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

Given an $n\times n\times p$ cost array $C$ we consider the problem $p$-P3AP which consists in finding $p$ pairwise disjoint permutations $\varphi_1,\varphi_2,\ldots,\varphi_p$ of $\{1,\ldots,n\}$ such that $\sum_{k=1}^{p}\sum_{i=1}^nc_{i\varphi_k(i)k}$ 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 $n\times n$ 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 $\le 4p-3$. 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.Comment: 16 pages, appendix will follow in v

Geometric versions of the 3-dimensional assignment problem under general norms

We discuss the computational complexity of special cases of the 3-dimensional (axial) assignment problem where the elements are points in a Cartesian space and where the cost coefficients are the perimeters of the corresponding triangles measured according to a certain norm. (All our results also carry over to the corresponding special cases of the 3-dimensional matching problem.) The minimization version is NP-hard for every norm, even if the underlying Cartesian space is 2-dimensional. The maximization version is polynomially solvable, if the dimension of the Cartesian space is fixed and if the considered norm has a polyhedral unit ball. If the dimension of the Cartesian space is part of the input, the maximization version is NP-hard for every $L_p$ norm; in particular the problem is NP-hard for the Manhattan norm $L_1$ and the Maximum norm $L_{\infty}$ which both have polyhedral unit balls.Comment: 21 pages, 9 figure