Computing Treewidth Via Exact And Heuristic Lists Of Minimal Separators

We develop practically efficient algorithms for computing the treewidth tw(G) of a graph G. The core of our approach is a new dynamic programming algorithm which, given a graph G, a positive integer k, and a set Delta of minimal separators of G, decides if G has a tree-decomposition of width at most k of a certain canonical form that uses minimal separators only from Delta, in the sense that the intersection of every pair of adjacent bags belongs to Delta. This algorithm is used to show a lower bound of k + 1 on tw(G), setting Delta to be the set of all minimal separators of cardinality at most k and to show an upper bound of k on tw(G), setting Delta to be some, hopefully rich, set of such minimal separators. Combining this algorithm with new algorithms for exact and heuristic listing of minimal separators, we obtain exact algorithms for treewidth which overwhelmingly outperform previously implemented algorithms.