Many optimization algorithms represent solutions as permutations. However, despite their apparent simplicity, permutations pose significant challenges—especially for Global Random Search (GRS) algorithms—due to the mutual-exclusivity constraint. This constraint complicates both the learning and sampling of probability distributions over the permutation space, often leading to computationally expensive procedures. A promising alternative involves transforming permutation-encoded solutions into integer vectors using bijective functions on the symmetric group Sn, resulting in what are known as inversion vectors. While inversion vectors have been studied for centuries, a unified and formal framework encompassing all their codifications has been lacking. In this paper, we introduce precise definitions and a unified notation for various types of inversion vector codifications. We establish bijective transformations between them, providing a formal characterization of their relationships and properties. Leveraging this theoretical foundation, we analyze and explain the behavior of GRS algorithms across different permutation problems when using different inversion vector representations.

Probability-based algorithms have proven to be a solid alternative for approaching optimization problems. Nevertheless, in many cases, using probabilistic models that efficiently exploit the characteristics of the problem involves large computational overheads, and therefore, lower complexity models such as those that are univariate are usually employed within approximation algorithms. With the motivation to address such an issue, in this article, we aim to introduce an iterative optimization framework that employs generative models to efficiently estimate the parameters of probability models for optimization problems. This allows the use of complex probabilistic models (or those that are appropriate for each problem) in a way that is feasible to apply them iteratively. Specifically, the framework is composed of three elements: a generative model, a probability model whose probability rule is differentiable, and a loss function. The possibility of modifying any of the three elements of the framework offers the flexibility to design algorithms that best adapt to the problem at hand. Experiments conducted on two case studies reveal that the presented approach has strong performance in terms of objective value and execution time when compared to other probability-based algorithms. Moreover, the experimental analysis demonstrates that the convergence of the algorithms is controllable by adjusting the components of the framework. For the sake of reproducibility, the source code, results, scripts, figures, and other material related to the manuscript are available at https://github.com/mikelma/nnco_lib .

As the issue of robustness in AI systems becomes vital, statistical learning techniques that are reliable even in presence of partly contaminated data have to be developed. Preference data, in the form of (complete) rankings in the simplest situations, are no exception and the demand for appropriate concepts and tools is all the more pressing given that technologies fed by or producing this type of data ($\textit{e.g.}$ search engines, recommending systems) are now massively deployed. However, the lack of vector space structure for the set of rankings ($\textit{i.e.}$ the symmetric group $\mathfrak{S}_n$) and the complex nature of statistics considered in ranking data analysis make the formulation of robustness objectives in this domain challenging. In this paper, we introduce notions of robustness, together with dedicated statistical methods, for $\textit{Consensus Ranking}$ the flagship problem in ranking data analysis, aiming at summarizing a probability distribution on $\mathfrak{S}_n$ by a $\textit{median}$ ranking. Precisely, we propose specific extensions of the popular concept of *breakdown point*, tailored to consensus ranking, and address the related computational issues. Beyond the theoretical contributions, the relevance of the approach proposed is supported by an experimental study.

This paper reports on the first international competition on AI for the traveling salesman problem (TSP) at the International Joint Conference on Artificial Intelligence 2021 (IJCAI-21). The TSP is one of the classical combinatorial optimization problems, with many variants inspired by real-world applications. This first competition asked the participants to develop algorithms to solve an orienteering problem with stochastic weights and time windows (OPSWTW). It focused on two learning approaches: surrogate-based optimization and deep reinforcement learning. In this paper, we describe the problem, the competition setup, and the winning methods, and give an overview of the results. The winning methods described in this work have advanced the state-of-the-art in using AI for stochastic routing problems. Overall, by organizing this competition we have introduced routing problems as an interesting problem setting for AI researchers. The simulator of the problem has been made open-source and can be used by other researchers as a benchmark for new learning-based methods. The instances and code for the competition are available at https://github.com/paulorocosta/ai-for-tsp-competition.

