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

We discuss the computational complexity of special cases of the three-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 three-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 LpLp norm; in particular the problem is NP-hard for the Manhattan norm L1L1 and the Maximum norm L∞L∞ which both have polyhedral unit balls.