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

TL;DR

This paper studies the computational complexity of a 3D assignment problem involving points in space and triangle perimeter costs under various norms, revealing NP-hardness and polynomial cases.

Contribution

It characterizes the complexity of the problem for different norms and space dimensions, including NP-hardness and polynomial solvability results.

Findings

Minimization is NP-hard for all norms, even in 2D.

Maximization is polynomial if dimension is fixed and norm has a polyhedral unit ball.

Maximization is NP-hard when dimension varies, including for L1 and L-infinity norms.

Abstract

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.