? We introduce this class of problems and present two algorithmic ideas for solving them, the branch-and-bound and cutting plane methods. Both capitalize heavily on the fact that linear programming is a well-solved class of problems. Many practical situations can be modeled as integer programs, as will be shown in Chap. 2. But in Chap. 1 we reflect on the consequences of the algorithmic ideas just mentioned. Algorithms raise the question of computational complexity. Which problems can be solved in “polynomial time”? The cutting plane approach also leads naturally to the notion of “convex hull” and the so-called polyhedral approach. We illustrate these notions in the case of 2-variable integer programs. Complementary to the connection with linear programming, there is also an interesting connection between integer programming and number theory. In particular, we show how to find an integer solution to a system of linear equations. Contrary to general integer linear programming problems which involve inequalities, this problem can be solved in polynomial time.KeywordsGeneral Integer Linear Programming ProblemCutting Plane ApproachNatural Linear Programming RelaxationSize EncodingHermite Normal FormThese 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.