A new efficiently solvable special case of the three-dimensional axial bottleneck assignment problem

Abstract

Given an n×n×n array C=(c _ijk ) of real numbers, the three-dimensional axial bottleneck assignment problem (3-BAP) is to find two permutations φ and ψ of {1, ..., n} such that max_i=1,...,n c _iφ(i)ψ(i) is minimized.

We first present two closely related conditions on the cost array C, the wedge property and the weak wedge property, which guarantee that an optimal solution of 3-BAP is obtained by setting φ and ψ to the identity permutation. In order to enlarge this class of efficiently solvable special cases of the 3-BAP, we then propose an O(n ³ log n) time algorithm which, given an n × n × n array C, either finds three permutations ρ, σ and τ such that the permuted array C _{ρ, σ, τ}=(c _{ρ(i)σ(j)τ(k)}) satisfies the wedge property, or proves that no such permutations exist.

This research has been partially supported by the Spezialforschungsbereich F 003 “Optimierung und Kontrolle”, Projektbereich Diskrete Optimierung. The second author acknowledges financial support by a research fellowship of the Euler Institute for Discrete Mathematics and its Applications.