The multiple knapsack problem (MKP) generalizes the classical knapsack problem by assigning items to multiple knapsacks subject to capacity constraints. It is used to model many real-world resource allocation and scheduling problems. In practice, these optimization problems often involve stochastic and dynamic components. Evolutionary algorithms provide a flexible framework for addressing such problems under uncertainty and dynamic changes. In this paper, we investigate a stochastic and dynamic variant of MKP with chance constraints, where the item weights are modeled as independent normally distributed random variables and knapsack capacities change during the optimization process. We formulate the problem as a bi-objective optimization formulation that balances profit maximization and probabilistic capacity satisfaction at a given confidence level. We conduct an empirical comparison of four widely used multi-objective evolutionary algorithms (MOEAs), representing both decomposition- and dominance-based search paradigms. The algorithms are evaluated under varying uncertainty levels, confidence thresholds, and dynamic change settings. The results provide comparative insights into the behavior of decomposition-based and dominance-based MOEAs for stochastic MKP under dynamic constraints.

How people spend their finite time budget of 24 hours on daily activities is linked to their wellbeing. Yet, how to best allocate time to optimise multi-dimensional wellbeing (physical, mental and cognitive) remains unknown. Here, we utilise a number of (objective) functions derived using compositional data analysis and a large child cohort ( \(n>1{,}000\) ), to predict how time allocation is associated with wellbeing outcomes such as body mass index, life satisfaction and cognition. We develop and advocate joint cumulative distribution function constraints to ensure the feasible solutions do not extrapolate the sampled data for which the objective function is derived from. Moreover, we incorporate quality diversity (QD) approaches to study these objective functions. We define two types of behavioural spaces (BSs), one based on the activities, called the variable-based behavioural space (VBS), and the other based on the objectives, called the objective-based behavioural space (OBS). The VBS allows us to generate a set of high-quality solutions with different activity durations, while the OBS allows us to tradeoff different wellbeing dimensions against each other. We also demonstrate a web application, Time allocation optimiser, for creating personalised, optimised time-use plans.

The chance constrained travelling thief problem (chance constrained TTP) has been introduced as a stochastic variation of the classical travelling thief problem (TTP) in an attempt to embody the effect of uncertainty in the problem definition. In this work, we characterise the chance constrained TTP using a limited number of weighted scenarios. Each scenario represents a similar TTP instance, differing slightly in the weight profile of the items and associated with a certain probability of occurrence. Collectively, the weighted scenarios represent a relaxed form of a stochastic TTP instance where the objective is to maximise the expected benefit while satisfying the knapsack constraint with a larger probability. We incorporate a set of evolutionary algorithms and heuristic procedures developed for the classical TTP, and formulate adaptations that apply to the weighted scenario-based representation of the problem. The analysis focuses on the performance of the algorithms on different settings and examines the impact of uncertainty on the quality of the solutions.

Bayesian optimisation (BO) is a surrogate-based optimisation technique that efficiently solves expensive black-box functions with small evaluation budgets. Recent studies consider trust regions to improve the scalability of BO approaches when the problem space scales to more dimensions. Motivated by this research, we explore the effectiveness of trust region-based BO algorithms for diversity optimisation in different dimensional black box problems. We propose diversity optimisation approaches extending TuRBO1, which is the first BO method that uses a trust region-based approach for scalability. We extend TuRBO1 as divTuRBO1, which finds an optimal solution while maintaining a given distance threshold relative to a reference solution set. We propose two approaches to find diverse solutions for black-box functions by combining divTuRBO1 runs in a sequential and an interleaving fashion. We conduct experimental investigations on the proposed algorithms and compare their performance with that of the baseline method, ROBOT (rank-ordered Bayesian optimisation with trust regions). We evaluate proposed algorithms on benchmark functions with dimensions 2 to 20. Experimental investigations demonstrate that the proposed methods perform well, particularly in larger dimensions, even with a limited evaluation budget.

Evolutionary diversity optimization using multi-objective indicators, aims to evolve diverse solutions considering multiple features simultaneously. In this paper, we evolve diverse discriminating instances for chance-constraint submodular problems using multi-objective quality indicators. For any pair of algorithms, discriminating instances are easy to solve by one algorithm and hard to solve by the other. These instances help with investigating the strengths and weaknesses of different algorithms in solving a given problem. Hence, the availability of diverse sets of discriminating instances for important problems is essential. We introduce a (μ + 1) evolutionary algorithm to evolve diverse differentiating instance sets for the chance-constrained maximum coverage problem. This problem contains stochastic costs on the vertices, each represented by its expected value and variance. In the selection process of our algorithm, we use inverted generational distance and hypervolume to optimize the diversity of the set. The experimental results demonstrate these indicators significantly improve the diversity of the set of instances in a multi-dimensional feature space while ensuring all of them are clearly differentiating.

Evolutionary multitasking is an emerging paradigm in optimisation that draws inspiration from cognitive multitasking in humans. It seeks to solve multiple optimisation tasks in parallel referring to a shared population of individuals. This approach uses underlying commonalities between tasks to accelerate the convergence, by leveraging the principles of knowledge transfer. The travelling thief problem (TTP) combines the characteristics of both the travelling salesman problem (TSP) and the 0–1 knapsack problem (KP), depicting the interdependency of multiple components that can be seen in real-world applications. In this study, we represent the TTP problem as a combination of multiple scenarios where each scenario combines a similar TSP component, yet a different KP component. We set up scenarios for the experiments using both fixed weights and weights generated uniformly at random. We follow an evolutionary multitasking optimisation approach to solve multiple scenarios in parallel. By experimenting with a variety of TTP instances, we compare the performance of basic and advanced multitasking approaches, against classical methods. The analysis shows that multitasking brings a competitive advantage over the classical methods when operating on small time budgets.

Evolutionary multitasking has gained significant attention in the evolutionary computation literature in recent years. Here an evolutionary algorithm is used to compute good or optimal solutions for not just a single but several (possibly related) tasks. We provide a first runtime analysis of evolutionary multitask algorithms and investigate generalized versions of OneMax, LeadingOnes, and Jump which are classical benchmark functions frequently studied in the area of runtime analysis. Our theoretical investigations point out significant speed ups when using evolutionary multitasking instead of several runs of the classical (1+1) EA. In particular, our analysis reveals how progress is shared between the different tasks using uniform crossover in an evolutionary multitasking algorithm. We complement our asymptotic theoretical analysis by experimental investigations which provide further insights into the actual speed ups dependent on the similarity of the given tasks for realistic problem sizes.

Variational quantum algorithms, such as the Recursive Quantum Approximate Optimization Algorithm (RQAOA), have become increasingly popular, offering promising avenues for employing Noisy Intermediate-Scale Quantum devices to address challenging combinatorial optimization tasks like the maximum cut problem. In this study, we utilize an evolutionary algorithm equipped with a unique fitness function. This approach targets hard maximum cut instances within the latent space of a Graph Autoencoder, identifying those that pose significant challenges or are particularly tractable for RQAOA, in contrast to the classic Goemans and Williamson algorithm. Our findings not only delineate the distinct capabilities and limitations of each algorithm but also expand our understanding of RQAOA's operational limits. Furthermore, the diverse set of graphs we have generated serves as a crucial benchmarking asset, emphasizing the need for more advanced algorithms to tackle combinatorial optimization challenges. Additionally, our results pave the way for new avenues in graph generation research, offering exciting opportunities for future explorations.

