Together with the NSGA-II, the SPEA2 is one of the most widely used domination-based multi-objective evolutionary algorithms. For both algorithms, the known runtime guarantees are linear in the population size; for the NSGA-II, matching lower bounds exist. With a careful study of the more complex selection mechanism of the SPEA2, we show that it has very different population dynamics. From these, we prove runtime guarantees for the OneMinMax, LeadingOnesTrailingZeros, and OneJumpZeroJump benchmarks that depend less on the population size. For example, we show that the SPEA2 with parent population size mu >= n - 2k + 3 and offspring population size lambda computes the Pareto front of the OneJumpZeroJump benchmark with gap size k in an expected number of O((lambda+mu)n + n^(k+1)) function evaluations. This shows that the best runtime guarantee of O(n^(k+1)) is not only achieved for mu = Theta(n) and lambda = O(n) but for arbitrary mu, lambda = O(n^k). Thus, choosing suitable parameters - a key challenge in using heuristic algorithms - is much easier for the SPEA2 than the NSGA-II.

The compact genetic algorithm (cGA) is one of the simplest estimation-of-distribution algorithms (EDAs). Next to the univariate marginal distribution algorithm (UMDA)– another simple EDA–, the cGA has been subject to extensive mathematical runtime analyses, often showcasing a similar or even superior performance to competing approaches. Surprisingly though, up to date and in contrast to the UMDA and many other heuristics, we lack a rigorous runtime analysis of the cGA on the LEADINGONES benchmark–one of the most studied theory benchmarks in the domain of evolutionary computation. We fill this gap in the literature by conducting a formal runtime analysis of the cGA on LEADINGONES. For the cGA’s single parameter–called the hypothetical population size–at least polylogarithmically larger than the problem size, we prove that the cGA samples the optimum of LEADINGONES with high probability within a number of function evaluations quasi-linear in the problem size and linear in the hypothetical population size. For the best hypothetical population size, our result matches, up to polylogarithmic factors, the typical quadratic runtime that many randomized search heuristics exhibit on LEADINGONES. Our analysis exhibits some noteworthy differences in the working principles of the two algorithms which were not visible in previous works.

The NSGA-II is the most prominent multi-objective evolutionary algorithm (cited more than 50,000 times). Very recently, a mathematical runtime analysis has proven that this algorithm can have enormous difficulties when the number of objectives is larger than two (Zheng, Doerr. IEEE Transactions on Evolutionary Computation (2024)). However, this result was shown only for the OneMinMax benchmark problem, which has the particularity that all solutions are on the Pareto front, a fact heavily exploited in the proof of this result. In this work, we show a comparable result for the LeadingOnesTrailingZeroes benchmark. This popular benchmark problem appears more natural in that most of its solutions are not on the Pareto front. With a careful analysis of the population dynamics of the NGSA-II optimizing this benchmark, we manage to show that when the population grows on the Pareto front, then it does so much faster by creating known Pareto optima than by spreading out on the Pareto front. Consequently, already when still a constant fraction of the Pareto front is unexplored, the crowding distance becomes the crucial selection mechanism, and thus the same problems arise as in the optimization of OneMinMax. With these and some further arguments, we show that the NSGA-II, with a population size by at most a constant factor larger than the Pareto front, cannot compute the Pareto front in less than exponential time.

The non-dominated sorting genetic algorithm II (NSGA-II) is the most popular multi-objective optimization heuristic. Recent mathematical runtime analyses have detected two shortcomings in discrete search spaces, namely, that the NSGA-II has difficulties with more than two objectives and that it is very sensitive to the choice of the population size. To overcome these difficulties, we analyze a simple tie-breaking rule in the selection of the next population. Similar rules have been proposed before, but have found only little acceptance. We prove the effectiveness of our tie-breaking rule via mathematical runtime analyses on the classic OneMinMax, LeadingOnesTrailingZeros, and OneJumpZeroJump benchmarks. We prove that this modified NSGA-II can optimize the three benchmarks efficiently also for many objectives, in contrast to the exponential lower runtime bound previously shown for OneMinMax with three or more objectives. For the bi-objective problems, we show runtime guarantees that do not increase when moderately increasing the population size over the minimum admissible size. For example, for the OneJumpZeroJump problem with representation length n and gap parameter k, we show a runtime guarantee of O(max {n^(k + 1), N n}) function evaluations when the population size is at least four times the size of the Pareto front. For population sizes larger than the minimal choice N = Θ(n), this result improves considerably over the Θ(N n^k) runtime of the classic NSGA-II.