The concept of median/consensus has been widely investigated in order to provide a statistical summary of ranking data, i.e. realizations of a random permutation $\Sigma$ of a finite set, $\{1,; \ldots,;{n}\}$ with $n\geq 1$ say. As it sheds light onto only one aspect of $\Sigma$’s distribution $P$, it may neglect other informative features. It is the purpose of this paper to define analogues of quantiles, ranks and statistical procedures based on such quantities for the analysis of ranking data by means of a metric-based notion of depth function on the symmetric group. Overcoming the absence of vector space structure on $\mathfrak{S}_n$, the latter defines a center-outward ordering of the permutations in the support of $P$ and extends the classic metric-based formulation of consensus ranking (medians corresponding then to the deepest permutations). The axiomatic properties that ranking depths should ideally possess are listed, while computational and generalization issues are studied at length. Beyond the theoretical analysis carried out, the relevance of the novel concepts and methods introduced for a wide variety of statistical tasks are also supported by numerous numerical experiments.

In Machine Learning, a benchmark refers to an ensemble of datasets associated with one or multiple metrics together with a way to aggregate different systems performances. They are instrumental in {\it (i)} assessing the progress of new methods along different axes and {\it (ii)} selecting the best systems for practical use. This is particularly the case for NLP with the development of large pre-trained models (\textit{e.g.} GPT, BERT) that are expected to generalize well on a variety of tasks. While the community mainly focused on developing new datasets and metrics, there has been little interest in the aggregation procedure, which is often reduced to a simple average over various performance measures. However, this procedure can be problematic when the metrics are on a different scale, which may lead to spurious conclusions. This paper proposes a new procedure to rank systems based on their performance across different tasks. Motivated by the social choice theory, the final system ordering is obtained through aggregating the rankings induced by each task and is theoretically grounded. We conduct extensive numerical experiments (on over 270k scores) to assess the soundness of our approach both on synthetic and real scores (\textit{e.g.} GLUE, EXTREM, SEVAL, TAC, FLICKR). In particular, we show that our method yields different conclusions on state-of-the-art systems than the mean-aggregation procedure while being both more reliable and robust.

Expensive black-box combinatorial optimization problems arise in practice when the objective function is evaluated by means of a simulator or a real-world experiment. Since each fitness evaluation is expensive in terms of time or resources, the number of possible evaluations is typically several orders of magnitude smaller than in non-expensive problems. Classical optimization methods are not useful in this scenario. In this paper, we propose and analyze UMM, an estimation-of-distribution (EDA) algorithm based on a Mallows probabilistic model and unbalanced rank aggregation (uBorda). Experimental results on black-box versions of LOP and PFSP show that UMM outperforms the solutions obtained by CEGO, a Bayesian optimization algorithm for combinatorial optimization. Nevertheless, a slight modification to CEGO, based on the different interpretations for rankings and orderings, significantly improves its performance, thus producing solutions that are slightly better than those of UMM and dramatically better than the original version. Another benefit of UMM is that its computational complexity increases linearly with both the number of function evaluations and the permutation size, which results in computation times an order of magnitude shorter than CEGO, making it specially useful when both computation time and number of evaluations are limited.

In this paper, we study mixtures of two Mallows models for top-$k$ rankings with equal location parameters but with different scale parameters (a mixture of concentric Mallows models). These models arise when we have a heterogeneous population of voters formed by two populations, one of which is a subpopulation of expert voters. We show the identifiability of both components and the learnability of their respective parameters. These results are based upon, first, bounding the sample complexity for the Borda algorithm with top-$k$ rankings. Second, we characterize the distances between rankings, showing that an off-the-shelf clustering algorithm separates the rankings by components with high probability -provided the scales are well-separated.As a by-product, we include an efficient sampling algorithm for Mallows top-$k$ rankings. Finally, since the rank aggregation will suffer from a large amount of noise introduced by the non-expert voters, we adapt the Borda algorithm to be able to recover the ground truth consensus ranking which is especially consistent with the expert rankings.

Since their introduction, Estimation of Distribution Algorithms (EDAs) have proved to be very competitive algorithms to solve many optimization problems. However, despite recent developments, in the case of permutation-based combinatorial optimization problems, there are still many aspects that deserve further research. One of them is the influence of the codification employed to represent the solutions on the overall performance of the algorithm. When considering classical EDAs, optimizing permutation problems is challenging, and specific mechanisms are needed to hold the restrictions associated with the permutation nature of solutions.In this paper, in addition to the permutation-vector codification, we investigate alternative representations to describe solutions of permutation problems in the context of EDAs. In order to evaluate their influence, we adopted a classical EDA and conducted an experimental study on two different permutation problems and representations for codifying solutions. The results revealed a narrow relationship between the type of combinatorial problem optimized and the selected representation used to codify its solutions. Moreover, the results point out that choosing the appropriate representation to codify solutions of the given permutation problem is critical for the performance of the algorithm.

