MNK landscapes are multi-objective combinatorial optimization problems. For MNK landscapes, computing the entire set of Pareto local optima and the Pareto front by full enumeration quickly becomes infeasible within a reasonable amount of time. Approximation methods are faster, but cannot guarantee that all Pareto local optima or Pareto non-dominated solutions are found. To address this, we propose a multi-objective gray-box optimization algorithm based on hyperplane elimination. By exploiting gray-box information, the number of evaluations is reduced by maintaining and updating changes in subfunction values of the objectives instead of re-evaluating the entire bitstring after each bit flip. In addition, hyperplanes of bitstrings can be eliminated from the search space by knowing all dependencies between the bits and the delta values. Since this algorithm identifies all Pareto local optima, the Pareto front can be obtained with very low additional cost. We provide theoretical proofs of correctness and experimental results on adjacent, random and p-correlated MNK landscapes showing speed-ups of several orders of magnitude. Further improvements on runtime are obtained by using a bit reordering heuristic and a prefix mechanism. Altogether, our proposed method enables the computation of exact Pareto sets of MNK landscapes even for large bit lengths.

Surrogates provide a cheap solution evaluation and offer significant leverage for optimizing computationally expensive problems. Usually, surrogates only approximate the original function. Recently, the perfect linear surrogates were proposed that ideally represent the original function. These surrogates do not mimic the original function. In fact, they are another (correct) representation of it and enable a wide range of possibilities, e.g., discovering the optimized function for problems where the direct transformation of the encoded solution into its evaluation is not available. However, many real-world problems can not be represented by linear models, making the aforementioned surrogates inapplicable. Therefore, we propose the Limited Monotonical Perfect Surrogate (LyMPuS), which overcomes this difficulty and enables the comparison of two solutions that differ by a single variable. Our proposition is suitable for limiting the cost of expensive local search procedures. The proposed surrogate is parameterless and can be trained on the fly without any separate surrogate-building step. It uses only the necessary fitness evaluations, and the already-paid costs are not wasted when the model is updated. Finally, it offers low-cost missing-linkage detection and low-cost linkage discovery, guaranteed to find a missing dependency in no more than $2\lceil\log_2(n)\rceil$ steps.

We demonstrate that there are strong patterns of periodicity in terms of how local optima are distributed across search spaces that can help to explain the "Big Valley" distribution of local optima. Examples of periodicity can be found by looking at several Partition Crossover events simultaneously and grouping together those that are nearer to each other in Hamming space. All of the local optima associated with one Partition Crossover event can be evaluated using a single linear equation. When looking at many Partition Crossover events simultaneously, linearity is preserved over complete and partially preserved recombining components.

Gray-box optimization proposes effective and efficient optimizers of general use. To this end, it leverages information about variable dependencies and the subfunction-based problem representation. These approaches were already shown effective by enabling tunnelling between local optima even if these moves require the modification of many dependent variables. Tunnelling is useful in solving the maximum satisfiability problem (MaxSat), which can be reformulated to Max3Sat. Since many real-world problems can be brought to solving the MaxSat/Max3Sat instances, it is important to solve them effectively and efficiently. Therefore, we focus on Max3Sat instances for which tunnelling fails to introduce improving moves between locally optimal high-quality solutions and the region of globally optimal solutions. We analyze the features of such instances on the ground of phase transitions. Based on these observations, we propose manipulating clause-satisfiability characteristics that allow connecting high-quality solutions distant in the solution space. We utilize multi-satisfiability characteristics in the optimizer built from typical gray-box mechanisms. The experimental study shows that the proposed optimizer can solve those Max3Sat instances that are out of the grasp of state-of-the-art gray-box optimizers. At the same time, it remains effective for instances that have already been successfully solved by gray-box.

Gray-box optimization employs Walsh decomposition to obtain non-linear variable dependencies and utilize them to propose masks of variables that have a joint non-linear influence on fitness value. These masks significantly improve the effectiveness of variation operators. In some problems, all variables are non-linearly dependent, making the aforementioned masks useless. We analyze the features of the real-world instances of such problems and show that many of their dependencies may have noise-like origins. Such noise-caused dependencies are irrelevant to the optimization process and can be ignored. To identify them, we propose extending the use of Walsh decomposition by measuring variable dependency strength that allows the construction of the weighted dynamic Variable Interaction Graph (wdVIG). wdVIGs adjust the dependency strength to mixed individuals. They allow the filtering of irrelevant dependencies and re-enable using dependency-based masks by variation operators. We verify the wdVIG potential on a large benchmark suite. For problems with noise, the wdVIG masks can improve the optimizer's effectiveness. If all dependencies are relevant for the optimization, i.e., the problem is not noised, the influence of wdVIG masks is similar to that of state-of-the-art structures of this kind.

A combination of recombination and local search can expose the existence of an exponential number of parallel lattices that span the search space for all classes of k-bounded pseudo-Boolean functions, including MAX-kSAT problems. These "parallel" lattices sometimes have identical evaluations shifted by a constant. We use Partition Crossover to aid in the discovery of lattices, which are sets of 2q possible offspring from recombination events, organized into q-dimensional hypercubes, where q is the number of recombining components given two parents. Finally, we show that recursively embedded subspace lattices display a fractal structure, which can be captured using rewrite rules based on a Lindenmayer system that accurately model how local optima are distributed across different size lattices.

