Test sets of integer programs

This article is a survey about recent developments in the area of test sets of families of linear integer programs. Test sets are finite subsets of the integer lattice that allow us to improve any given feasible non-optimal point of an integer program by one element in the set. There are various possible ways of defining test sets depending on the view that one takes: the Graver test set is naturally derived from a study of the integral vectors in cones; the Scarf test set (neighbors of the origin) is strongly connected to the study of lattice point free convex bodies; the so-called reduced Gröbner basis of an integer program is obtained from a study of generators of polynomial ideals. This explains why the study of test sets connects various branches of mathematics. We introduce in this paper these three kinds of test sets and discuss relations between them. We also illustrate on various examples such as the minimum cost flow problem, the knapsack problem and the matroid optimization problem how these test sets may be interpreted combinatorially. From the viewpoint of integer programming a major interest in test sets is their relation to the augmentation problem. This is discussed here in detail. In particular, we derive a complexity result of the augmentation problem, we discuss an algorithm for solving the augmentation problem by computing the Graver test set and show that, in the special case of an integer knapsack problem with 3 coefficients, the augmentation problem can be solved in polynomial time.