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

A. Custic, B. Klinz, G.J. Woeginger

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6 Citations (Scopus)
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Abstract

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 Lp norm; in particular the problem is NP-hard for the Manhattan norm L1 and the Maximum norm L8 which both have polyhedral unit balls. Keywords: Combinatorial optimization; Computational complexity; 3-dimensional assignment problem; 3-dimensional matching problem; Polyhedral norm
Original languageEnglish
Pages (from-to)38-55
JournalDiscrete Optimization
Volume18
DOIs
Publication statusPublished - 2015

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