No abstract

Estimation of Distribution Algorithms (EDAs) are a set of algorithms that belong to the field of Evolutionary Computation. Characterized by the use of probabilistic models to represent the solutions and the dependencies between the variables of the problem, these algorithms have been applied to a wide set of academic and real-world optimization problems, achieving competitive results in most scenarios. Nevertheless, there are some optimization problems, whose solutions can be naturally represented as permutations, for which EDAs have not been extensively developed. Although some work has been carried out in this direction, most of the approaches are adaptations of EDAs designed for problems based on integer or real domains, and only a few algorithms have been specifically designed to deal with permutation-based problems. In order to set the basis for a development of EDAs in permutation-based problems similar to that which occurred in other optimization fields (integer and real-value problems), in this paper we carry out a thorough review of state-of-the-art EDAs applied to permutation-based problems. Furthermore, we provide some ideas on probabilistic modeling over permutation spaces that could inspire the researchers of EDAs to design new approaches for these kinds of problems.

In this paper we present the R package PerMallows, which is a complete toolbox to work with permutations, distances and some of the most popular probability models for permutations: Mallows and the Generalized Mallows models. The Mallows model is an exponential location model, considered as analogous to the Gaussian distribution. It is based on the definition of a distance between permutations. The Generalized Mallows model is its best-known extension. The package includes functions for making inference, sampling and learning such distributions. The distances considered in PerMallows are Kendall's τ , Cayley, Hamming and Ulam.

The Mallows and Generalized Mallows Models are two of the most popular probability models for distributions on permutations. In this paper we consider both models under the Hamming distance. This models can be seen as models for matchings instead of models for rankings. These models can not be factorized, which contrasts with the popular MM and GMM under Kendall's-τ and Cayley distances. In order to overcome the computational issues that the models involve, we introduce a novel method for computing the partition function. By adapting this method we can compute the expectation, joint and conditional probabilities. All these methods are the basis for three sampling algorithms, which we propose and analyze. Moreover, we also propose a learning algorithm. All the algorithms are analyzed both theoretically and empirically, using synthetic and real data from the context of e-learning and Massive Open Online Courses (MOOC).

The Mallows and Generalized Mallows models are compact yet powerful and natural ways of representing a probability distribution over the space of permutations. In this paper we deal with the problems of sampling and learning (estimating) such distributions when the metric on permutations is the Cayley distance. We propose new methods for both operations, whose performance is shown through several experiments. We also introduce novel procedures to count and randomly generate permutations at a given Cayley distance both with and without certain structural restrictions. An application in the field of biology is given to motivate the interest of this model.