Randomized search heuristics have been applied successfully to a plethora of problems. This success is complemented by a large body of theoretical results. Unfortunately, the vast majority of these results regard problems with binary or continuous decision variables -- the theoretical analysis of randomized search heuristics for unbounded integer domains is almost nonexistent. To resolve this shortcoming, we start the runtime analysis of multi-objective evolutionary algorithms, which are among the most successful randomized search heuristics, for unbounded integer search spaces. We analyze single- and full-dimensional mutation operators with three different mutation strengths, namely changes by plus/minus one (unit strength), random changes following a law with exponential tails, and random changes following a power-law. The performance guarantees we prove on a recently proposed natural benchmark problem suggest that unit mutation strengths can be slow when the initial solutions are far from the Pareto front. When setting the expected change right (depending on the benchmark parameter and the distance of the initial solutions), the mutation strength with exponential tails yields the best runtime guarantees in our results -- however, with a wrong choice of this expectation, the performance guarantees quickly become highly uninteresting. With power-law mutation, which is an essentially parameter-less mutation operator, we obtain good results uniformly over all problem parameters and starting points. We complement our mathematical findings with experimental results that suggest that our bounds are not always tight. Most prominently, our experiments indicate that power-law mutation outperforms the one with exponential tails even when the latter uses a near-optimal parametrization. Hence, we suggest to favor power-law mutation for unknown problems in integer spaces.

Populations play a key role in the area of evolutionary computation to tackle complex optimization problems. Nevertheless, it is hard to understand the underlying population dynamics from a theoretical perspective, and only a limited number of theoretical results for population-based algorithms are available even for simple benchmark functions. In this paper, we study the classic (μ+1) EA on the benchmark problem BinVal, which allows for exponentially many function values. Previous methods for the analysis, based on fitness levels and multiplicative drift analysis, lead to runtime bounds for this function of size n that include an additive term of Θ(n2). We provide new insights into how this standard algorithm optimizes BinVal, and we provide runtime bounds that are polynomial in the population size μ and do not include this additive term. In particular, we prove bounds on the expected runtime that are O(μ5n log (n/μ4)) for standard bit mutation, which is O(n log n) for constant μ. Our analysis considers the population dynamics of the (μ+1) EA more closely, proving that copies created by mutation lead to a low diversity in short blocks of bits across all individuals. We extend this method to mutation operators that cannot create duplicates, and prove bounds similar to standard bit mutation.

Diffusion processes pervade numerous areas of AI, abstractly modeling the dynamics of exchanging, oftentimes volatile, information in networks. A central question is how long the information remains in the network, known as survival time. For the commonly studied SIS process, the expected survival time is at least super-polynomial in the network size already on star graphs, for a wide range of parameters. In contrast, the expected survival time of the SIRS process, which introduces temporary immunity, is always at most polynomial on stars and only known to be super-polynomial for far denser networks, such as expanders. However, this result relies on featuring full temporary immunity, which is not always present in actual processes. We introduce the cSIRS process, which incorporates gradually declining immunity such that the expected immunity at each point in time is identical to that of the SIRS process. We study the survival time of the cSIRS process rigorously on star graphs and expanders and show that its expected survival time is very similar to that of the SIS process, which features no immunity. This suggests that featuring gradually declining immunity is almost as having none at all.

Together with the NSGA-II and SMS-EMOA, the strength Pareto evolutionary algorithm 2 (SPEA2) is one of the most prominent dominance-based multi-objective evolutionary algorithms (MOEAs). Different from the NSGA-II, it does not employ the crowding distance (essentially the distance to neighboring solutions) to compare pairwise non-dominating solutions but a complex system of σ-distances that builds on the distances to all other solutions. In this work, we give a first mathematical proof showing that this more complex system of distances can be superior. More specifically, we prove that a simple steady-state SPEA2 can compute optimal approximations of the Pareto front of the OneMinMax benchmark in polynomial time. The best proven guarantee for a comparable variant of the NSGA-II only assures approximation ratios of roughly a factor of two, and both mathematical analyses and experiments indicate that optimal approximations are not found efficiently.

