Runtime Analysis of Quality Diversity Algorithms

Quality diversity~(QD) is a branch of evolutionary computation that gained increasing interest in recent years. The Map-Elites QD approach defines a feature space, i.e., a partition of the search space, and stores the best solution for each cell of this space. We study a simple QD algorithm in the context of pseudo-Boolean optimisation on the ``number of ones'' feature space, where the $i$th cell stores the best solution amongst those with a number of ones in $[(i-1)k, ik-1]$. Here $k$ is a granularity parameter $1 \leq k \leq n+1$. We give a tight bound on the expected time until all cells are covered for arbitrary fitness functions and for all $k$ and analyse the expected optimisation time of QD on \textsc{OneMax} and other problems whose structure aligns favourably with the feature space. On combinatorial problems we show that QD finds a ${(1-1/e)}$-approximation when maximising any monotone sub-modular function with a single uniform cardinality constraint efficiently. Defining the feature space as the number of connected components of a connected graph, we show that QD finds a minimum spanning tree in expected polynomial time

On Single-Objective Sub-Graph-Based Mutation for Solving the Bi-Objective Minimum Spanning Tree Problem

We contribute to the efficient approximation of the Pareto-set for the classical $\mathcal{NP}$-hard multi-objective minimum spanning tree problem (moMST) adopting evolutionary computation. More precisely, by building upon preliminary work, we analyse the neighborhood structure of Pareto-optimal spanning trees and design several highly biased sub-graph-based mutation operators founded on the gained insights. In a nutshell, these operators replace (un)connected sub-trees of candidate solutions with locally optimal sub-trees. The latter (biased) step is realized by applying Kruskal's single-objective MST algorithm to a weighted sum scalarization of a sub-graph. We prove runtime complexity results for the introduced operators and investigate the desirable Pareto-beneficial property. This property states that mutants cannot be dominated by their parent. Moreover, we perform an extensive experimental benchmark study to showcase the operator's practical suitability. Our results confirm that the sub-graph based operators beat baseline algorithms from the literature even with severely restricted computational budget in terms of function evaluations on four different classes of complete graphs with different shapes of the Pareto-front

On the Impact of Operators and Populations within Evolutionary Algorithms for the Dynamic Weighted Traveling Salesperson Problem

Evolutionary algorithms have been shown to obtain good solutions for complex optimization problems in static and dynamic environments. It is important to understand the behaviour of evolutionary algorithms for complex optimization problems that also involve dynamic and/or stochastic components in a systematic way in order to further increase their applicability to real-world problems. We investigate the node weighted traveling salesperson problem (W-TSP), which provides an abstraction of a wide range of weighted TSP problems, in dynamic settings. In the dynamic setting of the problem, items that have to be collected as part of a TSP tour change over time. We first present a dynamic setup for the dynamic W-TSP parameterized by different types of changes that are applied to the set of items to be collected when traversing the tour. Our first experimental investigations study the impact of such changes on resulting optimized tours in order to provide structural insights of optimization solutions. Afterwards, we investigate simple mutation-based evolutionary algorithms and study the impact of the mutation operators and the use of populations with dealing with the dynamic changes to the node weights of the problem

Evolving Diverse Sets of Tours for the Travelling Salesperson Problem

Evolving diverse sets of high quality solutions has gained increasing interest in the evolutionary computation literature in recent years. With this paper, we contribute to this area of research by examining evolutionary diversity optimisation approaches for the classical Traveling Salesperson Problem (TSP). We study the impact of using different diversity measures for a given set of tours and the ability of evolutionary algorithms to obtain a diverse set of high quality solutions when adopting these measures. Our studies show that a large variety of diverse high quality tours can be achieved by using our approaches. Furthermore, we compare our approaches in terms of theoretical properties and the final set of tours obtained by the evolutionary diversity optimisation algorithm.Comment: 11 pages, 3 tables, 3 figures, to be published in GECCO '2