A New Facet Class and a Polyhedral Method for the Three-Index Assignment Problem

Abstract

Since the number of variables of the three-dimensional assignment problem of order n is n3, an O(n 3) separation algorithm for a class of facets of the three-dimensional assignment poly tope is linear-time and its complexity is best possible. In [1] and [12], linear-time separation algorithms were given for four classes of facets. In this paper, we introduce a new class of facet-defining inequalities, with coefficients equal to 0 or 1 find right-hand side equal to a positive integer p. These inequalities are of Chvàtal rank 2. The special case when p = 1 has been identified by Balas and Saltzman in [2]. An O(n 3) separation algorithm is given for facets with p = 2 in this class. Based upon the linear-time separation algorithms for the five classes of facets found in [l], [12] and this paper, we construct a polyhedral procedure for solving the three-index assignment problem. Computational results are given to show that this procedure is efficient.KeywordsAssignment ProblemSeparation AlgorithmSubgradient OptimizationFacet ClassMultidimensional Assignment ProblemThese 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.