Adapting the directed grid theorem into an fpt algorithm

The grid theorem of Robertson and Seymour [J. Combin. Theory Ser. B, 41 (1986), pp. 92-114] is one of the most important tools in the field of structural graph theory, finding numerous applications in the design of algorithms for undirected graphs. An analogous version of the grid theorem in digraphs was conjectured by Johnson et al. [J. Combin. Theory Ser. B, 82 (2001), pp. 138-154] and proved by Kawarabayashi and Kreutzer [Proceedings of STOC, 2015, pp. 655-664]. Namely, they showed that there is a function f(k) such that every digraph of directed tree-width at least f(k) contains a cylindrical grid of order k as a butterfly minor, and stated that their proof can be turned into an XP algorithm, with parameter k, that either constructs a decomposition of the appropriate width or finds the claimed large cylindrical grid as a butterfly minor. In this paper, we adapt some of the steps of the proof of Kawarabayashi and Kreutzer to improve this XP algorithm into a fixed-parameter tractable (FPT) algorithm. Toward this, our main technical contributions are two FPT algorithms with parameter k. The first one either produces an arboreal decomposition of width 3k 2 or finds a haven of order k in a digraph D, improving on the original result for arboreal decompositions by Johnson et al. [J. Combin. Theory Ser. B, 82 (2001), pp. 138-154]. The second algorithm finds a well-linked set of order k in a digraph D of large directed tree-width. As tools to prove these results, we show how to solve a generalized version of the problem of finding balanced separators for a given set of vertices T in FPT time with parameter | T|, a result that we consider to be of its own interest.