The global simple evolutionary multi-objective optimizer (GSEMO) is a simple, yet often effective multi-objective evolutionary algorithm (MOEA). By only maintaining non-dominated solutions, it has a variable population size that automatically adjusts to the needs of the optimization process. The downside of the dynamic population size is that the population dynamics of this algorithm are harder to understand, resulting, e.g., in the fact that only sporadic tight runtime analyses exist. In this work, we significantly enhance our understanding of the dynamics of the GSEMO, in particular, for the classic CountingOnesCountingZeros (COCZ) benchmark. From this, we prove a lower bound of order Ω(n² log n), for the first time matching the seminal upper bounds known for over twenty years. We also show that the GSEMO finds any constant fraction of the Pareto front in time O(n²), improving over the previous estimate of O(n² log n) for the time to find the first Pareto optimum. Our methods extend to other classic benchmarks and yield, e.g., the first Ω(n^(k+1)) lower bound for the OJZJ benchmark in the case that the gap parameter is k ∈ {2,3}. We are therefore optimistic that our new methods will be useful in future mathematical analyses of MOEAs.

BACKGROUND: Bivariate monotonic classifiers (BMCs) are based on pairs of input features. Like many other models used for machine learning, they can capture nonlinear patterns in high-dimensional data. At the same time, they are simple and easy to interpret. Until now, the use of BMCs on a genome scale was hampered by the high computational complexity of the search for pairs of features with a high leave-one-out performance estimate. RESULTS: We introduce the fastBMC algorithm, which drastically speeds up the identification of BMCs. The algorithm is based on a mathematical bound for the BMC performance estimate while maintaining optimality. We show empirically that fastBMC speeds up the computation by a factor of at least 15 already for a small number of features, compared to the traditional approach. For two of the three smaller biomedical datasets that we consider here, the resulting possibility of considering much larger sets of features translates into significantly improved classification performance. As an example of the high degree of interpretability of BMCs, we discuss a straightforward interpretation of a BMC glioblastoma survival predictor, an immediate novel biomedical hypothesis, options for biomedical validation, and treatment implications. In addition, we study the performance of fastBMC on a larger and well-known breast cancer dataset, validating the benefits of the BMCs for biomarker identification and biomedical hypothesis generation. CONCLUSION: fastBMC enables the rapid construction of robust and interpretable ensemble models using BMC, facilitating the discovery of gene pairs predictive of relevant phenotypes and their interaction in that context. AVAILABILITY: We provide the first open-source implementation for learning BMCs, a Python implementation of fastBMC in particular, and Python code to reproduce the fastBMC results on real and simulated data in this paper, at https://github.com/oceanefrqt/fastBMC .

Information diffusion models on networks are at the forefront of AI research. The dynamics of such models typically follow stochastic models from epidemiology, used to model not only infections but various phenomena, including the behavior of computer viruses and viral marketing campaigns. A core question in this setting is how to efficiently detect the most influential vertices in the host graph such that the infection survives the longest. In processes that incorporate re-infection of the vertices, such as the SIS process, theoretical studies identify parameter thresholds where the survival time of the process rapidly transitions from logarithmic to super-polynomial. These results contradict the intuition that the starting configuration is relevant, since the process will always either die out fast or survive almost indefinitely. A shortcoming of these results is that models incorporating short-term immunity (or creative advertisement fatigue) have not been subjected to such a theoretical analysis so far. We reduce this gap in the literature by studying the SIRS process, a more realistic model, which besides re-infection additionally incorporates short-term immunity. On complex network models, we identify parameter regimes for which the process survives exponentially long, and we get a tight threshold for random graphs. Underlying these results is our main technical contribution, showing a threshold behavior for the survival time of the SIRS process on graphs with large expander subgraphs, such as social network models.

