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

TL;DR

This paper investigates the complexity and structure of the p-P3AP, a 3D assignment problem with layered Monge arrays, proving NP-hardness but also identifying polynomial-time solutions for fixed p.

Contribution

It proves NP-hardness of p-P3AP on layered Monge arrays and introduces a dynamic programming approach for fixed p cases.

Findings

p-P3AP remains NP-hard on layered Monge arrays.

Optimal solutions have bounded bandwidth, enabling polynomial-time algorithms for fixed p.

The structural properties facilitate efficient solution methods for specific cases.

Abstract

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.