Multitasking evolutionary algorithms can be effectively used to solve a number of problems with a single population. A key issue in deciding their effectiveness, is how to transfer good solutions from one problem instance to another problem instance which shares some characteristics. We investigate in this paper how to transfer solutions between different problem instances of the Travelling Salesperson Problem (TSP) based matching algorithms and introduce different transfer mechanisms based on matching the nodes between problem instances. In our experimental study, we examine how the different transfer approaches perform for different classes of TSP instances dependent on the characteristics of the considered problem instances.

Real-world optimization often demands diverse, high-quality solutions. Quality-Diversity (QD) optimization is a multifaceted approach in evolutionary algorithms that aims to generate a set of solutions that are both high-performing and diverse. QD algorithms have been successfully applied across various domains, providing robust solutions by exploring diverse behavioral niches. However, their application has primarily focused on static problems, with limited exploration in the context of dynamic combinatorial optimization problems. Furthermore, the theoretical understanding of QD algorithms remains underdeveloped, particularly when applied to learning heuristics instead of directly learning solutions in complex and dynamic combinatorial optimization domains, which introduces additional challenges. This paper introduces a novel QD framework for dynamic scheduling problems. We propose a map-building strategy that visualizes the solution space by linking heuristic genotypes to their behaviors, enabling their representation on a QD map. This map facilitates the discovery and maintenance of diverse scheduling heuristics. Additionally, we conduct experiments on both fixed and dynamically changing training instances to demonstrate how the map evolves and how the distribution of solutions unfolds over time. We also discuss potential future research directions that could enhance the learning process and broaden the applicability of QD algorithms to dynamic combinatorial optimization challenges.

Quality diversity (QD) algorithms, an extension of evolutionary algorithms, excel at generating diverse sets of high-quality solutions for complex problems in robotics, games, and combinatorial optimisation. Despite their success, the underlying mechanisms remain poorly understood due to a lack of a theoretical foundation. We address this gap by analysing QD algorithms on the all-pairs-shortest-paths (APSP) problem, a classical planning task that naturally seeks multiple solutions. Using Map-Elites, a prominent QD approach, we leverage its ability to evolve solutions across distinct regions of a behavioural space, which for APSP corresponds to all pairs of nodes in the graph. Our analysis rigorously demonstrates that evolutionary algorithms using Map-Elites efficiently compute shortest paths for all node pairs in parallel by exploiting synergies in the behavioural space. By appending edges to an existing shortest path, mutation can create optimal solutions in other regions of the behavioural space. Crossover is particularly effective, as it can combine optimal paths from two regions to produce an optimal path for a third region simply by concatenating two shortest paths. Finally, refining the parent selection to facilitate successful crossovers exhibits significant speed-ups compared to standard QD approaches.

Abstract Parameterized analysis provides powerful mechanisms for obtaining fine-grained insights into different types of algorithms. In this work, we combine this field with evolutionary algorithms and provide parameterized complexity analysis of evolutionary multi-objective algorithms for the W-separator problem, which is a natural generalization of the vertex cover problem. The goal is to remove the minimum number of vertices such that each connected component in the resulting graph has at most W vertices. We provide different multi-objective formulations involving two or three objectives that provably lead to fixed-parameter evolutionary algorithms with respect to the value of an optimal solution OPT and W. Of particular interest are kernelizations and the reducible structures used for them. We show that in expectation the algorithms make incremental progress in finding such structures and beyond. The current best known kernelization of the W-separator uses linear programming methods and requires non-trivial post-processing steps to extract the reducible structures. We provide additional structural features to show that evolutionary algorithms with appropriate objectives are also capable of extracting them. Our results show that evolutionary algorithms with different objectives guide the search and admit fixed parameterized runtimes to solve or approximate (even arbitrarily close) the W-separator problem.

Abstract Understanding how evolutionary algorithms perform on constrained problems has gained increasing attention in recent years. In this paper, we study how evolutionary algorithms optimize constrained versions of the classical LeadingOnes problem. We first provide a run time analysis for the classical (1+1) EA on the LeadingOnes problem with a deterministic cardinality constraint, giving $$\Theta (n (n-B)\log (B) + nB)$$ Θ ( n ( n - B ) log ( B ) + n B ) as the tight bound. Our results show that the behaviour of the algorithm is highly dependent on the constraint bound of the uniform constraint. Afterwards, we consider the problem in the context of stochastic constraints and provide insights using theoretical and experimental studies on how the ( $$\mu $$ μ +1) EA is able to deal with these constraints in a sampling-based setting.

Active Directory (AD) is the default security management system for Windows domain networks. An AD environment can be described as a cyber-attack graph, with nodes representing computers, accounts, and so forth, and edges indicating existing accesses or known exploits that enable attackers to move from one node to another. This article explores a Stackelberg game model between one attacker and one defender on an AD attack graph. The attacker’s goal is to maximize their chances of successfully reaching the destination before getting detected. The defender’s aim is to block a constant number of edges to minimize the attacker’s chance of success. The article shows that the problem is #P-hard and, therefore, intractable to solve exactly. To defend the AD graph from cyberattackers, this article proposes two defensive approaches. In the first approach, we convert the attacker’s problem to an exponential-sized Dynamic Program that is approximated by a neural network (NN). Once trained, the NN serves as an efficient fitness function for defender’s Evolutionary Diversity Optimization-based defensive policy. The diversity emphasis on the defender’s solution provides a diverse set of training samples, improving the training accuracy of our NN for modeling the attacker. In the second approach, we propose a RL-based policy to solve the attacker’s problem and Critic network-assisted Evolutionary Diversity Optimization-based defensive policy to solve defender’s problem. Experimental results on synthetic AD graphs show that the proposed defensive policies are scalable, highly effective, approximate attacker’s problem accurately and generate good defensive plans.

We propose a combinatorial optimisation model called Limited Query Graph Connectivity Test. We consider a graph whose edges have two possible states (On/Off). The edges' states are hidden initially. We could query an edge to reveal its state. Given a source s and a destination t, we aim to test s−t connectivity by identifying either a path (consisting of only On edges) or a cut (consisting of only Off edges). We are limited to B queries, after which we stop regardless of whether graph connectivity is established. We aim to design a query policy that minimizes the expected number of queries. Our model is mainly motivated by a cyber security use case where we need to establish whether attack paths exist in a given network, between a source (i.e., a compromised user node) and a destination (i.e., a high-privilege admin node). Edge query is resolved by manual effort from the IT admin, which is the motivation behind query minimization. Our model is highly related to Stochastic Boolean Function Evaluation (SBFE). There are two existing exact algorithms for SBFE that are prohibitively expensive. We propose a signifcantly more scalable exact algorithm. While previous exact algorithms only scale for trivial graphs (i.e., past works experimented on at most 20 edges), we empirically demonstrate that our algorithm is scalable for a wide range of much larger practical graphs (i.e., graphs representing Windows domain networks with tens of thousands of edges). We also propose three heuristics. Our best-performing heuristic is via limiting the planning horizon of the exact algorithm. The other two are via reinforcement learning (RL) and Monte Carlo tree search (MCTS). We also derive an algorithm for computing the performance lower bound. Experimentally, we show that all our heuristics are near optimal. The heuristic building on the exact algorithm outperforms all other heuristics, surpassing RL, MCTS and eight existing heuristics ported from SBFE and related literature.