International audience

The target set selection problem (TSS) asks for a set of vertices such that an influence spreading process started in these vertices reaches the whole graph. The current state of the art for this NP-hard problem are three recently proposed randomized search heuristics, namely a biased random-key genetic algorithm (BRKGA) obtained from extensive parameter tuning, a max-min ant system (MMAS), and a MMAS using Q-learning with a graph convolutional network.

We propose a new, flexible approach for dynamically maintaining successful mutation rates in evolutionary algorithms using k-bit flip mutations. The algorithm adds successful mutation rates to an archive of promising rates that are favored in subsequent steps. Rates expire when their number of unsuccessful trials has exceeded a threshold, while rates currently not present in the archive can enter it in two ways: (i) via user-defined minimum selection probabilities for rates combined with a successful step or (ii) via a stagnation detection mechanism increasing the value for a promising rate after the current bit-flip neighborhood has been explored with high probability. For the minimum selection probabilities, we suggest different options, including heavy-tailed distributions.

The majority of research on estimation-of-distribution algorithms (EDAs) concentrates on pseudo-Boolean optimization and permutation problems, leaving the domain of EDAs for problems in which the decision variables can take more than two values, but which are not permutation problems, mostly unexplored. To render this domain more accessible, we propose a natural way to extend the known univariate EDAs to this setting. Different from a naïve reduction to the binary case, our approach avoids additional constraints. Since understanding genetic drift is crucial for an optimal parameter choice, we extend the known quantitative analysis of genetic drift to EDAs for multi-valued, categorical variables. Roughly speaking, when the variables take r different values, the time for genetic drift to become significant is r times shorter than in the binary case. Consequently, the update strength of the probabilistic model has to be chosen r times lower now. To investigate how desired model updates take place in this framework, we undertake a mathematical runtime analysis on the r-valued LeadingOnes problem. We prove that with the right parameters, the multi-valued UMDA solves this problem efficiently in O(rln⁡(r)2n2ln⁡(n)) function evaluations. This bound is nearly tight as our lower bound Ω(rln⁡(r)n2ln⁡(n)) shows. Overall, our work shows that our good understanding of binary EDAs naturally extends to the multi-valued setting, and it gives advice on how to set the main parameters of multi-values EDAs.

With apparently all research on estimation-of-distribution algorithms (EDAs) concentrated on pseudo-Boolean optimization and permutation problems, we undertake the first steps towards using EDAs for problems in which the decision variables can take more than two values, but which are not permutation problems. To this aim, we propose a natural way to extend the known univariate EDAs to such variables. Different from a naïve reduction to the binary case, it avoids additional constraints.

Traffic congestion is a major issue that can be solved by suggesting drivers alternative routes they are willing to take. This concept has been formalized as a strategic routing problem in which a single alternative route is suggested to an existing one. We extend this formalization and introduce the multiple-routes (MRs) problem, which is given a start and destination and aims at finding up to$n$different routes that the drivers strategically disperse over, minimizing the overall travel time of the system. Due to theNP-hard nature of the problem, we introduce the MRs evolutionary algorithm (MREA) as a heuristic solver. We study several mutation and crossover operators and evaluate them on real-world data of Berlin, Germany. We find that a combination of all operators yields the best result, reducing the overall travel time by a factor between 1.8 and 3, in the median, compared to all drivers taking the fastest route. 6mm]Please cite reference[2]in the text of the paper. It was removed from the abstract as having reference in an abstract is contrary to IEEE journal style.For the base case$n=2$, we compare our MREA to the highly tailored optimal solver by Bläsius et al. (2020), and show that, in the median, our approach finds solutions of quality at least 99.69% of an optimal solution while only requiring 40% of the time.

