Analysis of Quality Diversity Algorithms for the Knapsack Problem

TL;DR

This paper explores the application of quality diversity algorithms to the knapsack problem, demonstrating their ability to compute optimal solutions efficiently and establishing parameter settings for approximation schemes.

Contribution

It provides the first runtime analysis of QD algorithms on the knapsack problem and identifies parameter settings for a fully polynomial randomized approximation scheme.

Findings

QD algorithms compute optimal solutions in expected pseudo-polynomial time.

Parameter settings can lead to a fully polynomial randomized approximation scheme.

Experimental results validate the effectiveness of QD approaches on benchmark sets.

Abstract

Quality diversity (QD) algorithms have been shown to be very successful when dealing with problems in areas such as robotics, games and combinatorial optimization. They aim to maximize the quality of solutions for different regions of the so-called behavioural space of the underlying problem. In this paper, we apply the QD paradigm to simulate dynamic programming behaviours on knapsack problem, and provide a first runtime analysis of QD algorithms. We show that they are able to compute an optimal solution within expected pseudo-polynomial time, and reveal parameter settings that lead to a fully polynomial randomised approximation scheme (FPRAS). Our experimental investigations evaluate the different approaches on classical benchmark sets in terms of solutions constructed in the behavioural space as well as the runtime needed to obtain an optimal solution.