Production scheduling and long-term planning are challenging in large-scale open pit mine operations. Proper planning ensures the maximum cash flow while utilizing mining resources. Most existing methodologies for solving open-pit mine scheduling problems are based on conventional approaches. However, these methods face many challenges in terms of computational cost due to the high dimensionality, and physical and operational constraints. Multi-objective evolutionary algorithms (MOEAs) have been successfully applied to a wide range of combinatorial optimization problems, as they often provide high-quality solutions to complex problems without significant design effort and computational cost. In this study, we investigate the effectiveness of the Non-dominated Sorting Genetic Algorithm II (NSGA-II) for the open-pit mine scheduling problem. We compare the effectiveness of this algorithm with the Global Simple Evolutionary Multi-Objective Optimizer (GSEMO) by analyzing well-known real-world mine deposits consisting of up to 112 687 blocks. We show that the NSGA-II algorithm has a clear advantage over GSEMO in obtaining better results. Further, we introduce the local search technique to enhance the performance of the NSGA-II.

Chance-constrained problems involve stochastic components in the constraints which can be violated with a small probability. We investigate the impact of different types of chance constraints on the performance of iterative search algorithms and study the classical maximum coverage problem in graphs with chance constraints. Our goal is to evolve reliable chance constraint settings for a given graph where the performance of algorithms differs significantly not just in expectation but with high confidence. This allows to better learn and understand how different types of algorithms can deal with different types of constraint settings and supports automatic algorithm selection. We develop an evolutionary algorithm that provides sets of chance constraints that differentiate the performance of two stochastic search algorithms with high confidence. We initially use traditional approximation ratio as the fitness function of (1+1) EA to evolve instances, which shows inadequacy to generate reliable instances. To address this issue, we introduce a new measure to calculate the performance difference for two algorithms, which considers variances of performance ratios. Our experiments show that our approach is highly successful in solving the instability issue of the performance ratios and leads to evolving reliable sets of chance constraints with significantly different performance for various types of algorithms.

Real-world optimization problems often require finding not only one good solution, but a diverse set of good solutions. Evolutionary algorithms (EAs) have been shown to suit well for such a task. However our theoretical understanding of their behavior remains unsatisfying, especially in the multi-objective domain.

We consider whether conditions exist under which block-coordinate descent is asymptotically efficient in evolutionary multi-objective optimization, addressing an open problem. Block-coordinate descent, where an optimization problem is decomposed into k blocks of decision variables and each of the blocks is optimized (with the others fixed) in a sequence, is a technique used in some large-scale optimization problems such as airline scheduling, however its use in multi-objective optimization is less studied. We propose a block-coordinate version of GSEMO and compare its running time to the standard GSEMO algorithm. Theoretical and empirical results on a bi-objective test function, a variant of LOTZ, serve to demonstrate the existence of cases where block-coordinate descent is faster. The result may yield wider insights into this class of algorithms.

The travelling thief problem (TTP) is a multi-component combinatorial optimization problem that has gained significant attention in the evolutionary computation and heuristic search literature. In this paper, we introduce the chance constrained TTP which involves stochastic weights. Our problem formulation captures the stochastic aspect of the knapsack in the form of a chance constraint. Such a constraint can only be violated with a small probability. We introduce surrogate and sampling-based approaches for the chance constrained TTP to optimize the expected objective score under the condition that the solution is feasible with a high probability. We use these approaches to evaluate the feasibility of solutions and incorporate our approaches into high-performing algorithms for deterministic TTP. In our experimental investigations, we compare the performance of these algorithms and show the impact of uncertainty in connection with the underlying stochastic model.

This study proposes Evolutionary Diversity Optimisation (EDO) to Lower the Probability of Detection (LPD) in directed wireless networks. LPD communication aims to communicate between authorised parties, however minimises the probability that an intruder can detect the communication. We represent the problem as a directed graph and our objective is to minimise the area of detectability in a network whilst avoiding adversary nodes to be in the area. We utilise EDO to produce a population of solutions that are of quality i.e., that minimise the area of detectability whilst providing solutions that are diverse. To produce a solution, we find the strongly connected components by running depth-first-search (DFS) twice and extracting the edges traversed from the DFS runs. We use 3 permutation operators (insert, swap, random) to produce different solutions. We propose 2 methods for survival selection - one based on the diversity value and the other on the edge population count. We control the sparsity of the directed graphs by implementing a maximal communication range. Our results show that sparser graphs had smaller areas of detectability, however the final population was less diverse. We also found controlling the maximal communication range an effective strategy to reduce the area of detectability.

How people spend their finite time budget of 24 hours on daily activities is linked to their wellbeing. Yet, how to best allocate time to optimise multi-dimensional wellbeing (physical, mental and cognitive) remains unknown. Here, we utilise a number of (objective) functions derived using compositional data analysis and a large child cohort (n > 1000), to predict how time allocation is associated with wellbeing outcomes such as body mass index, life satisfaction, and cognition. We develop and advocate joint cumulative distribution function constraints to ensure the feasible solutions do not extrapolate the sampled data for which the objective function is derived from. Moreover, we incorporate quality diversity (QD) approaches to study these objective functions. We define two types of behavioural spaces (BSs), one based on the activities, called the variable-based BS, and the other based on objectives. The variable-based BS aids in studying solution space and generating a set of high-quality solutions with different variable values, while the objective-based BS is beneficial in diversifying the objective values for a number of objective functions while optimising another. We also demonstrate a web application, Time allocation optimiser, providing personalised, optimised time-use plans.

Optimizing real-world problems often involves decision-making under uncertainty due to the presence of unknown or uncontrollable variables. Chance-constraints allow to model the optimization problem with stochastic components by ensuring the probabilistic constraint is satisfied with high probability. Multi-objective evolutionary algorithms (MOEAs) are successfully applied to chance constrained optimization problems to achieve high-quality results. Most of these algorithms are based on Pareto dominance for measuring the quality of solutions during their search. A very few algorithms are based on the decomposition approach which tries to optimize the aggregations of the objectives. Among them, multi-objective evolutionary algorithm based on decomposition (MOEA/D) is one of the efficient MOEAs which decomposes the multi-objective optimization problems (MOPs) into a number of scalar optimization problems and then optimizes these sub-problems simultaneously. In this paper, we investigate the effectiveness of the MOEA/D algorithm when solving 2- and 3-objective formulations of the chance constrained knapsack problem, where the weights of each item are stochastic. We compare its performance with global simple evolutionary multi-objective optimizer (GSEMO) across various benchmark scenarios. Overall, we demonstrate that the MOEA/D achieved high-quality solutions with lower computational complexity.

