Well-solvable cases of the QAP with block-structured matrices

TL;DR

This paper explores specific instances of the quadratic assignment problem with block-structured matrices, identifying conditions under which these instances are solvable in polynomial time or are NP-hard.

Contribution

It characterizes the complexity of QAPs with block-structured matrices, providing polynomial algorithms for some cases and NP-hardness results for others.

Findings

Polynomial time solvability for QAP with monotone anti-Monge matrices under certain conditions

NP-hardness of QAP with block structures under specific conditions

Identification of structural properties influencing computational complexity

Abstract

We investigate special cases of the quadratic assignment problem (QAP) where one of the two underlying matrices carries a simple block structure. For the special case where the second underlying matrix is a monotone anti-Monge matrix, we derive a polynomial time result for a certain class of cut problems. For the special case where the second underlying matrix is a product matrix, we identify two sets of conditions on the block structure that make this QAP polynomially solvable respectively NP-hard.