Federated learning is a paradigm that proposes protecting data privacy by sharing local models instead of raw data during each iteration of model training. However, these models can be large, with many parameters, provoking a substantial communication cost and having a notable environmental impact. Reducing communication overhead is paramount but conflictual to maintaining the model’s accuracy. Most research has dealt with the different factors influencing communication reduction separately without addressing their correlations. Moreover, most of them do not consider the heterogeneity of clients’ hardware. Finding the optimal configuration to fulfil all these training aspects can become intractable for classical techniques. This work explores the add-in that multi-objective evolutionary algorithms can provide for solving the communication overhead problem while achieving high accuracy. We do this by 1) realistically modelling and formulating this task as a multi-objective problem by considering the devices’ heterogeneity, 2) including all the communication-triggering aspects, and 3) applying a multi-objective evolutionary algorithm with an intensification operator to solve the problem. A simulated client–server architecture of four devices with four different processing speeds is studied. Both fully connected and convolutional neural network models are investigated with 33,400 and 887,530 weights, respectively. The experiments are performed using the MNIST and Fashion-MNIST datasets. A comparison is made between three approaches using an extensive set of metrics. Results prove that our approach obtains solutions with better accuracy than the full-communication setting and other methods while getting reductions in communications by around 1,000 times in most cases and up to 10,000 times in some cases compared to the maximum communication setting.

In pseudo-Boolean optimization, a variable interaction graph represents variables as vertices, and interactions between pairs of variables as edges. In black-box optimization, the variable interaction graph may be at least partially discovered by using empirical linkage learning techniques. These methods never report false variable interactions, but they are computationally expensive. The recently proposed local search with linkage learning discovers the partial variable interaction graph as a side-effect of iterated local search. However, information about the strength of the interactions is not learned by the algorithm. We propose local search with linkage learning 2, which builds a weighted variable interaction graph that stores information about the strength of the interaction between variables. The weighted variable interaction graph can provide new insights about the optimization problem and behavior of optimizers. Experiments with NK landscapes, knapsack problem, and feature selection show that local search with linkage learning 2 is able to efficiently build weighted variable interaction graphs. In particular, experiments with feature selection show that the weighted variable interaction graphs can be used for visualizing the feature interactions in machine learning. Additionally, new transformation operators that exploit the interactions between variables can be designed. We illustrate this ability by proposing a new perturbation operator for iterated local search.

When Partition Crossover is used to recombine two parents which are local optima, the offspring are all local optima in the smallest hyperplane subspace that contains the two parents. The offspring can also be organized into a non-planar hypercube "lattice." Furthermore, all of the offspring can be evaluated using a simple linear equation. When a child of Partition Crossover is a local optimum in the full search space, the linear equation exactly determines its evaluation. When a child of Partition Crossover can be improved by local search, the linear equation is an upper bound on the evaluation of the associated local optimum when minimizing. This theoretical result holds for all k-bounded Pseudo-Boolean optimization problems, including MAX-kSAT, QUBO problems, as well as random and adjacent NK landscapes. These linear equations provide a stronger explanation as to why the "Big Valley" distribution of local optima exists. We fully enumerate a sample of NK landscapes to collect frequency information to complement our theoretical results. We also introduce new algorithmic contributions that can 1) expand smaller lattices in order to find larger lattices that contain additional local optima, and 2) introduce an efficient method to find new improving moves in lattices using score vectors.

Specialized graybox local search and crossover have been successfully combined within the framework of the so-called Drils (Deterministic recombination and iterated local search) algorithm. As for any evolutionary algorithm, the initial design framework, and the underlying high-level choices and parameters, are crucially important. The Drils algorithm is no exception, and recent enhanced variants exist in the literature. In this paper, we aim at: (i) improving the performance of the latest variants of Drils, and (ii) providing a better principled understanding of graybox search behavior and dynamics. On the basis of a preliminary analysis using Local Optima Networks of small-size NKQ-landscapes, we first highlight the difference of using local search with and without crossover. We then propose to pipeline these two techniques in a simple two-phase like iterated local search scheme which is shown to provide substantial improvements over the latest Drils+ variant for large-size NKQ-landscapes. We further report a dedicated analysis in an attempt to provide new insights into the impact of local search and crossover on the phenotype and the genotype of the local optima encountered in the search trajectory.

In the context of pseudo-Boolean optimization, surrogate functions based on the Walsh-Hadamard transform have been recently proposed with great success. It has been shown that lower-order components of the Walsh-Hadamard transform have usually a larger influence on the value of the objective function. Thus, creating a surrogate model using the lower-order components of the transform can provide a good approximation to the objective function. The Walsh-Hadamard transform in pseudo-Boolean optimization is a particularization in the binary representation of a Fourier transform over a finite group, precisely defined in the framework of group representation theory. Using this more general definition, it is possible to define a Fourier transform for the functions over permutations. We propose in this paper the use of surrogate functions based on the Fourier transforms over the permutation space. We check how similar the proposed surrogate models are to the original objective function and we also apply regression to learn a surrogate model based on the Fourier transform. The experimental setting includes two permutation problems for which the exact Fourier transform is unknown based on the problem parameters: the Asteroid Routing Problem and the Single Machine Total Weighted Tardiness.

Landscape analysis is a very useful tool in optimization to understand the structure of the search space of a problem when there is some kind of distance or neighborhood defined over the solutions. Local Optima Networks (LON) have been proposed to serve as a summary of the landscape of a problem. LONs are graphs where the nodes are the local optima of the search space according to a particular neighborhood and edges join local optima when one can be reached from the other using some kind of perturbation followed by hill climbing. In this paper we enhance local optima networks to include precise information on the transition probabilities among local optima, yielding a Markov Chain for the visited local optima during the search. The new analysis tool, called Local Optima Markov Chain (LOMA), is built on top of the static landscape information depending on the problem and includes information about algorithm dynamics. We show how LOMAs can be used to compute metrics that are out of the reach of other landscape-aware tools, thus offering more information to understand algorithm dynamics.