Real-world optimization problems often involve stochastic and dynamic components. Evolutionary algorithms are particularly effective in these scenarios, as they can easily adapt to uncertain and changing environments but often uncertainty and dynamic changes are studied in isolation. In this paper, we explore the use of 3-objective evolutionary algorithms for the chance constrained knapsack problem with dynamic constraints. In our setting, the weights of the items are stochastic and the knapsack's capacity changes over time. We introduce a 3-objective formulation that is able to deal with the stochastic and dynamic components at the same time and is independent of the confidence level required for the constraint. This new approach is then compared to the 2-objective formulation which is limited to a single confidence level. We evaluate the approach using two different multi-objective evolutionary algorithms (MOEAs), namely the global simple evolutionary multi-objective optimizer (GSEMO) and the multi-objective evolutionary algorithm based on decomposition (MOEA/D), across various benchmark scenarios. Our analysis highlights the advantages of the 3-objective formulation over the 2-objective formulation in addressing the dynamic chance constrained knapsack problem.

Evolutionary algorithms are particularly effective for optimisation problems with dynamic and stochastic components. We propose multi-objective evolutionary approaches for the knapsack problem with stochastic profits under static and dynamic weight constraints. The chance-constrained problem model allows us to effectively capture the stochastic profits and associate a confidence level to the solutions' profits. We consider a bi-objective formulation that maximises expected profit and minimises variance, which allows optimising the problem independent of a specific confidence level on the profit. We derive a three-objective formulation by relaxing the weight constraint into an additional objective. We consider the GSEMO algorithm with standard and a sliding window-based parent selection to evaluate the objective formulations. Moreover, we modify fitness formulations and algorithms for the dynamic problem variant to store some infeasible solutions to cater to future changes. We conduct experimental investigations on both problems using the proposed problem formulations and algorithms. Our results show that three-objective approaches outperform approaches that use bi-objective formulations, and they further improve when GSEMO uses sliding window selection.

Recently surrogate functions based on the tail inequalities were developed to evaluate the chance constraints in the context of evolutionary computation and several Pareto optimization algorithms using these surrogates were successfully applied in optimizing chance-constrained monotone submodular problems. However, the difference in performance between algorithms using the surrogates and those employing the direct sampling-based evaluation remains unclear. Within the paper, a sampling-based method is proposed to directly evaluate the chance constraint. Furthermore, to address the problems with more challenging settings, an enhanced GSEMO algorithm integrated with an adaptive sliding window, called ASW-GSEMO, is introduced. In the experiments, the ASW-GSEMO employing the sampling-based approach is tested on the chance-constrained version of the maximum coverage problem with different settings. Its results are compared with those from other algorithms using different surrogate functions. The experimental findings indicate that the ASW-GSEMO with the sampling-based evaluation approach outperforms other algorithms, highlighting that the performances of algorithms using different evaluation methods are comparable. Additionally, the behaviors of ASW-GSEMO are visualized to explain the distinctions between it and the algorithms utilizing the surrogate functions.

Parameter control has succeeded in accelerating the convergence process of evolutionary algorithms. While empirical and theoretical studies have shed light on the behavior of algorithms for single-objective optimization, little is known about how self-adaptation influences multi-objective evolutionary algorithms. In this work, we contribute (1) extensive experimental analysis of the Global Simple Evolutionary Multi-objective Algorithm (GSEMO) variants on classic problems, such as OneMinMax, LOTZ, COCZ, and (2) a novel version of GSEMO with self-adjusting mutation rates.

In real-world optimisation, it is common to face several sub-problems interacting and forming the main problem. There is an inter-dependency between the sub-problems, making it impossible to solve such a problem by focusing on only one component. The travelling thief problem (TTP) belongs to this category and is formed by the integration of the travelling salesperson problem (TSP) and the knapsack problem (KP). In this paper, we investigate the inter-dependency of the TSP and the KP by means of quality diversity (QD) approaches. QD algorithms provide a powerful tool not only to obtain high-quality solutions but also to illustrate the distribution of high-performing solutions in the behavioural space. We introduce a multi-dimensional archive of phenotypic elites (MAP-Elites) based evolutionary algorithm using well-known TSP and KP search operators, taking the TSP and KP score as the behavioural descriptor. MAP-Elites algorithms are QD-based techniques to explore high-performing solutions in a behavioural space. Afterwards, we conduct comprehensive experimental studies that show the usefulness of using the QD approach applied to the TTP. First, we provide insights regarding high-quality TTP solutions in the TSP/KP behavioural space. Afterwards, we show that better solutions for the TTP can be obtained by using our QD approach, and it can improve the best-known solution for a number of TTP instances used for benchmarking in the literature.

Active Directory (AD) is the default security management system for Windows domain networks. An AD environment naturally describes an attack graph where nodes represent computers/accounts/security groups, and edges represent existing accesses/known exploits that allow the attacker to gain access from one node to another. Motivated by practical AD use cases, we study a Stackelberg game between one attacker and one defender. There are multiple entry nodes for the attacker to choose from and there is a single target (Domain Admin). Every edge has a failure rate. The attacker chooses the attack path with the maximum success rate. The defender can block a limited number of edges (i.e., revoke accesses) from a set of blockable edges, limited by budget. The defender's aim is to minimize the attacker's success rate. We exploit the tree-likeness of practical AD graphs to design scalable algorithms. We propose two novel methods that combine theoretical fixed parameter analysis and practical optimisation techniques. For graphs with small tree widths, we propose a tree decomposition based dynamic program. We then propose a general method for converting tree decomposition based dynamic programs to reinforcement learning environments, which leads to an anytime algorithm that scales better, but loses the optimality guarantee. For graphs with small numbers of non-splitting paths (a parameter we invent specifically for AD graphs), we propose a kernelization technique that significantly downsizes the model, which is then solved via mixed-integer programming. Experimentally, our algorithms scale to handle synthetic AD graphs with tens of thousands of nodes.

Submodular functions play a key role in the area of optimization as they allow to model many real-world problems that face diminishing returns. Evolutionary algorithms have been shown to obtain strong theoretical performance guarantees for a wide class of submodular problems under various types of constraints while clearly outperforming standard greedy approximation algorithms. This paper introduces a setup for benchmarking algorithms for submodular optimization problems with the aim to provide researchers with a framework to enhance and compare the performance of new algorithms for submodular problems. The focus is on the development of iterative search algorithms such as evolutionary algorithms with the implementation provided and integrated into IOHprofiler which allows for tracking and comparing the progress and performance of iterative search algorithms. We present a range of submodular optimization problems that have been integrated into IOHprofiler and show how the setup can be used for analyzing and comparing iterative search algorithms in various settings.

Mine planning is a complex task that involves many uncertainties. During early stage feasibility, available mineral resources can only be estimated based on limited sampling of ore grades from sparse drilling, leading to large uncertainty in under-sampled parts of the deposit. Planning the extraction schedule of ore over the life of a mine is crucial for its economic viability. We introduce a new approach for determining an “optimal schedule under uncertainty” that provides probabilistic bounds on the profits obtained in each period. This treatment of uncertainty within an economic framework reduces previously difficult-to-use models of variability into actionable insights. The new method discounts profits based on uncertainty within an evolutionary algorithm, sacrificing economic optimality of a single geological model for improving the downside risk over an ensemble of equally likely models. We provide experimental studies using Maptek's mine planning software Evolution. Our results show that our new approach is successful for effectively making use of uncertainty information in the mine planning process.

