An Integer Labeling Algorithm for Solving a Class of Integer Programming Problems

Abstract

In this chapter we consider the following problem. Given an arbitrary simplex P, for example, the convex hull of m + 1 (0 ≤ m ≤ n) affinely independent vectors of ℝⁿ, the problem is that of determining whether or not P contains an integer point. This problem is not quite as innocent as it appears. In fact we first prove that it is intractable in the sense that it is NP-complete. We then develop an algorithm to solve the problem. The algorithm is based on both a specific integer labeling rule and the K _l-triangulation of ℝⁿ. The main feature of the algorithm can be described as follows. We first subdivide ℝⁿ into n-dimensional simplices such that all integer points of ℝⁿ are the vertices of the triangulation, and then assigns an integer to each integer point of ℝⁿ according to the labeling rule. Starting from an arbitrarily chosen integer point, the algorithm generates a sequence of adjacent simplices of varying dimension and terminates with either the YES or (exclusively) NO answer within a finite number of steps. In the YES case, the algorithm finds an integer point in P. The NO answer shows that there is no integer point in P. The algorithm is derived from the algorithm of van der Laan and Talman [1979] (see Chapter 5) although their algorithm was introduced to compute a fixed point of a continuous function.