Next-generation genetic algorithms (GAs) should explore information from the problem structure whenever possible. Variable interactions can be inferred using linkage learning. Statistical linkage learning techniques were shown to improve GAs' effectiveness significantly in many problems, but may eventually report false linkages. On the other hand, empirical linkage learning (ELL) techniques discover only true variable dependencies. However, traditional ELL techniques are computationally expensive. We introduce the genetic algorithm with linkage learning (GAwLL), which discovers an empirical weighted variable interaction graph (VIGw) as a side-effect of the optimization performed by a GA, making it a no-cost ELL technique. Vertices of the VIGw represent decision variables and weights indicate the strength of the interaction between variables. The VIGw allows us to obtain new insights about the optimization problem and can be used to design genetic operators that efficiently explore the information about variable dependencies. Experiments with NK landscapes show that GAwLL is able to efficiently build the empirical VIGw. We also present an interesting machine learning application, where the VIGw represents a feature interaction network. By using GAwLL, the feature interaction network is built as a side-effect of evolutionary feature selection.

Discrete-variable gate-model quantum machines are promising quantum systems considering their wide applicability. Being in their noisy-intermediate-scale era, they allow executing only circuits of limited complexity and fitting the machines' features. Thus, such systems implement a key and unavoidable tailoring process to produce the most possible compact and device-compliant circuit. The qubits' initialisation is a primary and complex step that can ease/jeopardise the tailoring process and restrict/extend the machine's computational capacities. Ultimately, this bottleneck can be responsible of making quantum leaps like quantum supremacy. As a step towards the former, this work investigates how evolutionary algorithms can enhance the qubits' initialisation by tackling it as a single-objective problem using a genetic algorithm. The experiments used instances representing 19 real IBM quantum machines with 7 to 65 qubits and 9 different qubit topologies. Also, 76 GHZ circuits of sizes 7-65 qubits and 25%-100% of entanglement were created and studied. Extensive standard and statistical comparisons have been made against the IBM qubit initialiser that is currently used in real quantum machines. Results showed that the proposal outperforms IBM in 64 instances and is similar to it in 10 ones, with an average circuit-compression gain up to 46%.

This document presents an extended version of the article: Pedro Delgado-Pérez and Francisco Chicano. An experimental and practical study on the equivalent mutant connection: an evolutionary approach (August 2020). https://doi.org/10.1016/j.infsof.2020.106317

An optimal recombination operator for two-parent solutions provides the best solution among those that take the value for each variable from one of the parents (gene transmission property). If the solutions are bit strings, the offspring of an optimal recombination operator is optimal in the smallest hyperplane containing the two parent solutions. Exploring this hyperplane is computationally costly, in general, requiring exponential time in the worst case. However, when the variable interaction graph of the objective function is sparse, exploration can be done in polynomial time. In this article, we present a recombination operator, called Dynastic Potential Crossover (DPX), that runs in polynomial time and behaves like an optimal recombination operator for low-epistasis combinatorial problems. We compare this operator, both theoretically and experimentally, with traditional crossover operators, like uniform crossover and network crossover, and with two recently defined efficient recombination operators: partition crossover and articulation points partition crossover. The empirical comparison uses NKQ Landscapes and MAX-SAT instances. DPX outperforms the other crossover operators in terms of quality of the offspring and provides better results included in a trajectory and a population-based metaheuristic, but it requires more time and memory to compute the offspring.

In contrast with random uniform instances, industrial SAT instances of large size are solvable today by state-of-the-art algorithms. It is believed that this is the consequence of the non-random structure of the distribution of variables into clauses. In order to produce benchmark instances resembling those of real-world formulas with a given structure, generative models have been proposed. In this paper we study the MAX-3SAT problem with model-generated instances having a power-law distribution. Specifically, we target the regions in which computational difficulty undergoes an easy/hard phase transition as a function of clause density and of the power-law exponent. Our approach makes use of a sampling technique to build a graph model (a local optima network) in which nodes are local optima and directed edges are transitions between optima basins. The objective is to relate the structure of the instance fitness landscape with problem difficulty through the transition. We succeed in associating the transition with straightforward network metrics, thus providing a novel and original fitness landscape view of the computational features of the power-law model and its phase transition.

Partition crossover (PX) is an efficient recombination operator for gray-box optimization. PX is applied in problems where the objective function can be written as a sum of subfunctions fl(.). In PX, the variable interaction graph (VIG) is decomposed by removing vertices with common variables. Parent variables are inherited together during recombination if they are part of the same connected recombining component of the decomposed VIG. A new way of generating the recombination graph is proposed here. The VIG is decomposed by removing edges associated with subfunctions fl(.) that have similar evaluation for combinations of variables inherited from the parents. By doing so, the partial evaluations of fl(.) are taken into account when decomposing the VIG. This allows the use of partition crossover in continuous optimization. Results of experiments where local optima are recombined indicate that more recombining components are found. When the proposed epsilon-PX (ePX) is compared with other recombination operators in Genetic Algorithms and Differential Evolution, better performance is obtained when the epistasis degree is low.

In multiobjective optimization, the result of an optimization algorithm is a set of efficient solutions from which the decision maker selects one. It is common that not all the efficient solutions can be computed in a short time and the search algorithm has to be stopped prematurely to analyze the solutions found so far. A set of efficient solutions that are well-spread in the objective space is preferred to provide the decision maker with a great variety of solutions. However, just a few exact algorithms in the literature exist with the ability to provide such a well-spread set of solutions at any moment: we call them anytime algorithms. We propose a new exact anytime algorithm for multiobjective combinatorial optimization combining three novel ideas to enhance the anytime behavior. We compare the proposed algorithm with those in the state-of-the-art for anytime multiobjective combinatorial optimization using a set of 480 instances from different well-known benchmarks and four different performance measures: the overall non-dominated vector generation ratio, the hypervolume, the general spread and the additive epsilon indicator. A comprehensive experimental study reveals that our proposal outperforms the previous algorithms in most of the instances.

