Understanding instance hardness for optimisation algorithms: Methodologies, open challenges and post-quantum implications

Abstract

This paper reviews efforts to characterise the hardness of optimisation problem instances, and to develop improved methodologies for empirical testing of the strengths and weaknesses of algorithms, based on comprehensive and unbiased sets of test instances whose hardness can be understood. Using the Travelling Salesperson Problem (TSP) as an illustrative example throughout the paper, efforts during the 20th century to solve optimisation problems with both exact and heuristic algorithms are briefly reviewed. From the early 21st century however, with a growing number of available algorithms (from nature-inspired to quantum), the focus has naturally shifted to exploring how the characteristics of particular problem instances impact algorithm performance, and whether empirical testing is rigorous, unbiased, conclusive and trustworthy based on the choice of test instances. Instance Space Analysis (ISA) was pioneered by the author as a methodology to visualize explainable strengths and weaknesses of algorithms across the broadest set of test instances, and to scrutinize the diversity and suitability of test instances chosen for empirical testing of algorithms. The ISA methodology is reviewed and illustrated using the TSP to show how insights into instance hardness for particular algorithms can be obtained. The paper then turns attention to quantum algorithms for optimisation, and argues that key lessons learned about testing optimisation algorithms over recent decades are highly relevant to current efforts to explore quantum advantage for optimisation. The implications of these lessons for quantum optimisation, including applications such as machine learning, are summarised. Given the difficulties faced by quantum algorithms for solving the TSP, the idea of designing extremely hard TSP instances as the basis for new post-quantum cryptography protocols is discussed. Finally, ten open research challenges are posed to provide fruitful directions for future research.