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 maxi=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.