The evolutionary diversity optimization aims at finding a diverse set of solutions which satisfy some constraint on their fitness. In the context of multi-objective optimization this constraint can require solutions to be Pareto-optimal. In this paper we study how the GSEMO algorithm with additional diversity-enhancing heuristic optimizes a diversity of its population on a bi-objective benchmark problem OneMinMax, for which all solutions are Pareto-optimal. We provide a rigorous runtime analysis of the last step of the optimization, when the algorithm starts with a population with a second-best diversity, and prove that it finds a population with optimal diversity in expected time O(n2), when the problem size n is odd. For reaching our goal, we analyse the random walk of the population, which reflects the frequency of changes in the population and their outcomes.

Understanding how evolutionary algorithms perform on constrained problems has gained increasing attention in recent years. In this paper, we study how evolutionary algorithms optimize constrained versions of the classical LeadingOnes problem. We first provide a run time analysis for the classical (1+1) EA on the LeadingOnes problem with a deterministic cardinality constraint, giving Θ(n(n - B) log(B) + n2) as the tight bound. Our results show that the behaviour of the algorithm is highly dependent on the constraint bound of the uniform constraint. Afterwards, we consider the problem in the context of stochastic constraints and provide insights using experimental studies on how the (μ+1) EA is able to deal with these constraints in a sampling-based setting.

Parameterized analysis provides powerful mechanisms for obtaining fine-grained insights into different types of algorithms. In this work, we combine this field with evolutionary algorithms and provide parameterized complexity analysis of evolutionary multiobjective algorithms for the W-separator problem, which is a natural generalization of the vertex cover problem. The goal is to remove the minimum number of vertices such that each connected component in the resulting graph has at most W vertices. We provide different multi-objective formulations involving two or three objectives that provably lead to fixed-parameter evolutionary algorithms with respect to the value of an optimal solution OPT and W. Of particular interest are kernelizations and the reducible structures used for them. We show that in expectation the algorithms make incremental progress in finding such structures and beyond. The current best known kernelization of the W-separator uses linear programming methods and requires a non-trivial post-process to extract the reducible structures. We provide additional structural features to show that evolutionary algorithms with appropriate objectives are also capable of extracting them. Our results show that evolutionary algorithms with different objectives guide the search and admit fixed parameterized runtimes to solve or approximate (even arbitrarily close) the W-separator 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.

We study a Stackelberg game between one attacker and one defender in a configurable environment. The defender picks a specific environment configuration. The attacker observes the configuration and attacks via Reinforcement Learning (RL trained against the observed environment). The defender's goal is to find the environment with minimum achievable reward for the attacker. We apply Evolutionary Diversity Optimization (EDO) to generate diverse population of environments for training. Environments with clearly high rewards are killed off and replaced by new offsprings to avoid wasting training time. Diversity not only improves training quality but also fits well with our RL scenario: RL agents tend to improve gradually, so a slightly worse environment earlier on may become better later. We demonstrate the effectiveness of our approach by focusing on a specific application, Active Directory (AD). AD is the default security management system for Windows domain networks. AD environment describes an attack graph, where nodes represent computers/accounts/etc., and edges represent accesses. The attacker aims to find the best attack path to reach the highest-privilege node. The defender can change the graph by removing a limited number of edges (revoke accesses). Our approach generates better defensive plans than the existing approach and scales better.

Generating new instances via evolutionary methods is commonly used to create new benchmarking data-sets, with a focus on attempting to cover an instance-space as completely as possible. Recent approaches have exploited Quality-Diversity methods to evolve sets of instances that are both diverse and discriminatory with respect to a portfolio of solvers, but these methods can be challenging when attempting to find diversity in a high-dimensional feature-space. We address this issue by training a model based on Principal Component Analysis on existing instances to create a low-dimension projection of the high-dimension feature-vectors, and then apply Novelty Search directly in the new low-dimension space. We conduct experiments to evolve diverse and discriminatory instances of Knapsack Problems, comparing the use of Novelty Search in the original feature-space to using Novelty Search in a low-dimensional projection, and repeat over a given set of dimensions. We find that the methods are complementary: if treated as an ensemble, they collectively provide increased coverage of the space. Specifically, searching for novelty in a low-dimension space contributes 56% of the filled regions of the space, while searching directly in the feature-space covers the remaining 44%.

In recent years, computing diverse sets of high quality solutions for an optimization problem has become an important topic. The goal of computing diverse sets of high quality solutions is to provide a variety of options to decision makers, allowing them to choose the best solution for their particular problem. We consider the problem of constructing a wireless communication network for a given set of entities. Our goal is to minimize the area covered by the senders' transmissions while also avoiding adversaries that may observe the communication. We provide evolutionary diversity optimization (EDO) algorithms for this problem. We provide a formulation based on minimum spanning forests that are used as a representation and show how this formulation can be turned into a wireless communication network that avoids a given set of adversaries. We evaluate our EDO approach based on a number of benchmark instances and compare the diversity of the obtained populations in respect to the quality criterion of the given solutions as well as the chosen algorithm parameters. Our results demonstrate the effectiveness of our EDO approaches for the detection and concealment of communication networks both in terms of the quality and the diversity of the obtained solutions.

Finding diverse solutions to optimization problems has been of practical interest for several decades, and recently enjoyed increasing attention in research. While submodular optimization has been rigorously studied in many fields, its diverse solutions extension has not. In this study, we consider the most basic variants of submodular optimization, and propose two simple greedy algorithms, which are known to be effective at maximizing monotone submodular functions. These are equipped with parameters that control the trade-off between objective and diversity. Our theoretical contribution shows their approximation guarantees in both objective value and diversity, as functions of their respective parameters. Our experimental investigation with maximum vertex coverage instances demonstrates their empirical differences in terms of objective-diversity trade-offs.

We study the multi-objective minimum weight base problem, an abstraction of classical NP-hard combinatorial problems such as the multi-objective minimum spanning tree problem. We prove some important properties of the convex hull of the non-dominated front, such as its approximation quality and an upper bound on the number of extreme points. Using these properties, we give the first run-time analysis of the MOEA/D algorithm for this problem, an evolutionary algorithm that effectively optimizes by decomposing the objectives into single-objective components. We show that the MOEA/D, given an appropriate decomposition setting, finds all extreme points within expected fixed-parameter polynomial time, in the oracle model. Experiments are conducted on random bi-objective minimum spanning tree instances, and the results agree with our theoretical findings. Furthermore, compared with a previously studied evolutionary algorithm for the problem GSEMO, MOEA/D finds all extreme points much faster across all instances.

Active Directory is the default security management system for Windows domain networks. We study the shortest path edge interdiction problem for defending Active Directory style attack graphs. The problem is formulated as a Stackelberg game between one defender and one attacker. The attack graph contains one destination node and multiple entry nodes. The attacker's entry node is chosen by nature. The defender chooses to block a set of edges limited by his budget. The attacker then picks the shortest unblocked attack path. The defender aims to maximize the expected shortest path length for the attacker, where the expectation is taken over entry nodes. We observe that practical Active Directory attack graphs have small maximum attack path length and are structurally close to trees. We first show that even if the maximum attack path length is a constant, the problem is still w[1]-hard with respect to the defender's budget. Having a small maximum attack path length and a small budget is not enough to design fixed-parameter algorithms. If we further assume that the number of entry nodes is small, then we derive a fixed-parameter tractable algorithm. We then propose two other fixed-parameter algorithms by exploiting the tree-like features. One is based on tree decomposition and requires a small tree width. The other assumes a small number of splitting nodes (nodes with multiple out-going edges). Finally, the last algorithm is converted into a graph convolutional neural network based heuristic, which scales to larger graphs with more splitting nodes.

