Planar 3-dimensional assignment problems with Monge-like cost arrays

Given an n× n× p cost array C we consider the problem p-P3AP which consists in finding p pairwise disjoint permutations φ1, φ2, . . . , φp of {1, . . . , n} such that ∑p k=1 ∑n i=1 ciφk(i)k is minimized. For the case p = n the planar 3-dimensional assignment problem P3AP results. Our main result concerns the p-P3AP on cost arrays C that are layered Monge arrays. In a layered Monge array all n×n matrices that result from fixing the third index k are Monge matrices. We prove that the p-P3AP and the P3AP remain NP-hard for layered Monge arrays. Furthermore, we show that in the layered Monge case there always exists an optimal solution of the p-3PAP which can be represented as matrix with bandwidth ≤ 4p − 3. This structural result allows us to provide a dynamic programming algorithm that solves the p-P3AP in polynomial time on layered Monge arrays when p is fixed.