Theoretical Analysis of Quality Diversity Algorithms for a Classical Path Planning Problem
Duc-Cuong Dang, Aneta Neumann, Frank Neumann, Andre Opris, Dirk Sudholt
TL;DR
The paper studies quality diversity (QD) algorithms, especially Map-Elites, on the all-pairs shortest-path (APSP) problem, defining the behavioural space as all source–destination pairs. It provides the first runtime analyses showing that a QD-GA can obtain a shortest path for every pair in parallel, with bounds depending on graph parameters such as maximum degree $\Delta$ and diameter $\ell$. Mutation-only and crossover-enabled variants yield different speeds: mutation-only achieves $O(n^2\Delta \max\{\ell,\log n\})$ fitness evaluations, while crossover-enabled QD-GA attains $O(n^{3.5}\sqrt{\log n})$, improved to $O(n^{3.25}\sqrt[4]{\log n})$, and a compatible-parent version reaches $O(n^3\log n)$. A fast variant, Fast QD-APSP, achieves $O(\min\{\Delta n^2 \max\{\ell,\log n\},\; n^3 \log n\})$, with discussion of tightness and instances where the bounds are realized, highlighting practical implications for planning problems.
Abstract
Quality diversity (QD) algorithms have shown to provide sets of high quality solutions for challenging problems in robotics, games, and combinatorial optimisation. So far, theoretical foundational explaining their good behaviour in practice lack far behind their practical success. We contribute to the theoretical understanding of these algorithms and study the behaviour of QD algorithms for a classical planning problem seeking several solutions. We study the all-pairs-shortest-paths (APSP) problem which gives a natural formulation of the behavioural space based on all pairs of nodes of the given input graph that can be used by Map-Elites QD algorithms. Our results show that Map-Elites QD algorithms are able to compute a shortest path for each pair of nodes efficiently in parallel. Furthermore, we examine parent selection techniques for crossover that exhibit significant speed ups compared to the standard QD approach.