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In this paper we pursue the use of the Fourier transform for a general analysis of combinatorial optimization problems. While combinatorial optimization problems are defined by means of different notions like weights in a graph, set of numbers, distance between cities, etc., the Fourier transform allows to put all of them in the same framework, the Fourier coefficients. This permits its comparison looking for similarities and differences. Particularly, the Walsh transform has recently been used over pseudo-boolean functions in order to design new surrogate models in the black-box scenario and to generate new algorithms for the linkage discovery problem, among others, presenting very promising results. In this paper we focus on binary problems and the Walsh transform. After presenting the Walsh decomposition and some main properties, we compute the transform of the Unconstrained Binary Quadratic Problem and several particular cases of this problem such as the Max-Cut Problem and the Number Partitioning Problem. The obtained Walsh coefficients not only reinforce the similarities and differences among the problems which are known in the literature, but given a set of Walsh coefficients we can say whether or not they are produced by any of the problems analyzed. Finally, a geometrical interpretation of the space of Walsh coefficients with maximum order 2 and the subspace of each analyzed problem is presented., by Spanish Ministry of Science and Innovation through the projects PID2019-104966GB-I00/AEI/10.13039/501100011033, PID2019-104933GBI00/ AEI/10.13039/501100011033, PID2019-106453GA-I00/AEI/10.13039/501100011033 and BCAM Severo Ochoa accreditation SEV-2017-0718; and by the Basque Government through the program BERC 2018-2021 and the projects IT1244-19 and IT-1252-19; and by UPV/EHU through the project GIU20/054. Imanol holds a grant from the Department of Education of the Basque Government (PRE_2020_2_0166).

Estimation of Distribution Algorithms have been successfully used to solve permutation-based Combinatorial Optimization Problems. In this case, the algorithms use probabilistic models specifically designed for codifying probability distributions over permutation spaces. One class of these probability models are distance-based exponential models, and one example of this class is the Mallows model. In spite of its practical success, the theoretical analysis of Estimation of Distribution Algorithms for permutation-based Combinatorial Optimization Problems has not been developed as extensively as it has been for binary problems. With this motivation, this paper presents a first mathematical analysis of the convergence behavior of Estimation of Distribution Algorithms based on Mallows models. The model removes the randomness of the algorithm in order to associate a dynamical system to it. Several scenarios of increasing complexity with different fitness functions and initial probability distributions are analyzed. The obtained results show: a) the strong dependence of the final results on the initial population, and b) the possibility to converge to non-degenerate distributions even in very simple scenarios, which has not been reported before in the literature., Spanish Ministry of Science and Innovation through the projects PID2019-104966GB-I00/AEI/10.13039/501100011033, PID2019-104933GB-I00/AEI/10.13039/501100011033, PID2019-106453GA-I00/AEI/10.13039/501100011033 and BCAM Severo Ochoa accreditation SEV-2017-0718; and by the Basque Government through the program BERC 2022-2025 and the projects IT1504-22 and IT1494-22; and by UPV/EHU through the project GIU20/054. Imanol holds a grant from the Department of Education of the Basque Government (PRE_2021_2_0224).