Metaheuristic algorithms and tree decomposition

Abstract

This chapter deals with the application of evolutionary approaches and other metaheuristic techniques for generating tree decompositions. Tree decomposition is a concept introduced by Robertson and Seymour [64.1] and it is used to characterize the di?culty of constraint satisfaction and NP-hard problems that can be represented as a graph. Although, in general, no polynomial algorithms have been found for such problems, particular instances can be solved in polynomial time if the treewidth of their corresponding graph is bounded by a constant. The process of solving problems based on tree decomposition comprises two phases. First, a decomposition with small width is generated. Basically in this phase the problem is divided into several subproblems, each included in one of the nodes of the tree decomposition. The second phase includes solving a problem (based on the generated tree decomposition) with a particular algorithm such as dynamic programming. The main idea is that by decomposing a problem into subproblems of limited size, the whole problem can be solved more e?ciently. The time for solving the problem based on its tree decomposition usually depends on the width of the tree decomposition. Thus, it is of high interest to generate tree decompositions having small widths. Finding the treewidth of a graph is an NP-hard problem [64.2]. In order to solve this problem, different algorithms have been proposed in the literature. Exact methods such as branch and bound techniques can be used only for small graphs. Therefore, metaheuristic algorithms based on genetic algorithms [64.3], simulated annealing [64.4], tabu search [64.5], iterated local search [64.6], and ant colony optimization.