The quadratic assignment problem is easy for Robinsonian matrices with Toeplitz structure

TL;DR

This paper identifies a new polynomially solvable case of the Quadratic Assignment Problem when matrices are Robinsonian and one is Toeplitz, simplifying solutions in certain seriation contexts.

Contribution

It proves that the identity permutation is optimal for QAP when matrices are Robinsonian and one is Toeplitz, a novel solvable case in combinatorial optimization.

Findings

The identity permutation is optimal under specified conditions.

The problem is polynomially solvable in this case.

Applicable to seriation problems with Robinsonian matrices.

Abstract

We present a new polynomially solvable case of the Quadratic Assignment Problem in Koopmans-Beckman form $QAP(A,B)$, by showing that the identity permutation is optimal when $A$ and $B$ are respectively a Robinson similarity and dissimilarity matrix and one of $A$ or $B$ is a Toeplitz matrix. A Robinson (dis)similarity matrix is a symmetric matrix whose entries (increase) decrease monotonically along rows and columns when moving away from the diagonal, and such matrices arise in the classical seriation problem.