Permutation problems are combinatorial optimization problems whose solutions are naturally codified as permutations. Due to their complexity, motivated principally by the factorial cardinality of the search space of solutions, they have been a recurrent topic for the artificial intelligence and operations research community. Recently, among the vast number of metaheuristic algorithms, new advances on estimation of distribution algorithms (EDAs) have shown outstanding performance when solving some permutation problems. These novel EDAs implement distance-based exponential probability models such as the Mallows and Generalized Mallows models. In this article, we present a Matlab package, perm_mateda, of estimation of distribution algorithms on permutation problems, which has been implemented as an extension to the Mateda-2.0 toolbox of EDAs. Particularly, we provide implementations of the Mallows and Generalized Mallows EDAs under the Kendall’s-τ, Cayley, and Ulam distances. In addition, four classical permutation problems have also been implemented: Traveling Salesman Problem, Permutation Flowshop Scheduling Problem, Linear Ordering Problem, and Quadratic Assignment Problem.

Recently, probability models on rankings have been proposed in the field of estimation of distribution algorithms in order to solve permutation-based combinatorial optimisation problems. Particularly, distance-based ranking models, such as Mallows and Generalized Mallows under the Kendall's-τ distance, have demonstrated their validity when solving this type of problems. Nevertheless, there are still many trends that deserve further study. In this paper, we extend the use of distance-based ranking models in the framework of EDAs by introducing new distance metrics such as Cayley and Ulam. In order to analyse the performance of the Mallows and Generalized Mallows EDAs under the Kendall, Cayley and Ulam distances, we run them on a benchmark of 120 instances from four well known permutation problems. The conducted experiments showed that there is not just one metric that performs the best in all the problems. However, the statistical test pointed out that Mallows-Ulam EDA is the most stable algorithm among the studied proposals.

The aim of this paper is two-fold. First, we introduce a novel general estimation of distribution algorithm to deal with permutation-based optimization problems. The algorithm is based on the use of a probabilistic model for permutations called the generalized Mallows model. In order to prove the potential of the proposed algorithm, our second aim is to solve the permutation flowshop scheduling problem. A hybrid approach consisting of the new estimation of distribution algorithm and a variable neighborhood search is proposed. Conducted experiments demonstrate that the proposed algorithm is able to outperform the state-of-the-art approaches. Moreover, from the 220 benchmark instances tested, the proposed hybrid approach obtains new best known results in 152 cases. An in-depth study of the results suggests that the successful performance of the introduced approach is due to the ability of the generalized Mallows estimation of distribution algorithm to discover promising regions in the search space.

Haplotype data are especially important in the study of complex diseases since it contains more information than genotype data. However, obtaining haplotype data is technically difficult and costly. Computational methods have proved to be an effective way of inferring haplotype data from genotype data. One of these methods, the haplotype inference by pure parsimony approach (HIPP), casts the problem as an optimization problem and as such has been proved to be NP-hard. We have designed and developed a new preprocessing procedure for this problem. Our proposed algorithm works with groups of haplotypes rather than individual haplotypes. It iterates searching and deleting haplotypes that are not helpful in order to find the optimal solution. This preprocess can be coupled with any of the current solvers for the HIPP that need to preprocess the genotype data. In order to test it, we have used two state-of-the-art solvers, RTIP and GAHAP, and simulated and real HapMap data. Due to the computational time and memory reduction caused by our preprocess, problem instances that were previously unaffordable can be now efficiently solved.

A haplotype is a DNA sequence that is inherited from one parent. They are especially important in the study of complex diseases since they contain more information than genotype data, so the next high priority phase in human genomics involves the development of a full haplotype map of human genome. However, obtaining haplotype data is technically difficult and expensive. One of the computational methods for obtaining haplotype data from genotype data is the pure parsimony criterion, an approach known as haplotype inference by pure parsimony (HIPP). It has been proved to be an NP-hard problem. We present a new preprocessing method which drastically decreases the number of relevant haplotypes. Several algorithms need to preprocess data; for big problem instances this key procedure is even more important than the process. This preprocessing was eventually tested on real and simulated data applying a tabu search, and the performance of the resulting algorithm showed it to be competitive with the best actual solvers.