Multi-structure problems: Difficult model learning in discrete EDAs

Decomposition to smaller sub-problems is a general approach in problem solving. Many of the real-world problems can be decomposed into a number of sub-problems which may be solved easier. Appropriate decomposition is a significant issue specially for optimization problems where the optimal solution is usually obtained by combining the solutions of sub-problems. Estimation of distribution algorithms (EDAs) are a type of evolutionary algorithms that learn a model of problem from the population of candidate solutions. This model is intended to capture the interactions between problem variables, thus facilitating problem decomposition and is used to generate new solutions. In this paper, a novel type of problems is presented that is designed to challenge the model building process in discrete EDAs. The main idea is to propose a set of problems that their candidate solutions can be simultaneously decomposed into different sub-problems. This means that the candidate solution of a problem may be interpreted by two or more different structure where only one is true, resulting in the optimal solution to that problem. Some of these decompositions or structures may be more likely according to the low-order statistics collected from the population of candidate solutions, but may not necessarily lead to the optimal solution. Learning the correct structure/decomposition is a challenge for the model building process in EDA. The experimental results show that the proposed problems are indeed difficult for EDAs even when expressive models such as Bayesian networks are used to capture the interactions in the problem.