In this work, we consider the problem of finding a set of tours to a traveling salesperson problem (TSP) instance maximizing diversity, while satisfying a given cost constraint. This study aims to investigate the effectiveness of applying niching to maximize diversity rather than simply maintaining it. To this end, we introduce a 2-stage approach where a simple niching memetic algorithm (NMA), derived from a state-of-the-art for multi-solution TSP, is combined with a baseline diversifying algorithm. The most notable feature of the proposed NMA is the use of randomized improvement-first local search instead of 2-opt. Our experiment on TSPLIB instances shows that while the populations evolved by our NMA tend to contain clusters at tight quality constraints, they frequently occupy distant basins of attraction rather than close-by regions, improving on the baseline diversification in terms of sum-sum diversity. Compared to the original NMA, ours, despite its simplicity, finds more distant solutions of higher quality within less running time, by a large margin.

Active Directory (AD) is the default security management system for Windows domain networks. We study a Stackelberg game model between one attacker and one defender on an AD attack graph. The attacker initially has access to a set of entry nodes. The attacker can expand this set by strategically exploring edges. Every edge has a detection rate and a failure rate. The attacker aims to maximize their chance of successfully reaching the destination before getting detected. The defender's task is to block a constant number of edges to decrease the attacker's chance of success. We show that the problem is #P-hard and, therefore, intractable to solve exactly. We convert the attacker's problem to an exponential sized Dynamic Program that is approximated by a Neural Network (NN). Once trained, the NN provides an eficient fitness function for the defender's Evolutionary Diversity Optimisation (EDO). The diversity emphasis on the defender's solution provides a diverse set of training samples, which improves the training accuracy of our NN for modelling the attacker. We go back and forth between NN training and EDO. Experimental results show that for R500 graph, our proposed EDO based defense is less than 1% away from the optimal defense.

Computing diverse sets of high quality solutions for a given optimization problem has become an important topic in recent years. In this paper, we introduce a coevolutionary Pareto Diversity Optimization approach which builds on the success of reformulating a constrained single-objective optimization problem as a bi-objective problem by turning the constraint into an additional objective. Our new Pareto Diversity optimization approach uses this bi-objective formulation to optimize the problem while also maintaining an additional population of high quality solutions for which diversity is optimized with respect to a given diversity measure. We show that our standard co-evolutionary Pareto Diversity Optimization approach outperforms the recently introduced DIVEA algorithm which obtains its initial population by generalized diversifying greedy sampling and improving the diversity of the set of solutions afterwards. Furthermore, we study possible improvements of the Pareto Diversity Optimization approach. In particular, we show that the use of inter-population crossover further improves the diversity of the set of solutions.

In real-world optimisation, it is common to face several sub-problems interacting and forming the main problem. There is an inter-dependency between the sub-problems, making it impossible to solve such a problem by focusing on only one component. The traveling thief problem (TTP) belongs to this category and is formed by the integration of the traveling salesperson problem (TSP) and the knapsack problem (KP). In this paper, we investigate the inter-dependency of the TSP and the KP by means of quality diversity (QD) approaches. QD algorithms provide a powerful tool not only to obtain high-quality solutions but also to illustrate the distribution of high-performing solutions in the behavioural space. We introduce a MAP-Elite based evolutionary algorithm using well-known TSP and KP search operators, taking the TSP and KP score as behavioural descriptor. Afterwards, we conduct comprehensive experimental studies that show the usefulness of using the QD approach applied to the TTP. First, we provide insights regarding high-quality TTP solutions in the TSP/KP behavioural space. Afterwards, we show that better solutions for the TTP can be obtained by using our QD approach and it can improve the best-known solution for a wide range of TTP instances used for benchmarking in the literature.

There has been a growing interest in the evolutionary computation community to compute a diverse set of high-quality solutions for a given optimisation problem. This can provide the practitioners with invaluable information about the solution space and robustness against imperfect modelling and minor problems' changes. It also enables the decision-makers to involve their interests and choose between various solutions. In this study, we investigate for the first time a prominent multi-component optimisation problem, namely the Traveling Thief Problem (TTP), in the context of evolutionary diversity optimisation. We introduce a bi-level evolutionary algorithm to maximise the structural diversity of the set of solutions. Moreover, we examine the inter-dependency among the components of the problem in terms of structural diversity and empirically determine the best method to obtain diversity. We also conduct a comprehensive experimental investigation to examine the introduced algorithm and compare the results to another recently introduced framework based on the use of Quality Diversity (QD). Our experimental results show a significant improvement of the QD approach in terms of structural diversity for most TTP benchmark instances.

In this paper, we consider the subset selection problem for function f with constraint bound B which changes over time. We point out that adaptive variants of greedy approaches commonly used in the area of submodular optimization are not able to maintain their approximation quality. Investigating the recently introduced POMC Pareto optimization approach, we show that this algorithm efficiently computes a φ = (αf/2)(1− α1f )-approximation, where αf is the sube modularity ratio of f, for each possible constraint bound b ≤ B. Furthermore, we show that POMC is able to adapt its set of solutions quickly in the case that B increases. Our experimental investigations for the influence maximization in social networks show the advantage of POMC over generalized greedy algorithms.

Generating diverse populations of high-quality solutions has gained interest as a promising extension to the traditional optimization tasks. This work contributes to this line of research with an investigation on evolutionary diversity optimization for three of the most well-studied permutation problems: the Traveling Salesperson Problem (TSP), both symmetric and asymmetric variants, and the Quadratic Assignment Problem (QAP). It includes an analysis of the worst-case performance of a simple mutation-only evolutionary algorithm with different mutation operators, using an established diversity measure. Theoretical results show that many mutation operators for these problems guarantee convergence to maximally diverse populations of sufficiently small size within cubic to quartic expected runtime. On the other hand, the results regarding QAP suggest that strong mutations give poor worst-case performance, as mutation strength contributes exponentially to the expected runtime. Additionally, experiments are carried out on QAPLIB and synthetic instances in unconstrained and constrained settings, and reveal much more optimistic practical performances while corroborating the theoretical findings regarding mutation strength. These results should serve as a baseline for future studies.

The Stockpile blending problem is an important component of mine production scheduling, where stockpiles are used to store and blend raw material. The goal of blending material from stockpiles is to create parcels of concentrate which contain optimal metal grades based on the material available. The volume of material that each stockpile provides to a given parcel is dependent on a set of mine schedule conditions and customer demands. Therefore, the problem can be formulated as a continuous optimization problem. In the real-world application, there are several constraints required to guarantee parcels that meet the demand of downstream customers. It is a challenge in solving the stockpile blending problems since its scale can be very large. We introduce two repaired operators for the problems to convert the infeasible solutions into the solutions without violating the two tight constraints. Besides, we introduce a multi-component fitness function for solving the large-scale stockpile blending problem which can maximize the volume of metal over the plan and maintain the balance between stockpiles according to the usage of metal. Furthermore, we investigate the well-known approach in this paper, which is used to solve optimization problems over continuous space, namely the differential evolution (DE) algorithm. The experimental results show that the DE algorithm combined with two proposed duration repair methods is significantly better in terms of the values of results than the results on real-world instances for both one-month problems and large-scale problems.