The goal in Robust Optimization is to optimize not only the quality of the solutions but also the variation of this quality with the uncertain parameters of the optimization problem . We propose a robust model for the bi-objective shortest path problem applied in a smart mobility context: Finding routes for cars in a city to minimize travel time and gas emissions. Our proposal treats robustness from a multi-objective point of view. We model the parameters that define each instance as random variables , described through their mean and variance. In this way, we can obtain efficient solutions that are also less sensitive to changes in the environment. We run different types of algorithms in multiple instances to solve this problem so that we obtain a global view of the behavior of different techniques. All experimentation uses a scenario based on real data : The province of Malaga, Spain. This realistic settlement for our study allows us to test the applicability of our model in final systems for the citizens. The results clearly state the interest of our proposal for tackling robustness and represents a new state-of-the-art in smart mobility, an always appealing feature of works, that could lead to an industrial prototype.

There are two important challenges for local search algorithms when applied to Maximal Satisfiability (MAXSAT). 1) Local search spends a great deal of time blindly exploring plateaus in the search space and 2) local search is less effective on application instances. This second problem may be related to local search's inability to exploit problem structure. We propose a genetic recombination operator to address both of these issues. On problems with well defined local optima, partition crossover is able to "tunnel" between local optima to discover new local optima in O(n) time. The PXSAT algorithm combines partition crossover and local search to produce a new way to escape plateaus. Partition crossover locally decomposes the evaluation function for a given instance into independent components, and is guaranteed to find the best solution among an exponential number of candidate solutions in O(n) time. Empirical results on an extensive set of application instances show that the proposed framework substantially improves two of best local search solvers, AdaptG2WSAT and Sparrow, on many application instances. PXSAT combined with AdaptG2WSAT is also able to outperform CCLS, winner of several recent MAXSAT competitions.

Partition Crossover is a recombination operator for pseudo-Boolean optimization with the ability to explore an exponential number of solutions in linear or square time. It decomposes the objective function as a sum of subfunctions, each one depending on a different set of variables. The decomposition makes it possible to select the best parent for each subfunction independently and the operator provides the best out of 2q solutions, where q is the number of sub-functions in the decomposition. These subfunctions are defined over the connected components of the recombination graph: a subgraph of the objective function variable interaction graph containing only the differing variables in the two parents. In this paper, we advance further and propose a new way to increase the number of linearly independent subfunctions by analyzing the articulation points of the recombination graph. These points correspond to variables that, once flipped, increase the number of connected components. The presence of a connected component with an articulation point increases the number of explored solutions by a factor of, at least, 4. We evaluate the new operator using Iterated Local Search combined with Partition Crossover to solve NK Landscapes and MAX-SAT.

With millions of smartphones sold every year, the development of mobile apps has grown substantially. The battery power limitation of mobile devices has push developers and researchers to search for methods to improve the energy efficiency of mobile apps. We propose a multiobjective refactoring approach to automatically improve the architecture of mobile apps, while controlling for energy efficiency. In this extended abstract we briefly summarize our work.

Accepted version of publication "NK Hybrid Genetic Algorithm for Clustering", published in IEEE Transactions on Evolutionary Computation

The energy consumption of mobile apps is a trending topic and researchers are actively investigating the role of coding practices on energy consumption. Recent studies suggest that design choices can conflict with energy consumption. Therefore, it is important to take into account energy consumption when evolving the design of a mobile app. In this paper, we analyze the impact of eight type of anti-patterns on a testbed of 20 android apps extracted from F-Droid. We propose EARMO, a novel anti-pattern correction approach that accounts for energy consumption when refactoring mobile anti-patterns. We evaluate EARMO using three multiobjective search-based algorithms. The obtained results show that EARMO can generate refactoring recommendations in less than a minute, and remove a median of 84 percent of anti-patterns. Moreover, EARMO extended the battery life of a mobile phone by up to 29 minutes when running in isolation a refactored multimedia app with default settings (no Wi-Fi, no location services, and minimum screen brightness). Finally, we conducted a qualitative study with developers of our studied apps, to assess the refactoring recommendations made by EARMO. Developers found 68 percent of refactorings suggested by EARMO to be very relevant.

In gray-box optimization, the search algorithms have access to the variable interaction graph (VIG) of the optimization problem. For Mk Landscapes (and NK Landscapes) we can use the VIG to identify an improving solution in the Hamming neighborhood in constant time. In addition, using the VIG, deterministic Partition Crossover is able to explore an exponential number of solutions in a time that is linear in the size of the problem. Both methods have been used in isolation in previous search algorithms. We present two new gray-box algorithms that combine Partition Crossover with highly efficient local search. The best algorithms are able to locate the global optimum on Adjacent NK Landscape instances with one million variables. The algorithms are compared with a state-of-the-art algorithm for pseudo-Boolean optimization: Gray-Box Parameterless Population Pyramid. The results show that the best algorithm is always one combining Partition Crossover and highly efficient local search. But the results also illustrate that the best optimizer differs on Adjacent and Random NK Landscapes.

