A four-decomposition strategies for hierarchically modeling combinatorial optimization problems: framework, conditions and relations

Abstract

We address the problem of modeling combinatorial optimization problems (COP). COPs are generally complex problems to solve. So a good modeling step is fundamental to make the solution easier. Our approach orients researches to choose the best modeling strategy from the beginning to avoid any problem in the solving process. This paper aims at proposing a new approach dealing with hard COPs particularly when the decomposition process leads to some well-known and canonical optimization sub-problems. We tried to draw a clear framework that will help to model hierarchical optimization problems. The framework will be composed by four decomposition strategies which are: objective based decomposition; constraints based decomposition, semantic decomposition and data partitioning strategy. For each strategy, we present supporting examples from the literature where it was applied. But, not all combinatorial problems can be benefit from the outcomes and benefits of modeling problems hierarchically, rather only particular problems can be modeled like a hierarchical optimization problem. Thus, we propose a set of decomposability conditions for decomposing COPs. Furthermore, we define the types of relationships between obtained sub-problems and how partial solutions can be merged to obtain the final solution.