Evolutionary algorithms based on edge assembly crossover (EAX) constitute some of the best performing incomplete solvers for the well-known traveling salesperson problem (TSP). Often, it is desirable to compute not just a single solution for a given problem, but a diverse set of high quality solutions from which a decision maker can choose one for implementation. Currently, there are only a few approaches for computing a diverse solution set for the TSP. Furthermore, almost all of them assume that the optimal solution is known. In this paper, we introduce evolutionary diversity optimisation (EDO) approaches for the TSP that find a diverse set of tours when the optimal tour is known or unknown. We show how to adopt EAX to not only find a high-quality solution but also to maximise the diversity of the population. The resulting EAX-based EDO approach, termed EAX-EDO is capable of obtaining diverse high-quality tours when the optimal solution for the TSP is known or unknown. A comparison to existing approaches shows that they are clearly outperformed by EAX-EDO.

In practise, it is often desirable to provide the decision-maker with a rich set of diverse solutions of decent quality instead of just a single solution. In this paper we study evolutionary diversity optimization for the knapsack problem (KP). Our goal is to evolve a population of solutions that all have a profit of at least (1 - ε) · OPT, where OPT is the value of an optimal solution. Furthermore, they should differ in structure with respect to an entropy-based diversity measure. To this end we propose a simple (μ + 1)-EA with initial approximate solutions calculated by a well-known FPTAS for the KP. We investigate the effect of different standard mutation operators and introduce biased mutation and crossover which puts strong probability on flipping bits of low and/or high frequency within the population. An experimental study on different instances and settings shows that the proposed mutation operators in most cases perform slightly inferior in the long term, but show strong benefits if the number of function evaluations is severely limited.

Generating diverse populations of high quality solutions has gained interest as a promising extension to the traditional optimization tasks. We contribute to this line of research by studying evolutionary diversity optimization for two of the most prominent permutation problems, namely the Traveling Salesperson Problem (TSP) and Quadratic Assignment Problem (QAP). We explore the worst-case performance of a simple mutation-only evolutionary algorithm with different mutation operators, using an established diversity measure. Theoretical results show most mutation operators for both problems ensure production of maximally diverse populations of sufficiently small size within cubic expected run-time. We perform experiments on QAPLIB instances in unconstrained and constrained settings, and reveal much more optimistic practical performances. Our results should serve as a baseline for future studies.

Submodular functions allow to model many real-world optimisation problems. This paper introduces approaches for computing diverse sets of high quality solutions for submodular optimisation problems with uniform and knapsack constraints. We first present diversifying greedy sampling approaches and analyse them with respect to the diversity measured by entropy and the approximation quality of the obtained solutions. Afterwards, we introduce an evolutionary diversity optimisation (EDO) approach to further improve diversity of the set of solutions. We carry out experimental investigations on popular submodular benchmark problems and analyse trade-offs in terms of solution quality and diversity of the resulting solution sets.

Computing diverse sets of high-quality solutions has gained increasing attention among the evolutionary computation community in recent years. It allows practitioners to choose from a set of high-quality alternatives. In this paper, we employ a population diversity measure, called the high-order entropy measure, in an evolutionary algorithm to compute a diverse set of high-quality solutions for the Traveling Salesperson Problem. In contrast to previous studies, our approach allows diversifying segments of tours containing several edges based on the entropy measure. We examine the resulting evolutionary diversity optimisation approach precisely in terms of the final set of solutions and theoretical properties. Experimental results show significant improvements compared to a recently proposed edge-based diversity optimisation approach when working with a large population of solutions or long segments.

The stockpile blending problem seeks to determine how many tonnages of ore that stockpiles provide and to which parcels. This scheduling model maximizes the volume of valuable material in the final production subject to resource capacities, constraints for mill machines, and customer requirements. Motivated by the uncertainty in the geologic input date which affects optimization, we consider the stockpile blending problem with uncertainty in material grades. A non-linear continuous optimization model developed here to integrate uncertain variables to optimization. We introduce chance constraints that are used to guarantee the constraint is violated with a small probability to tackle the stochastic material grades. We investigate a well-known approach in this paper, which is used to solve optimization problems over continuous space, namely the differential evolution (DE) algorithm. In the experiment section, we compare the performance of the algorithm with the deterministic model and three chance constraint models by using a synthetic benchmark. We also evaluate the effectiveness of different chance constraints.

Addressing a complex real-world optimization problem is a challenging task. The chance-constrained knapsack problem with correlated uniform weights plays an important role in the case where dependent stochastic components are considered. We perform runtime analysis of a randomized search algorithm (RSA) and a basic evolutionary algorithm (EA) for the chance-constrained knapsack problem with correlated uniform weights. We prove bounds for both algorithms for producing a feasible solution. Furthermore, we investigate the behaviour of the algorithms and carry out analyses on two settings: uniform profit value and the setting in which every group shares an arbitrary profit profile. We provide insight into the structure of these problems and show how the weight correlations and the different profit profiles influence the runtime behavior of both algorithms in the chance-constrained setting.

Submodular optimization plays a key role in many real-world problems. In many real-world scenarios, it is also necessary to handle uncertainty, and potentially disruptive events that violate constraints in stochastic settings need to be avoided. In this paper, we investigate submodular optimization problems with chance constraints. We provide a first analysis on the approximation behavior of popular greedy algorithms for submodular problems with chance constraints. Our results show that these algorithms are highly effective when using surrogate functions that estimate constraint violations based on Chernoff bounds. Furthermore, we investigate the behavior of the algorithms on popular social network problems and show that high quality solutions can still be obtained even if there are strong restrictions imposed by the chance constraint.

Real-world combinatorial optimization problems are often stochastic and dynamic.Therefore, it is essential to make optimal and reliable decisions with a holistic approach.In this paper, we consider the dynamic chance-constrained knapsack problem where the weight of each item is stochastic, the capacity constraint changes dynamically over time, and the objective is to maximize the total profit subject to the probability that total weight exceeds the capacity.We make use of prominent tail inequalities such as Chebyshev's inequality, and Chernoff bound to approximate the probabilistic constraint.Our key contribution is to introduce an additional objective which estimates the minimal capacity bound for a given stochastic solution that still meets the chance constraint.This objective helps to cater for dynamic changes to the stochastic problem.We apply single-and multi-objective evolutionary algorithms to the problem and show how bi-objective optimization can help to deal with dynamic chance-constrained problems.

We study the problem of maximizing a non-monotone submodular function under multiple knapsack constraints.We propose a simple discrete greedy algorithm to approach this problem, and prove that it yields strong approximation guarantees for functions with bounded curvature.In contrast to other heuristics, this does not require problem relaxation to continuous domains and it maintains a constant-factor approximation guarantee in the problem size.In the case of a single knapsack, our analysis suggests that the standard greedy can be used in non-monotone settings.Additionally, we study this problem in a dynamic setting, in which knapsacks change during the optimization process.We modify our greedy algorithm to avoid a complete restart at each constraint update.This modification retains the approximation guarantees of the static case.We evaluate our results experimentally on a video summarization and sensor placement task.We show that our proposed algorithm competes with the state-of-the-art in static settings.Furthermore, we show that in dynamic settings with tight computational time budget, our modified greedy yields significant improvements over starting the greedy from scratch, in terms of the solution quality achieved.