Efficient hill climbers have been recently proposed for single- and multi-objective pseudo-Boolean optimization problems. For k-bounded pseudo-Boolean functions where each variable appears in at most a constant number of subfunctions, it has been theoretically proven that the neighborhood of a solution can be explored in constant time. These hill climbers, combined with a high-level exploration strategy, have shown to improve state of the art methods in experimental studies and open the door to the so-called Gray Box Optimization, where part, but not all, of the details of the objective functions are used to better explore the search space. One important limitation of all the previous proposals is that they can only be applied to unconstrained pseudo-Boolean optimization problems. In this work, we address the constrained case for multi-objective k-bounded pseudo-Boolean optimization problems. We find that adding constraints to the pseudo-Boolean problem has a linear computational cost in the hill climber.

The use of good evaluation functions is essential when evolutionary algorithms are employed for clustering. The NK internal clustering validation measure is proposed for hard partitional clustering. The evaluation function is composed of N subfunctions, where N is the number of objects in the dataset. Each subfunction is influenced by a group of K+1 objects. By using neighbourhood relations among connected small groups, density-based regions can be identified. The NK internal clustering validation measure allows the application of partition crossover (PX). PX for hard partitional clustering is also proposed in this work. By using PX, the evaluation function can be decomposed in q partial evaluations. As a consequence, PX deterministically finds the best of 2q possible offspring at the cost of evaluating 2 solutions. In the experiments, the application of PX resulted in a high number of successful recombinations. It was able to improve partitions defined by the best parents.

Anti-patterns are poor design choices that hinder code evolution, and understandability. Practitioners perform refactoring, that are semantic-preserving-code transformations, to correct anti-patterns and to improve design quality. However, manual refactoring is a consuming task and a heavy burden for developers who have to struggle to complete their coding tasks and maintain the design quality of the system at the same time. For that reason, researchers and practitioners have proposed several approaches to bring automated support to developers, with solutions that ranges from single anti-patterns correction, to multiobjective solutions. The latter approaches attempted to reduce refactoring effort, or to improve semantic similarity between classes and methods in addition to removing anti-patterns. To the best of our knowledge, none of the previous approaches have considered the impact of refactoring on another important aspect of software development, which is the testing effort. In this paper, we propose a novel search-based multiobjective approach for removing five well-known anti-patterns and minimizing testing effort. To assess the effectiveness of our proposed approach, we implement three different multiobjective metaheuristics (NSGA-II, SPEA2, MOCell) and apply them to a benchmark comprised of four open-source systems. Results show that MOCell is the metaheuristic that provides the best performance.

First paragraph: In combinatorial optimization, the goal is to find an optimal solution, according to some objective function, from a discrete search space. These problems arise widely in industry and academia and, unfortunately, many of them are NP-hard and no polynomial time algorithm can guarantee their solution to a certified optimality unless. Therefore, in the last decades researchers have investigated the use of stochastic search algorithms to find near optimal solutions to these problems. In particular, great research efforts have been devoted to the development and application of metaheuristic algorithms to solve combinatorial optimization problems.

This article investigates Gray Box Optimization for pseudo-Boolean optimization problems composed of M subfunctions, where each subfunction accepts at most k variables. We will refer to these as Mk Landscapes. In Gray Box Optimization, the optimizer is given access to the set of M subfunctions. We prove Gray Box Optimization can efficiently compute hyperplane averages to solve non-deceptive problems in [Formula: see text] time. Bounded separable problems are also solved in [Formula: see text] time. As a result, Gray Box Optimization is able to solve many commonly used problems from the evolutional computation literature in [Formula: see text] evaluations. We also introduce a more general class of Mk Landscapes that can be solved using dynamic programming and discuss properties of these functions. For certain type of problems Gray Box Optimization makes it possible to enumerate all local optima faster than brute force methods. We also provide evidence that randomly generated test problems are far less structured than those found in real-world problems.

A partition crossover operator is introduced for use with NK landscapes, MAX-kSAT and for all k-bounded pseudo-Boolean functions. By definition, these problems use a bit representation. Under partition crossover, the evaluation of offspring can be directly obtained from partial evaluations of substrings found in the parents. Partition crossover explores the variable interaction graph of the pseudo-Boolean functions in order to partition the variables of the solution vector. Proofs are presented showing that if the differing variable assignments found in the two parents can be partitioned into q non-interacting sets, partition crossover can be used to find the best of 2q possible offspring. Proofs are presented which show that parents that are locally optimal will always generate offspring that are locally optimal with respect to a (more restricted) hyperplane subspace. Empirical experiments show that parents that are locally optimal generate offspring that are locally optimal in the full search space more than 80 percent of the time. Experimental results also show the effectiveness of the proposed crossover when used in combination with a hybrid genetic algorithm.

Local optima networks are a recent model of fitness landscapes. They compress the landscape by representing local optima as nodes, and search transitions among them as edges. Previous local optima networks considered transitions based on mutation; this study looks instead at transitions based on deterministic recombination. We define and analyse networks based on the recently proposed partition crossover for k-bounded pseudo-Boolean functions, using NKq landscapes as a case study. Partition crossover was initially proposed for the travelling salesman problem, where it was found to ``tunnel" between local optima, i.e., jump from local optimum to local optimum. Our network analysis shows that this also happens for NK landscapes: local optima are densely connected via partition crossover. We found marked differences between the adjacent and random interaction NK models. Surprisingly, with the random model, instances have a lower number of local optima on average, but their networks are more sparse and decompose into several clusters. There is also large variability in the size and pattern of connectivity of instances coming from the same landscape parameter values. These network features offer new insight informing why some instances are harder to solve than others.

