Analysis of Evolutionary Diversity Optimisation for Permutation Problems

TL;DR

This paper investigates evolutionary diversity optimization for permutation problems like TSP and QAP, analyzing theoretical performance and conducting experiments to understand mutation effects and practical outcomes.

Contribution

It provides a theoretical analysis of mutation operators for diversity optimization in permutation problems and compares these with experimental results on QAP instances.

Findings

Many mutation operators guarantee convergence to diverse populations within polynomial time.

Strong mutations can lead to poor worst-case performance due to exponential run-time growth.

Practical experiments show more optimistic results than worst-case theoretical bounds.

Abstract

Generating diverse populations of high quality solutions has gained interest as a promising extension to the traditional optimization tasks. This work contributes to this line of research with an investigation on evolutionary diversity optimization for three of the most well-studied permutation problems, namely the Traveling Salesperson Problem (TSP), both symmetric and asymmetric variants, and Quadratic Assignment Problem (QAP). It includes an analysis of the worst-case performance of a simple mutation-only evolutionary algorithm with different mutation operators, using an established diversity measure. Theoretical results show many mutation operators for these problems guarantee convergence to maximally diverse populations of sufficiently small size within cubic to quartic expected run-time. On the other hand, the result on QAP suggests that strong mutations give poor worst-case performance, as mutation strength contributes exponentially to the expected run-time. Additionally, experiments are carried out on QAPLIB and synthetic instances in unconstrained and constrained settings, and reveal much more optimistic practical performances, while corroborating the theoretical finding regarding mutation strength. These results should serve as a baseline for future studies.