Computing Equilibria and Fixed Points pp 115-145 | Cite as

# 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 ℝ^{ n }, 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 ℝ^{ n }. The main feature of the algorithm can be described as follows. We first subdivide ℝ^{ n } into *n*-dimensional simplices such that all integer points of ℝ^{ n } are the vertices of the triangulation, and then assigns an integer to each integer point of ℝ^{ n } 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.

## Keywords

Convex Hull Standard Form Integer Point Integer Programming Problem Dimensional Simplex## Preview

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