@inproceedings{48872, abstract = {{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 ith 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 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.}}, author = {{Bossek, Jakob and Sudholt, Dirk}}, booktitle = {{Proceedings of the Genetic and Evolutionary Computation Conference}}, isbn = {{9798400701191}}, keywords = {{quality diversity, runtime analysis}}, pages = {{1546–1554}}, publisher = {{Association for Computing Machinery}}, title = {{{Runtime Analysis of Quality Diversity Algorithms}}}, doi = {{10.1145/3583131.3590383}}, year = {{2023}}, } @article{48871, abstract = {{Most runtime analyses of randomised search heuristics focus on the expected number of function evaluations to find a unique global optimum. We ask a fundamental question: if additional search points are declared optimal, or declared as desirable target points, do these additional optima speed up evolutionary algorithms? More formally, we analyse the expected hitting time of a target set OPT{$\cup$}S where S is a set of non-optimal search points and OPT is the set of optima and compare it to the expected hitting time of OPT. We show that the answer to our question depends on the number and placement of search points in S. For all black-box algorithms and all fitness functions with polynomial expected optimisation times we show that, if additional optima are placed randomly, even an exponential number of optima has a negligible effect on the expected optimisation time. Considering Hamming balls around all global optima gives an easier target for some algorithms and functions and can shift the phase transition with respect to offspring population sizes in the (1,{$\lambda$}) EA on OneMax. However, for the one-dimensional Ising model the time to reach Hamming balls of radius (1/2-{$ϵ$})n around optima does not reduce the asymptotic expected optimisation time in the worst case. Finally, on functions where search trajectories typically join in a single search point, turning one search point into an optimum drastically reduces the expected optimisation time.}}, author = {{Bossek, Jakob and Sudholt, Dirk}}, issn = {{0304-3975}}, journal = {{Theoretical Computer Science}}, keywords = {{Evolutionary algorithms, pseudo-Boolean functions, runtime analysis}}, pages = {{113757}}, title = {{{Do Additional Target Points Speed Up Evolutionary Algorithms?}}}, doi = {{10.1016/j.tcs.2023.113757}}, year = {{2023}}, } @inproceedings{48860, abstract = {{In the area of evolutionary computation the calculation of diverse sets of high-quality solutions to a given optimization problem has gained momentum in recent years under the term evolutionary diversity optimization. Theoretical insights into the working principles of baseline evolutionary algorithms for diversity optimization are still rare. In this paper we study the well-known Minimum Spanning Tree problem (MST) in the context of diversity optimization where population diversity is measured by the sum of pairwise edge overlaps. Theoretical results provide insights into the fitness landscape of the MST diversity optimization problem pointing out that even for a population of {$\mu$} = 2 fitness plateaus (of constant length) can be reached, but nevertheless diverse sets can be calculated in polynomial time. We supplement our theoretical results with a series of experiments for the unconstrained and constraint case where all solutions need to fulfill a minimal quality threshold. Our results show that a simple ({$\mu$} + 1)-EA can effectively compute a diversified population of spanning trees of high quality.}}, author = {{Bossek, Jakob and Neumann, Frank}}, booktitle = {{Proceedings of the Genetic and Evolutionary Computation Conference}}, isbn = {{978-1-4503-8350-9}}, keywords = {{evolutionary algorithms, evolutionary diversity optimization, minimum spanning tree, runtime analysis}}, pages = {{198–206}}, publisher = {{Association for Computing Machinery}}, title = {{{Evolutionary Diversity Optimization and the Minimum Spanning Tree Problem}}}, doi = {{10.1145/3449639.3459363}}, year = {{2021}}, } @inbook{48862, abstract = {{Most runtime analyses of randomised search heuristics focus on the expected number of function evaluations to find a unique global optimum. We ask a fundamental question: if additional search points are declared optimal, or declared as desirable target points, do these additional optima speed up evolutionary algorithms? More formally, we analyse the expected hitting time of a target set OPT {$\cup$} S where S is a set of non-optimal search points and OPT is the