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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.

In this paper we propose a tunable generator of instances of permutationbased Combinatorial Optimization Problems. Our approach is based on a probabilistic model for permutations, called the Generalized Mallows model. The generator depends on a set of parameters that permits the control of the properties of the output instances. Specifically, in order to create an instance, we solve a linear programing problem in the parameters, where the restrictions allow the instance to have a fixed number of local optima and the linear function encompasses qualitative characteristics of the instance. We exemplify the use of the generator by giving three distinct linear functions that produce three landscapes with different qualitative properties. After that, our generator is tested in two different ways. Firstly, we test the flexibility of the model by producing instances similar to benchmark instances. Secondly, we account for the capacity of the generator to create different types of instances according to the difficulty for population-based algorithms. We study the influence of the input parameters in the behavior of these algorithms, giving an example of a property that can be used to analyze their performance.

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