Jungles, bundles, and fixed parameter tractability

We give a fixed-parameter tractable (FPT) approximation algorithm computing the pathwidth of a tournament, and more generally, of a semi-complete digraph. Based on this result, we prove the following. • The Topological Containment problem is FPT on semi-complete digraphs. More precisely, given a semi-complete n-vertex digraph T and a digraph H, one can check in time f(|H|)n3 log n, where f is some elementary function, whether T contains a subdivision of H as a subgraph. The previous known algorithm for this problem was due to Fradkin and Seymour and was of running time nm(|H|), where m is a quadruple-exponential function. • The Rooted Immersion problem is FPT on semi-complete digraphs. The complexity of this problem was left open by Fradkin and Seymour. Our algorithm solves it in time g(|H|)n4 log n, for some elementary function g. • Vertex deletion distance to every immersion-closed class of semi-complete digraphs is fixed-parameter tractable. More precisely, for every immersion-closed class Π of semi-complete digraphs, there exists an algorithm with running time h(k)n3 log n that checks, whether one can remove at most k vertices from a semi-complete n-vertex digraph to obtain a digraph from class Π. Here, h is some function depending on the class Π