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.

The chance-constrained knapsack problem is a variant of the classical knapsack problem where each item has a weight distribution instead of a deterministic weight. The objective is to maximize the total profit of the selected items under the condition that the weight of the selected items only exceeds the given weight bound with a small probability of α. In this paper, we consider problem-specific single-objective and multi-objective approaches for the problem. We examine the use of heavy-tail mutations and introduce a problem-specific crossover operator to deal with the chance-constrained knapsack problem. Empirical results for single-objective evolutionary algorithms show the effectiveness of our operators compared to the use of classical operators. Moreover, we introduce a new effective multi-objective model for the chance-constrained knapsack problem. We use this model in combination with the problem-specific crossover operator in multi-objective evolutionary algorithms to solve the problem. Our experimental results show that this leads to significant performance improvements when using the approach in evolutionary multi-objective algorithms such as GSEMO and NSGA-II.

We present a study demonstrating how random walk algorithms can be used for evolutionary image transition. We design different mutation operators based on uniform and biased random walks and study how their combination with a baseline mutation operator can lead to interesting image transition processes in terms of visual effects and artistic features. Using feature-based analysis we investigate the evolutionary image transition behaviour with respect to different features and evaluate the images constructed during the image transition process. Afterwards, we investigate how modifications of our biased random walk approaches can be used for evolutionary image painting. We introduce an evolutionary image painting approach whose underlying biased random walk can be controlled by a parameter influencing the bias of the random walk and thereby creating different artistic painting effects.

In this paper, we consider the subset selection problem for function f with constraint bound B which changes over time. We point out that adaptive variants of greedy approaches commonly used in the area of submodular optimization are not able to maintain their approximation quality. Investigating the recently introduced POMC Pareto optimization approach, we show that this algorithm efficiently computes a φ = (αf/2)(1− α1f )-approximation, where αf is the sube modularity ratio of f, for each possible constraint bound b ≤ B. Furthermore, we show that POMC is able to adapt its set of solutions quickly in the case that B increases. Our experimental investigations for the influence maximization in social networks show the advantage of POMC over generalized greedy algorithms.

Evolutionary algorithms have successfully been applied to evolve problem instances that exhibit a significant difference in performance for a given algorithm or a pair of algorithms inter alia for the Traveling Salesperson Problem (TSP). Creating a large variety of instances is crucial for successful applications in the blooming field of algorithm selection. In this paper, we introduce new and creative mutation operators for evolving instances of the TSP. We show that adopting those operators in an evolutionary algorithm allows for the generation of benchmark sets with highly desirable properties: (1) novelty by clear visual distinction to established benchmark sets in the field, (2) visual and quantitative diversity in the space of TSP problem characteristics, and (3) significant performance differences with respect to the restart versions of heuristic state-of-the-art TSP solvers EAX and LKH. The important aspect of diversity is addressed and achieved solely by the proposed mutation operators and not enforced by explicit diversity preservation.

Evolutionary diversity optimization aims to compute a set of solutions that are diverse in the search space or instance feature space, and where all solutions meet a given quality criterion. With this paper, we bridge the areas of evolutionary diversity optimization and evolutionary multi-objective optimization. We show how popular indicators frequently used in the area of multi-objective optimization can be used for evolutionary diversity optimization. Our experimental investigations for evolving diverse sets of TSP instances and images according to various features show that two of the most prominent multi-objective indicators, namely the hypervolume indicator and the inverted generational distance, provide excellent results in terms of visualization and various diversity indicators.

Evolutionary algorithms have been widely used for a range of stochastic optimization problems. In most studies, the goal is to optimize the expected quality of the solution. Motivated by real-world problems where constraint violations have extremely disruptive effects, we consider a variant of the knapsack problem where the profit is maximized under the constraint that the knapsack capacity bound is violated with a small probability of at most α. This problem is known as chance-constrained knapsack problem and chance-constrained optimization problems have so far gained little attention in the evolutionary computation literature. We show how to use popular deviation inequalities such as Chebyshev's inequality and Chernoff bounds as part of the solution evaluation when tackling these problems by evolutionary algorithms and compare the effectiveness of our algorithms on a wide range of chance-constrained knapsack instances.

Evolutionary algorithms have been widely used in the area of creativity in order to help create art and music. We consider the recently introduced evolutionary image composition approach based on feature covariance matrices [1] which allows composing two images into a new one based on their feature characteristics. When using evolutionary image composition it is important to obtain a good weighting of interesting regions of the two images. We use colour-based segmentation based on K-Means clustering to come up with such a weighting of the images. Our results show that this preserves the chosen colour regions of the images and leads to composed images that preserve colours better than the previous approach based on saliency masks [1]. Furthermore, we evaluate our composed images in terms of aesthetic feature and show that our approach based on colour-based segmentation leads to higher feature values for most of the investigated features.

Diversity plays a crucial role in evolutionary computation. While diversity has been mainly used to prevent the population of an evolutionary algorithm from premature convergence, the use of evolutionary algorithms to obtain a diverse set of solutions has gained increasing attention in recent years. Diversity optimization in terms of features on the underlying problem allows to obtain a better understanding of possible solutions to the problem at hand and can be used for algorithm selection when dealing with combinatorial optimization problems such as the Traveling Salesperson Problem.

Multi-objective evolutionary algorithms (MOEAs) based on the concept of Pareto-dominance have been successfully applied to many real-world optimisation problems. Recently, research interest has shifted towards indicator-based methods to guide the search process towards a good set of trade-off solutions. One commonly used approach of this nature is the indicator-based evolutionary algorithm (IBEA). In this study, we highlight the solution distribution issues within IBEA and propose a modification of the original approach by embedding an additional Pareto-dominance based component for selection. The improved performance of the proposed modified IBEA (mIBEA) is empirically demonstrated on the well-known DTLZ set of benchmark functions. Our results show that mIBEA achieves comparable or better hypervolume indicator values and epsilon approximation values in the vast majority of our cases (13 out of 14 under the same default settings) on DTLZ1-7. The modification also results in an over 8-fold speed-up for larger populations.

Measures aimed to improve the diversity of images and image features in evolutionary art help to direct search toward more novel and creative parts of the artistic search domain. To date such measures have not focused on selecting from all individuals based on their contribution to diversity of feature metrics. In recent work on TSP problem instance classification, selection based on a direct measure of each individual's contribution to diversity was successfully used to generate hard and easy TSP instances. In this work we use this search framework to evolve diverse variants of a source image in one and two feature dimensions. The resulting images show the spectrum of effects from transforming images to score across the range of each feature. The results also reveal interesting correlations between feature values in two dimensions.

Evolutionary algorithms have recently been used to create a wide range of artistic work. In this paper, we propose a new approach for the composition of new images from existing ones, that retain some salient features of the original images. We introduce evolutionary algorithms that create new images based on a fitness function that incorporates feature covariance matrices associated with different parts of the images. This approach is very flexible in that it can work with a wide range of features and enables targeting specific regions in the images. For the creation of the new images, we propose a population-based evolutionary algorithm with mutation and crossover operators based on random walks. Our experimental results reveal a spectrum of aesthetically pleasing images that can be obtained with the aid of our evolutionary process.