Bit-flip mutation is a common mutation operator for evolutionary algorithms applied to optimize functions over binary strings. In this paper, we develop results from the theory of landscapes and Krawtchouk polynomials to exactly compute the probability distribution of fitness values of a binary string undergoing uniform bit-flip mutation. We prove that this probability distribution can be expressed as a polynomial in p, the probability of flipping each bit. We analyze these polynomials and provide closed-form expressions for an easy linear problem (Onemax), and an NP-hard problem, MAX-SAT. We also discuss a connection of the results with runtime analysis.

Software Product Lines (SPLs) are families of related software products, each with its own set of feature combinations. Their commonly large number of products poses a unique set of challenges for software testing as it might not be technologically or economically feasible to test of all them individually. SPL pairwise testing aims at selecting a set of products to test such that all possible combinations of two features are covered by at least one selected product. Most approaches for SPL pairwise testing have focused on achieving full coverage of all pairwise feature combinations with the minimum number of products to test. Though useful in many contexts, this single-objective perspective does not reflect the prevailing scenario where software engineers do face trade-offs between the objectives of maximizing the coverage or minimizing the number of products to test. In contrast and to address this need, our work is the first to propose a classical multi-objective formalisation where both objectives are equally important. In this paper, we study the application to SPL pairwise testing of four classical multi-objective evolutionary algorithms. We developed three seeding strategies — techniques that leverage problem domain knowledge — and measured their performance impact on a large and diverse corpus of case studies using two well-known multi-objective quality measures. Our study identifies the performance differences among the algorithms and corroborates that the more domain knowledge leveraged the better the search results. Our findings enable software engineers to select not just one solution (as in the case of single-objective techniques) but instead to select from an array of test suite possibilities the one that best matches the economical and technological constraints of their testing context.

Hill climbing algorithms are at the core of many approaches to solve optimization problems. Such algorithms usually require the complete enumeration of a neighborhood of the current solution. In the case of problems defined over binary strings of length n, we define the r-ball neighborhood as the set of solutions at Hamming distance r or less from the current solution. For r ll n this neighborhood contains Θ(nr) solutions. In this paper efficient methods areintroduced to locate improving moves in the r-ball neighborhood for problems that can be written as a sum of a linear number of subfunctions depending on a bounded number of variables. NK-landscapes and MAX-kSAT are examples of these problems. If the number of subfunctions depending on any given variable is also bounded, then we prove that the method can explore the neighborhood in constant time, despite the fact that the number of solutions in the neighborhood is polynomial in n. We develop a hill climber based on our exploration method and we analyze its efficiency and efficacy using experiments with NKq-landscapes instances.

Software Product Lines (SPLs) are families of related software systems, which provide different feature combinations. Different SPL testing approaches have been proposed. However, despite the extensive and successful use of evolutionary computation techniques for software testing, their application to SPL testing remains largely unexplored. In this paper we present the Parallel Prioritized product line Genetic Solver (PPGS), a parallel genetic algorithm for the generation of prioritized pairwise testing suites for SPLs. We perform an extensive and comprehensive analysis of PPGS with 235 feature models from a wide range of number of features and products, using 3 different priority assignment schemes and 5 product prioritization selection strategies. We also compare PPGS with the greedy algorithm prioritized-ICPL. Our study reveals that overall PPGS obtains smaller covering arrays with an acceptable performance difference with prioritized-ICPL.

Lopez-Herrejon, R. E., Chicano F., Ferrer J., Egyed A., & Alba E. (2013). Multi-objective Optimal Test Suite Computation for Software Product Line Pairwise Testing. 2013 IEEE International Conference on Software Maintenance, Eindhoven, The Netherlands, September 22-28, 2013. 404–407.

The frequency distribution of a fitness function over regions of its domain is an important quantity for understanding the behavior of algorithms that employ randomized sampling to search the function. In general, exactly characterizing this distribution is at least as hard as the search problem, since the solutions typically live in the tails of the distribution. However, in some cases it is possible to efficiently retrieve a collection of quantities (called moments) that describe the distribution. In this paper, we consider functions of bounded epistasis that are defined over length-n strings from a finite alphabet of cardinality q. Many problems in combinatorial optimization can be specified as search problems over functions of this type. Employing Fourier analysis of functions over finite groups, we derive an efficient method for computing the exact moments of the frequency distribution of fitness functions over Hamming regions of the q-ary hypercube. We then use this approach to derive equations that describe the expected fitness of the offspring of any point undergoing uniform mutation. The results we present provide insight into the statistical structure of the fitness function for a number of combinatorial problems. For the graph coloring problem, we apply our results to efficiently compute the average number of constraint violations that lie within a certain number of steps of any coloring. We derive an expression for the mutation rate that maximizes the expected fitness of an offspring at each fitness level. We also apply the results to the slightly more complex frequency assignment problem, a relevant application in the domain of the telecommunications industry. As with the graph coloring problem, we provide formulas for the average value of the fitness function in Hamming regions around a solution and the expectation-optimal mutation rate.

Uniform crossover is a popular operator used in genetic algorithms to combine two tentative solutions of a problem represented as binary strings. We use the Walsh decomposition of pseudo-Boolean functions and properties of Krawtchouk matrices to exactly compute the expected value for the fitness of a child generated by uniform crossover from two parent solutions. We prove that this expectation is a polynomial in Á, the probability of selecting the best-parent bit. We provide efficient algorithms to compute this polynomial for ONEMAX and MAX-kSAT problems, but the results also hold for domains such as NK-Landscapes.

