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(k⁵mn + f(k)) algorithm for the problem on a graph with n vertices and m edges. In particular, this implies that the problem is fixed 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 fixed 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.