Heuristic and metaheuristic methods for computing graph treewidth

The notion of treewidth is of considerable interest in relation to NP-hard problems. Indeed, several studies have shown that the tree-decomposition method can be used to solve many basic optimization problems in polynomial time when treewidth is bounded, even if, for arbitrary graphs, computing the treewidth is NP-hard. Several papers present heuristics with computational experiments. For many graphs the discrepancy between the heuristic results and the best lower bounds is still very large. The aim of this paper is to propose two new methods for computing the treewidth of graphs: a heuristic and a metaheuristic. The heuristic returns good results in a short computation time, whereas the metaheuristic (a Tabu search method) returns the best results known to have been obtained so far for all the DIMACS vertex coloring / treewidth benchmarks (a well-known collection of graphs used for both vertex coloring and treewidth problems.) Our results actually improve on the previous best results for treewidth problems in 53% of the cases. Moreover, we identify properties of the triangulation process to optimize the computing time of our method.