Tractability of parameterized completion problems on chordal and interval graphs: Minimum Fill-in and Physical Mapping

We study the parameterized complexity of several NP-Hard graph completion problems: The MINIMUM FILL-IN problem is to decide if a graph can be triangulated by adding at most k edges. We develop an O(k 5 mn + f(k)) algorithm for the problem on a graph with n vertices and m edges. In particular, this implies that the problem is xed parameter tractable (FPT). PROPER INTERVAL GRAPH COMPLETION problems, motivated by molecular biology, ask for adding edges in order to obtain a proper interval graph, so that a parameter in that graph does not exceed k. We show that the problem is FPT when k is the number of added edges. For the problem where k is the clique size, we give an O(f(k)n k?1) algorithm, so it is polynomial for xed k. On the other hand, we prove its hardness in the parameterized hierarchy, so it is probably not FPT. Those results are obtained even when a set of edges which should not be added is given. That set can be given either explicitly or by a proper vertex coloring which the added edges should respect.