The Wiener maximum quadratic assignment problem

TL;DR

This paper studies a specific NP-hard quadratic assignment problem related to chemical graph theory, providing a pseudo-polynomial solution and solving an open problem about maximizing the Wiener index in trees with given degree sequences.

Contribution

It introduces the Wiener maximum quadratic assignment problem, proves its NP-hardness, and offers a polynomial-time solution for a related problem in chemical graph theory.

Findings

The problem is NP-hard in the general case.

A pseudo-polynomial time algorithm is developed for the special case.

The paper solves an open problem about maximizing the Wiener index in trees with prescribed degrees.

Abstract

We investigate a special case of the maximum quadratic assignment problem where one matrix is a product matrix and the other matrix is the distance matrix of a one-dimensional point set. We show that this special case, which we call the Wiener maximum quadratic assignment problem, is NP-hard in the ordinary sense and solvable in pseudo-polynomial time. Our approach also yields a polynomial time solution for the following problem from chemical graph theory: Find a tree that maximizes the Wiener index among all trees with a prescribed degree sequence. This settles an open problem from the literature.