The Fourier transform over finite groups has proved to be a useful tool for analyzing combinatorial optimization problems. However, few heuristic and meta-heuristic 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 meta-heuristic 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 hyper-parameters.

Comparing combinatorial optimization problems is a difficult task. They are defined using different criteria and terms: weights, flows, distances, etc. In spite of this apparent discrepancy, on many occasions, they tend to produce problem instances with similar properties. One avenue to compare different problems is to project them onto the same space, in order to have homogeneous representations. Expressing the problems in a unified framework could also lead to the discovery of theoretical properties or the design of new algorithms. This article proposes the use of the Fourier transform over the symmetric group as the tool to project different permutation-based combinatorial optimization problems onto the same space. Based on a previous study (Kondor, 2010 ), which characterized the Fourier coefficients of the quadratic assignment problem, we describe the Fourier coefficients of three other well-known problems: the symmetric and nonsymmetric traveling salesperson problem and the linear ordering problem. This transformation allows us to gain a better understanding of the intersection between the problems, as well as to bound their intrinsic dimension.

Solving combinatorial optimization problems efficiently requires the development of algorithms that consider the specific properties of the problems. In this sense, local search algorithms are designed over a neighborhood structure that partially accounts for these properties. Considering a neighborhood, the space is usually interpreted as a natural landscape, with valleys and mountains. Under this perception, it is commonly believed that, if maximizing, the solutions located in the slopes of the same mountain belong to the same attraction basin, with the peaks of the mountains being the local optima. Unfortunately, this is a widespread erroneous visualization of a combinatorial landscape. Thus, our aim is to clarify this aspect, providing a detailed analysis of, first, the existence of plateaus where the local optima are involved, and second, the properties that define the topology of the attraction basins, picturing a reliable visualization of the landscapes. Some of the features explored in this article have never been examined before. Hence, new findings about the structure of the attraction basins are shown. The study is focused on instances of permutation-based combinatorial optimization problems considering the 2-exchange and the insert neighborhoods. As a consequence of this work, we break away from the extended belief about the anatomy of attraction basins.

The solution of many combinatorial optimization problems is carried out by metaheuristics, which generally make use of local search algorithms. These algorithms use some kind of neighborhood structure over the search space. The performance of the algorithms strongly depends on the properties that the neighborhood imposes on the search space. One of these properties is the number of local optima. Given an instance of a combinatorial optimization problem and a neighborhood, the estimation of the number of local optima can help not only to measure the complexity of the instance, but also to choose the most convenient neighborhood to solve it. In this paper we review and evaluate several methods to estimate the number of local optima in combinatorial optimization problems. The methods reviewed not only come from the combinatorial optimization literature, but also from the statistical literature. A thorough evaluation in synthetic as well as real problems is given. We conclude by providing recommendations of methods for several scenarios.