Permutation problems have captured the attention of the combinatorial optimization community for decades due to the challenge they pose. Although their solutions are naturally encoded as permutations, in each problem, the information to be used to optimize them can vary substantially. In this paper, we consider the Quadratic Assignment Problem (QAP) as a case study, and propose using Doubly Stochastic Matrices (DSMs) under the framework of Estimation of Distribution Algorithms. To that end, we design efficient learning and sampling schemes that enable an effective iterative update of the probability model. Conducted experiments on commonly adopted benchmarks for the QAP prove doubly stochastic matrices to be preferred to the other four models for permutations, both in terms of effectiveness and computational efficiency. Moreover, additional analyses performed on the structure of the QAP and the Linear Ordering Problem (LOP) show that DSMs are good to deal with assignment problems, but they have interesting capabilities to deal also with ordering problems such as the LOP. The paper concludes with a description of the potential uses of DSMs for other optimization paradigms, such as genetic algorithms or model-based gradient search.

The Fourier transform over finite groups has proved to be a useful tool for analyzing combinatorial optimization problems. However, few heuristic and metaheuristic algorithms have been proposed in the literature that utilize the information provided by this technique to guide the search process. In this work, we attempt to address this research gap by considering the case study of the Linear Ordering Problem (LOP). Based on the Fourier transform, we propose an instance decomposition strategy that divides any LOP instance into the sum of two LOP instances associated with a P and an NP-Hard optimization problem. By linearly aggregating the instances obtained from the decomposition, it is possible to create artificial instances with modified proportions of the P and NP-Hard components. Conducted experiments show that increasing the weight of the P component leads to a less rugged fitness landscape suitable for local search-based optimization. We take advantage of this phenomenon by presenting a new metaheuristic algorithm called P-Descent Search (PDS). The proposed method, first, optimizes a surrogate instance with a high proportion of the P component, and then, gradually increases the weight of the NP-Hard component until the original instance is reached. The multi-start version of PDS shows a promising and predictable performance that appears to be correlated to specific characteristics of the problem, which could open the door to an automatic tuning of its hyperparameters.

In the last decade, many works in combinatorial optimisation have shown that, due to the advances in multi-objective optimisation, the algorithms from this field could be used for solving single-objective problems as well. In this sense, a number of papers have proposed multi-objectivising single-objective problems in order to use multi-objective algorithms in their optimisation. In this article, we follow up this idea by presenting a methodology for multi-objectivising combinatorial optimisation problems based on elementary landscape decompositions of their objective function. Under this framework, each of the elementary landscapes obtained from the decomposition is considered as an independent objective function to optimise. In order to illustrate this general methodology, we consider four problems from different domains: the quadratic assignment problem and the linear ordering problem (permutation domain), the 0-1 unconstrained quadratic optimisation problem (binary domain), and the frequency assignment problem (integer domain). We implemented two widely known multi-objective algorithms, NSGA-II and SPEA2, and compared their performance with that of a single-objective GA. The experiments conducted on a large benchmark of instances of the four problems show that the multi-objective algorithms clearly outperform the single-objective approaches. Furthermore, a discussion on the results suggests that the multi-objective space generated by this decomposition enhances the exploration ability, thus permitting NSGA-II and SPEA2 to obtain better results in the majority of the tested instances.