Combinatorial Interaction Testing (CIT) is a technique used to discover faults caused by parameter interactions in highly configurable systems. These systems tend to be large and exhaustive testing is generally impractical. Indeed, when the resources are limited, prioritization of test cases is a must. Important test cases are assigned a high priority and should be executed earlier. On the one hand, the prioritization of test cases may reveal faults in early stages of the testing phase. But, on the other hand the generation of minimal test suites that fulfill the demanded coverage criteria is an NP-hard problem. Therefore, search based approaches are required to find the (near) optimal test suites. In this work we present a novel evolutionary algorithm to deal with this problem. The experimental analysis compares five techniques on a set of benchmarks. It reveals that the evolutionary approach is clearly the best in our comparison. The presented algorithm can be integrated into a professional tool for CIT.

SUMMARY Automatic test data generation is a very popular domain in the field of search‐based software engineering. Traditionally, the main goal has been to maximize coverage. However, other objectives can be defined, such as the oracle cost, which is the cost of executing the entire test suite and the cost of checking the system behavior. Indeed, in very large software systems, the cost spent to test the system can be an issue, and then it makes sense by considering two conflicting objectives: maximizing the coverage and minimizing the oracle cost. This is what we did in this paper. We mainly compared two approaches to deal with the multi‐objective test data generation problem: a direct multi‐objective approach and a combination of a mono‐objective algorithm together with multi‐objective test case selection optimization. Concretely, in this work, we used four state‐of‐the‐art multi‐objective algorithms and two mono‐objective evolutionary algorithms followed by a multi‐objective test case selection based on Pareto efficiency. The experimental analysis compares these techniques on two different benchmarks. The first one is composed of 800 Java programs created through a program generator. The second benchmark is composed of 13 real programs extracted from the literature. In the direct multi‐objective approach, the results indicate that the oracle cost can be properly optimized; however, the full branch coverage of the system poses a great challenge. Regarding the mono‐objective algorithms, although they need a second phase of test case selection for reducing the oracle cost, they are very effective in maximizing the branch coverage. Copyright © 2011 John Wiley & Sons, Ltd.

Bit-flip mutation is a common operation when a genetic algorithm is applied to solve a problem with binary representation. We use in this paper some results of landscapes theory and Krawtchouk polynomials to exactly compute the expected value of the fitness of a mutated solution. We prove that this expectation is a polynomial in p, the probability of flipping a single bit. We analyze these polynomials and propose some applications of the obtained theoretical results.

The Software Project Scheduling (SPS) problem relates to the decision of who does what during a software project lifetime. This problem has a capital importance for software companies. In the SPS problem, the total budget and human resources involved in software development must be optimally managed in order to end up with a successful project. Companies are mainly concerned with reducing both the duration and the cost of the projects, and these two goals are in conflict with each other. A multi-objective approach is therefore the natural way of facing the SPS problem. In this paper, a number of multi-objective metaheuristics have been used to address this problem. They have been thoroughly compared over a set of 36 publicly available instances that cover a wide range of different scenarios. The resulting project schedulings of the algorithms have been analyzed in order to show their relevant features. The algorithms used in this paper and the analysis performed may assist project managers in the difficult task of deciding who does what in a software project.

The Software Project Scheduling (SPS) problem relates to the decision of who does what during a software project lifetime. This problem has a capital importance for software companies, where the total budget and human resources involved in software development must be managed optimally in order to end up with a successful project. Companies are mainly concerned with reducing both the duration and the cost of the projects, and these two goals are in conflict with each other. A multi-objective approach is therefore the natural way of facing the SPS problem and multi-objective metaheuristics have been used to solve the problem in the past. Nowadays, software projects faced by the large companies are increasing in size and we need algorithms that are able to deal with the new large instances of the SPS problem. In this paper we analyze the scalability of four multi-objective algorithms when they are applied to the SPS problem using instances of increasing size. The algorithms are a genetic algorithm (NSGA-II), an evolution strategy (PAES), a differential evolution (DEPT) and a firefly algorithm (MO-FA). The results suggest that PAES is the algorithm with the best scalability behaviour.

A small number of combinatorial optimization problems have search spaces that correspond to elementary landscapes, where the objective function f is an eigenfunction of the Laplacian that describes the neighborhood structure of the search space. Many problems are not elementary; however, the objective function of a combinatorial optimization problem can always be expressed as a superposition of multiple elementary landscapes if the underlying neighborhood used is symmetric. This paper presents theoretical results that provide the foundation for algebraic methods that can be used to decompose the objective function of an arbitrary combinatorial optimization problem into a sum of subfunctions, where each subfunction is an elementary landscape. Many steps of this process can be automated, and indeed a software tool could be developed that assists the researcher in finding a landscape decomposition. This methodology is then used to show that the subset sum problem is a superposition of two elementary landscapes, and to show that the quadratic assignment problem is a superposition of three elementary landscapes.

The Quadratic Assignment Problem (QAP) is a well-known NP-hard combinatorial optimization problem that is at the core of many real-world optimization problems. We prove that QAP can be written as the sum of three elementary landscapes when the swap neighborhood is used. We present a closed formula for each of the three elementary components and we compute bounds for the autocorrelation coefficient.

We analyze various forms of the Frequency Assignment Problem using the theory of elementary landscapes. We show that three variants of the Frequency Assignment Problem are either directly an Elementary Landscape, or are a superposition of two Elementary Landscapes. We also examine the computability of neighborhood averages for partial neighborhoods.

Most of the software developed in the world follows the object-oriented (OO) paradigm. However, the existing work on evolutionary testing is mainly targeted to procedural languages. All this work can be used with small changes on OO programs, but object orientation introduces new features that are not present in procedural languages. Some important issues are polymorphism and inheritance. In this paper we want to make a contribution to the inheritance field by proposing some approaches that use the information of the class hierarchy for helping test case generators to better guide the search. To the best of our knowledge, no work exists using this information to propose test cases. In this work we define a branch distance for logical expressions containing the instanceof operator in Java programs. In addition to the distance measure, we propose two mutation operators based on the distance. We study the behaviour of the mutation operators on a benchmark set composed of nine OO programs. The results show that the information collected from the class hierarchy helps in the search for test cases.

