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We show that there are O(1.8899 n ) time algorithms to compute the treewidth and the minimum fill-in of each graph G on n vertices. Our result is based on a combinatorial proof that each graph on n vertices has at most n · 1.7087 n minimal separators and that all potential maximal cliques can be listed in O(1.8899 n ) time. For the class of AT-free graphs we obtain O(1.4142 n ) time algorithms to compute treewidth and minimum fill-in.doi:10.1137/050643350 fatcat:vp2ffushcfhcnprue4fftvhalu