New special cases of the Quadratic Assignment Problem with diagonally structured coefficient matrices

TL;DR

This paper identifies new polynomially solvable special cases of the Quadratic Assignment Problem involving diagonally structured coefficient matrices, expanding the class of efficiently solvable instances.

Contribution

It introduces new polynomially solvable cases of QAP with specific diagonal matrix structures and provides a polynomial-time recognition algorithm for these cases.

Findings

New polynomially solvable cases of QAP with diagonal structures.

Recognition algorithm for a class of Robinson matrices.

Expanded class of solvable QAP instances.

Abstract

We consider new polynomially solvable cases of the well-known Quadratic Assignment Problem involving coefficient matrices with a special diagonal structure. By combining the new special cases with polynomially solvable special cases known in the literature we obtain a new and larger class of polynomially solvable special cases of the QAP where one of the two coefficient matrices involved is a Robinson matrix with an additional structural property: this matrix can be represented as a conic combination of cut matrices in a certain normal form. The other matrix is a conic combination of a monotone anti-Monge matrix and a down-benevolent Toeplitz matrix. We consider the recognition problem for the special class of Robinson matrices mentioned above and show that it can be solved in polynomial time.