Model checking is a well-known and fully automatic technique for checking software properties, usually given as temporal logic formulae on the program variables. Most of model checkers found in the literature use exact deterministic algorithms to check the properties. These algorithms usually require huge amounts of memory if the checked model is large. We propose here the use of an algorithm based on ACOhg, a new kind of ant colony optimization model, to search for liveness property violations in concurrent systems. This algorithm has been previously applied to the search for safety errors with very good results and we apply it here for the first time to liveness errors. The results state that our algorithmic proposal, called ACOhg-live, is able to obtain very short error trails in faulty concurrent systems using a low amount of resources, outperforming by far the results of nested-DFS, the traditional algorithm used for this task in the model checking community and implemented in most of the explicit state model checkers. This fact makes ACOhg-live a very suitable algorithm for finding liveness errors in large faulty concurrent systems, in which traditional techniques fail because of the model size.

Liveness properties in concurrent systems are, informally, those properties that stipulate that something good eventually happens during execution. In order to prove that a given system satisfies a liveness property, model checking techniques are utilized. However, most of the model checkers found in the literature use exhaustive deterministic algorithms that require huge amounts of memory if the concurrent system is large. Here we propose the use of an algorithm based on ACOhg, a new kind of Ant Colony Optimization algorithm, for searching for liveness property violations in concurrent systems. We also take into account the structure of the liveness property in order to improve the efficacy and efficiency of the search. The results state that our algorithmic proposal, called ACOhg-live, is able to obtain very short error trails in faulty concurrent systems using a low amount of resources, outperforming by far the results of Nested-DFS and Improved-Nested-DFS, two algorithms used in the literature for this task in the model checking community. This fact makes ACOhg-live a very suitable algorithm for finding liveness errors in large faulty concurrent systems, in which traditional techniques fail because of the model size.

Model checking is a fully automatic technique for check-ing concurrent software properties in which the states of a concurrent system are explored in an explicit or implicit way. However, the state explosion problem limits the size of the models that are possible to check. Genetic Algorithms (GAs) are metaheuristic techniques that have obtained good results in problems in which exhaustive techniques fail due to the size of the search space. Unlike exact techniques, metaheuristic techniques can not be used to verify that a program satisfies a given property, but they can find errors on the software using a lower amount of resources than exact techniques. In this paper, we compare a GA against clas-sical exact techniques and we propose a new operator for this problem, called memory operator, that allows the GA to explore even larger search spaces. We implemented our ideas in the Java Pathfinder (JPF) model checker to validate them and present our results. To the best of our knowledge, this is the first implementation of a Genetic Algorithm in this model checker.

Ant Colony Optimization (ACO) has been successfully applied to those combinatorial optimization problems which can be translated into a graph exploration. Artificial ants build solutions step by step adding solution components that are represented by graph nodes. The existing ACO algorithms are suitable when the graph is not very large (thousands of nodes) but is not useful when the graph size can be a challenge for the computer memory and cannot be completely generated or stored in it. In this paper we study a new ACO model that overcomes the difficulties found when working with a huge construction graph. In addition to the description of the model, we analyze in the experimental section one technique used for dealing with this huge graph exploration. The results of the analysis can help to understand the meaning of the new parameters introduced and to decide which parameterization is more suitable for a given problem. For the experiments we use one real problem with capital importance in Software Engineering: refutation of safety properties in concurrent systems. This way, we foster an innovative research line related to the application of ACO to formal methods in Software Engineering.

Model Checking is a well-known and fully automatic technique forchecking software properties, usually given as temporal logicformulae on the program variables. Most model checkers found inthe literature use exact deterministic algorithms to check theproperties. These algorithms usually require huge amounts ofcomputational resources if the checked model is large. We proposehere the use of a new kind of Ant Colony Optimization (ACO) model, ACOhg, to refute safety properties in concurrent systems. ACO algorithms are stochastic techniques belonging to the class of metaheuristic algorithms and inspired by the foraging behaviour of real ants. The traditional ACO algorithms cannot deal with the model checking problem and thus we use ACOhg to tackle it. The results state that ACOhg algorithms find optimal or near optimal error trails in faulty concurrent systems with a reduced amount of resources, outperforming algorithms that are the state-of-the-art in model checking. This fact makes them suitable for checking safety properties in large concurrent systems, in which traditional techniques fail to find errors because of the model size.

No abstract available.

Radio network design (RND) is a fundamental problem in cellular networks for telecommunications. In these networks, the terrain must be covered by a set of base stations (or antennae), each of which defines a covered area called cell. The problem may be reduced to figure out the optimal placement of antennae out of a list of candidate sites trying to satisfy two objectives: to maximize the area covered by the radio signal and to reduce the number of used antennae. Consequently, RND is a bi-objective optimization problem. Previous works have solved the problem by using single-objective techniques which combine the values of both objectives. The used techniques have allowed to find optimal solutions according to the defined objective, thus yielding a unique solution instead of the set of Pareto optimal solutions. In this paper, we solve the RND problem using a multi-objective version of the algorithm CHC, which is the metaheuristic having reported the best results when solving the single-objective formulation of RND. This new algorithm, called MOCHC, is compared against a binary-coded NSGA-II algorithm and also against the provided results in the literature. Our experiments indicate that MOCHC outperfoms NSGA-II and, more importantly, it is more efficient finding the optimal solutions than single-objectives techniques.