Run time analysis of evolutionary algorithms recently makes significant progress in linking algorithm performance to algorithm parameters. However, settings that study the impact of problem parameters are rare. The recently proposed W-model provides a good framework for such analyses, generating pseudo-Boolean optimization problems with tunable properties. We initiate theoretical research of the W-model by studying how one of its properties—neutrality—influences the run time of random local search (RLS). Neutrality creates plateaus in the search space by first performing a majority vote for subsets of the solution candidate and then evaluating the smaller dimensional string via a low-level fitness function. We prove upper bounds for the expected run time of RLS on this$\mathrm{M{\scriptstyle AJORITY}}$problem for its entire parameter spectrum. To this end, we provide a theorem, applicable to many optimization algorithms, that links the run time of$\mathrm{M{\scriptstyle AJORITY}}$with its symmetric version$\mathrm{H{\scriptstyle AS}M{\scriptstyle AJORITY}}$, where a sufficient majority is needed to optimize the subset. We also introduce a generalized version of classic drift theorems as well as a generalized version of Wald's equation, both of which we believe to be of independent interest.

It has long been observed that the performance of evolutionary algorithms and other randomized search heuristics can benefit from a non-static choice of the parameters that steer their optimization behavior. Mechanisms that identify suitable configurations on the fly ("parameter control") or via a dedicated training process ("dynamic algorithm configuration") are thus an important component of modern evolutionary computation frameworks. Several approaches to address the dynamic parameter setting problem exist, but we barely understand which ones to prefer for which applications. As in classical benchmarking, problem collections with a known ground truth can offer very meaningful insights in this context. Unfortunately, settings with well-understood control policies are very rare.

The hard-sphere model is one of the most extensively studied models in statistical physics. It describes the continuous distribution of spherical particles, governed by hard-core interactions. An important quantity of this model is the normalizing factor of this distribution, called the partition function. We propose a Markov chain Monte Carlo algorithm for approximating the grand canonical partition function of the hard-sphere model in $d$ dimensions. Up to a fugacity of $\lambda < e/2^d$, the runtime of our algorithm is polynomial in the volume of the system. Key to our approach is to define a discretization that closely approximates the partition function of the continuous model. This results in a discrete hard-core instance that is exponential in the size of the initial hard-sphere model. Our approximation bound follows directly from the correlation decay threshold of an infinite regular tree with degree equal to the maximum degree of our discretization. To cope with the exponential blow-up of the discrete instance, we use block dynamics, a Markov chain that generalizes the more frequently studied Glauber dynamics by grouping the vertices of the graph into blocks and updating an entire block instead of a single vertex in each step. We prove rapid mixing of block dynamics, based on disjoint cliques as blocks, up to the tree threshold of the univariate hard-core model. This is achieved by adapting the spectral expansion method, which was recently used for bounding the mixing time of Glauber dynamics within the same parameter regime.

Traffic congestion is a major issue that can be solved by suggesting drivers alternative routes they are willing to take. This concept has been formalized as a strategic routing problem in which a single alternative route is suggested to an existing one. We extend this formalization and introduce the Multiple-Routes problem, which is given a start and destination and aims at finding up to n different routes that the drivers strategically disperse over, minimizing the overall travel time of the system.

In their recent work, Lehre and Nguyen (2019) show that the univariate marginal distribution algorithm (UMDA) needs time exponential in the parent populations size to optimize the DeceptiveLeadingBlocks (DLB) problem. They conclude from this result that univariate EDAs have difficulties with deception and epistasis. In this work, we show that this negative finding is caused by the choice of the parameters of the UMDA. When the population sizes are chosen large enough to prevent genetic drift, then the UMDA optimizes the DLB problem with high probability with at most λ(n2+2elnn) fitness evaluations. Since an offspring population size λ of order nlogn can prevent genetic drift, the UMDA can solve the DLB problem with O(n2logn) fitness evaluations. In contrast, for classic evolutionary algorithms no better runtime guarantee than O(n3) is known (which we prove to be tight for the (1+1) EA), so our result rather suggests that the UMDA can cope well with deception and epistatis. From a broader perspective, our result shows that the UMDA can cope better with local optima than many classic evolutionary algorithms; such a result was previously known only for the compact genetic algorithm. Together with the lower bound of Lehre and Nguyen, our result for the first time rigorously proves that running EDAs in the regime with genetic drift can lead to drastic performance losses.

Finding a large set of optima in a multimodal optimization landscape is a challenging task. Classical population-based evolutionary algorithms (EAs) typically converge only to a single solution. While this can be counteracted by applying niching strategies, the number of optima is nonetheless trivially bounded by the population size.

