A Remark on the Manhattan Distance Matrix of a Rectangular Grid

TL;DR

This paper extends recent lower bounds for the Quadratic Assignment Problem to cases where the distance matrix is spherical Euclidean, encompassing Manhattan and Hamming distances, broadening the applicability of these bounds.

Contribution

It generalizes previous results to include spherical Euclidean distance matrices, unifying Manhattan and Hamming distances under a common framework.

Findings

Lower bounds for QAP extend to spherical Euclidean distance matrices.

Manhattan and Hamming distances are special cases of spherical Euclidean distances.

The results unify different distance metrics within a common theoretical framework.

Abstract

Consider the Quadratic Assignment Problem (QAP): given two matrices A and D, minimize {trace AXDX^T: X is a permutation matrix}. New lower bounds were obtained recently (Mittelmann and peng [8]) for the QAP where D is either the Manhattan distance matrix of a rectangular grid, or the Hamming distance of a hypercube. In this note, we show that the results in [8,11] extend to the case where D is a spherical Euclidean distance matrix, which includes the Manhattan distance matrix and the Hamming distance matrix as special cases.