New Heuristic Methods for Tree Decompositions and Generalized Hypertree Decompositions

Many practical hard problems in mathematics and computer science may be formulated as constraint satisfaction problems (CSPs). Tree and generalized hypertree decompositions are two important concepts which can be used for identifying and solving tractable classes of CSPs. Unfortunately the task of finding an optimal tree or generalized hypertree decomposition is an NP-complete problem. Thus many heuristic methods have been developed for finding tree decompositions and generalized hypertree decompositions of small width. In this master thesis we present new heuristic methods for tree and generalized hypertree decompositions. For that purpose we examine already existing heuristic methods for tree decompositions and extend them to an A* algorithm and a genetic algorithm for tree decompositions and to a genetic algorithm and a self-adaptive genetic algorithm for generalized hypertree decompositions. Furthermore we prove that the set of all elimination orderings may act as a search space for the generalized hypertree width and we develop a lower bound heuristic for the generalized hypertree width, which combines lower bound heuristics for tree decompositions with lower bound heuristics for the kset cover problem. Moreover we show how existing reduction and pruning techniques, for shrinking the search space for the optimal tree decomposition, may also be used for generalized hypertree decompositions. Based on these results we propose a branch and bound algorithm and an A* algorithm for generalized hypertree decompositions. Computational experiments show that the heuristic methods presented in this thesis are able to compete with other heuristic methods for tree and generalized hypertree decompositions. For many benchmark instances the genetic algorithms and the branch and bound algorithm return improved upper bounds on the treewidth and generalized hypertree width and for some instances the A* algorithms and the branch and bound algorithm are able to fix the exact treewidth and generalized hypertree width.