Estimation-of-distribution algorithms (EDAs) are randomized search heuristics that create a probabilistic model of the solution space, which is updated iteratively, based on the quality of the solutions sampled according to the model. As previous works show, this iteration-based perspective can lead to erratic updates of the model, in particular, to bit-frequencies approaching a random boundary value. In order to overcome this problem, we propose a new EDA based on the classic compact genetic algorithm (cGA) that takes into account a longer history of samples and updates its model only with respect to information which it classifies as statistically significant. We prove that this significance-based cGA (sig-cGA) optimizes the commonly regarded benchmark functions OneMax (OM), LeadingOnes, and BinVal all in quasilinear time, a result shown for no other EDA or evolutionary algorithm so far. For the recently proposed stable compact genetic algorithm - an EDA that tries to prevent erratic model updates by imposing a bias to the uniformly distributed model - we prove that it optimizes OM only in a time exponential in its hypothetical population size. Similarly, we show that the convex search algorithm cannot optimize OM in polynomial time.

Estimation-of-distribution algorithms (EDAs) are randomized search heuristics that maintain a stochastic model of the solution space. This model is updated from iteration to iteration based on the quality of the solutions sampled according to the model. As previous works show, this short-term perspective can lead to erratic updates of the model, in particular, to bit-frequencies approaching a random boundary value. This can lead to significant performance losses.

Modern routing algorithms reduce query time by depending heavily on preprocessed data. The recently developed Navigation Data Standard (NDS) enforces a separation between algorithms and map data, rendering preprocessing inapplicable. Furthermore, map data is partitioned into tiles with respect to their geographic coordinates. With the limited memory found in portable devices, the number of tiles loaded becomes the major factor for run time. We study routing under these restrictions and present new algorithms as well as empirical evaluations. Our results show that, on average, the most efficient algorithm presented uses more than 20 times fewer tile loads than a normal A.

Population diversity is essential for avoiding premature convergence in genetic algorithms (GAs) and for the effective use of crossover. Yet the dynamics of how diversity emerges in populations are not well understood. We use rigorous runtime analysis to gain insight into population dynamics and GA performance for the (μ + 1) GA and the Jump test function. We show that the interplay of crossover followed by mutation may serve as a catalyst leading to a sudden burst of diversity. This leads to significant improvements of the expected optimization time compared to mutation-only algorithms like the (1 + 1) evolutionary algorithm. Moreover, increasing the mutation rate by an arbitrarily small constant factor can facilitate the generation of diversity, leading to even larger speedups. Experiments were conducted to complement our theoretical findings and further highlight the benefits of crossover on the function class.

The Univariate Marginal Distribution Algorithm (UMDA), a popular estimation of distribution algorithm, is studied from a run time perspective. On the classical OneMax benchmark function, a lower bound of Ω(μ√n + n log n), where μ is the population size, on its expected run time is proved. This is the first direct lower bound on the run time of the UMDA. It is stronger than the bounds that follow from general black-box complexity theory and is matched by the run time of many evolutionary algorithms. The results are obtained through advanced analyses of the stochastic change of the frequencies of bit values maintained by the algorithm, including carefully designed potential functions. These techniques may prove useful in advancing the field of run time analysis for estimation of distribution algorithms in general.

Practical optimization problems frequently include uncertainty about the quality measure, for example, due to noisy evaluations. Thus, they do not allow for a straightforward application of traditional optimization techniques. In these settings, randomized search heuristics such as evolutionary algorithms are a popular choice because they are often assumed to exhibit some kind of resistance to noise. Empirical evidence suggests that some algorithms, such as estimation of distribution algorithms (EDAs) are robust against a scaling of the noise intensity, even without resorting to explicit noise-handling techniques such as resampling. In this paper, we want to support such claims with mathematical rigor. We introduce the concept of graceful scaling in which the run time of an algorithm scales polynomially with noise intensity. We study a monotone fitness function over binary strings with additive noise taken from a Gaussian distribution. We show that myopic heuristics cannot efficiently optimize the function under arbitrarily intense noise without any explicit noise-handling. Furthermore, we prove that using a population does not help. Finally, we show that a simple EDA called the compact genetic algorithm can overcome the shortsightedness of mutation-only heuristics to scale gracefully with noise. We conjecture that recombinative genetic algorithms also have this property.