set of optima and compare it to the expected hitting time of OPT. We show that the answer to our question depends on the number and placement of search points in S. For all black-box algorithms and all fitness functions we show that, if additional optima are placed randomly, even an exponential number of optima has a negligible effect on the expected optimisation time. Considering Hamming balls around all global optima gives an easier target for some algorithms and functions and can shift the phase transition with respect to offspring population sizes in the (1,{$\lambda$}) EA on One-Max. Finally, on functions where search trajectories typically join in a single search point, turning one search point into an optimum drastically reduces the expected optimisation time.}}, author = {{Bossek, Jakob and Sudholt, Dirk}}, booktitle = {{Proceedings of the 16th ACM/SIGEVO Conference on Foundations of Genetic Algorithms}}, isbn = {{978-1-4503-8352-3}}, keywords = {{evolutionary algorithms, pseudo-boolean functions, runtime analysis, theory}}, pages = {{1–11}}, publisher = {{Association for Computing Machinery}}, title = {{{Do Additional Optima Speed up Evolutionary Algorithms?}}}, year = {{2021}}, } @inproceedings{48895, abstract = {{Evolutionary algorithms (EAs) are general-purpose problem solvers that usually perform an unbiased search. This is reasonable and desirable in a black-box scenario. For combinatorial optimization problems, often more knowledge about the structure of optimal solutions is given, which can be leveraged by means of biased search operators. We consider the Minimum Spanning Tree (MST) problem in a single- and multi-objective version, and introduce a biased mutation, which puts more emphasis on the selection of edges of low rank in terms of low domination number. We present example graphs where the biased mutation can significantly speed up the expected runtime until (Pareto-)optimal solutions are found. On the other hand, we demonstrate that bias can lead to exponential runtime if "heavy" edges are necessarily part of an optimal solution. However, on general graphs in the single-objective setting, we show that a combined mutation operator which decides for unbiased or biased edge selection in each step with equal probability exhibits a polynomial upper bound - as unbiased mutation - in the worst case and benefits from bias if the circumstances are favorable.}}, author = {{Roostapour, Vahid and Bossek, Jakob and Neumann, Frank}}, booktitle = {{Proceedings of the 2020 Genetic and Evolutionary Computation Conference}}, isbn = {{978-1-4503-7128-5}}, keywords = {{biased mutation, evolutionary algorithms, minimum spanning tree problem, runtime analysis}}, pages = {{551–559}}, publisher = {{Association for Computing Machinery}}, title = {{{Runtime Analysis of Evolutionary Algorithms with Biased Mutation for the Multi-Objective Minimum Spanning Tree Problem}}}, doi = {{10.1145/3377930.3390168}}, year = {{2020}}, } @inproceedings{48870, abstract = {{The edge coloring problem asks for an assignment of colors to edges of a graph such that no two incident edges share the same color and the number of colors is minimized. It is known that all graphs with maximum degree {$\Delta$} can be colored with {$\Delta$} or {$\Delta$} + 1 colors, but it is NP-hard to determine whether {$\Delta$} colors are sufficient. We present the first runtime analysis of evolutionary algorithms (EAs) for the edge coloring problem. Simple EAs such as RLS and (1+1) EA efficiently find (2{$\Delta$} - 1)-colorings on arbitrary graphs and optimal colorings for even and odd cycles, paths, star graphs and arbitrary trees. A partial analysis for toroids also suggests efficient runtimes in bipartite graphs with many cycles. Experiments support these findings and investigate additional graph classes such as hypercubes, complete graphs and complete bipartite graphs. Theoretical and experimental results suggest that simple EAs find optimal colorings for all these graph classes in expected time O({$\Delta\mathscrl$}2m log m), where m is the number of edges and {$\mathscrl$} is the length of the longest simple path in the graph.}}, author = {{Bossek, Jakob and Sudholt, Dirk}}, booktitle = {{Proceedings of the 15th ACM/SIGEVO Conference on Foundations of Genetic Algorithms}}, isbn = {{978-1-4503-6254-2}}, keywords = {{edge coloring problem, runtime analysis}}, pages = {{102–115}}, publisher = {{Association for Computing Machinery}}, title = {{{Time Complexity Analysis of RLS and (1 + 1) EA for the Edge Coloring Problem}}}, doi = {{10.1145/3299904.3340311}}, year = {{2019}}, }