The Minimum Fill-in problem is to decide if a graph can be triangulated by adding at most k edges. Kaplan, Shamir, and Tarjan [FOCS 1994] have shown that the problem is solvable in time O(2 O(k) + k 2 nm) on graphs with n vertices and m edges and thus is fixed parameter tractable. Here, we give the first subexponential parameterized algorithm solving Minimum Fillin in time O(2 O( √ k log k) + k 2 nm). This substantially lowers the complexity of the problem. Techniques developed for Minimum

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... in can be used to obtain subexponential parameterized algorithms for several related problems including Minimum Chain Completion, Chordal Graph Sandwich, and Triangulating Colored Graph.