An estimated 30% of urban traffic is caused by search for parking spots [Shoup, 2005]. Suggesting routes along highly probable parking spots could reduce traffic. In this paper, we formalize parking search as a probabilistic problem on a road graph and show that it is NP-complete. We explore heuristics that optimize for the driving duration and the walking distance to the destination. Routes are constrained to reach a certain probability threshold of finding a spot. Empirically estimated probabilities of successful parking attempts are provided by TomTom on a per-street basis. We release these probabilities as a dataset of about 80,000 roads covering the Berlin area. This allows to evaluate parking search algorithms on a real road network with realistic probabilities for the first time. However, for many other areas, parking probabilities are not openly available. Because they are effortful to collect, we propose an algorithm that relies on conventional road attributes only. Our experiments show that this algorithm comes close to the baseline by a factor of 1.3 in our cost measure. This leads to the conclusion that conventional road attributes may be sufficient to compute reasonably good parking search routes.

Population diversity is essential for the effective use of any crossover operator. We compare seven commonly used diversity mechanisms and prove rigorous run time bounds for the (μ+1) GA using uniform crossover on the fitness function Jumpk. All previous results in this context only hold for unrealistically low crossover probability pc=O(k/n), while we give analyses for the setting of constant pc < 1 in all but one case. Our bounds show a dependence on the problem size~$n$, the jump length k, the population size μ, and the crossover probability pc. For the typical case of constant k > 2 and constant pc, we can compare the resulting expected optimisation times for different diversity mechanisms assuming an optimal choice of μ:

Estimation of Distribution Algorithms (EDAs) work by iteratively updating a distribution over the search space with the help of samples from each iteration. Up to now, theoretical analyses of EDAs are scarce and present run time results for specific EDAs. We propose a new framework for EDAs that captures the idea of several known optimizers, including PBIL, UMDA, λ -MMASIB, cGA, and (1, λ)-EA. Our focus is on analyzing two core features of EDAs: a balanced EDA is sensitive to signals in the fitness; a stable EDA remains uncommitted under a biasless fitness function. We prove that no EDA can be both balanced and stable.

Different works have shown how crossover can help with building block assembly. Typically, crossover might get lucky to select good building blocks from each parent, but these lucky choices are usually rare. In this work we consider a crossover operator which works on three parent individuals. In each component, the offspring inherits the value present in the majority of the parents; thus, we call this crossover operator majority vote. We show that, if good components are sufficiently prevalent in the individuals, majority vote creates an optimal individual with high probability. Furthermore, we show that this process can be amplified: as long as components are good independently and with probability at least 1/2+δ, we require only O(log 1/δ + log log n) successive stages of majority vote to create an optimal individual with high probability!

Recently, ant colony optimization (ACO) algorithms have proven to be efficient in uncertain environments, such as noisy or dynamically changing fitness functions. Most of these analyses have focused on combinatorial problems such as path finding. We rigorously analyze an ACO algorithm optimizing linear pseudo-Boolean functions under additive posterior noise. We study noise distributions whose tails decay exponentially fast, including the classical case of additive Gaussian noise. Without noise, the classical [Formula: see text] EA outperforms any ACO algorithm, with smaller [Formula: see text] being better; however, in the case of large noise, the [Formula: see text] EA fails, even for high values of [Formula: see text] (which are known to help against small noise). In this article, we show that ACO is able to deal with arbitrarily large noise in a graceful manner; that is, as long as the evaporation factor [Formula: see text] is small enough, dependent on the variance [Formula: see text] of the noise and the dimension n of the search space, optimization will be successful. We also briefly consider the case of prior noise and prove that ACO can also efficiently optimize linear functions under this noise model.

Recently Ant Colony Optimization (ACO) algorithms have been proven to be efficient in uncertain environments, such as noisy or dynamically changing fitness functions. Most of these analyses focus on combinatorial problems, such as path finding.