Revisiting Where are the Hard Knapsack Problems? via Instance Space Analysis

Abstract In 2005, David Pisinger asked the question “where are the hard knapsack problems?”. Noting that the classical benchmark test instances were limited in difficulty due to their selected structure, he proposed a set of new test instances for the 0-1 knapsack problem with characteristics that made them more challenging for dynamic programming and branch-and-bound algorithms. This important work highlighted the influence of diversity in test instances to draw reliable conclusions about algorithm performance. In this paper, we revisit the question in light of recent methodological advances – in the form of Instance Space Analysis – enabling the strengths and weaknesses of algorithms to be visualised and assessed across the broadest possible space of test instances. We show where the hard instances lie, and objectively assess algorithm performance across the instance space to articulate the strengths and weaknesses of algorithms. Furthermore, we propose a method to fill the instance space with diverse and challenging new test instances with controllable properties to support greater insights into algorithm selection, and drive future algorithmic innovations.