AG 1   Info

AG Audience

English

30 Minutes

Saarbrücken

The Grid Theorem of Robertson and Seymour [JCTB, 1986], 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. [JCTB, 2001], and proved by Kawarabayashi and Kreutzer [STOC, 2015]. 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 work, we adapt some of the steps of the proof of Kawarabayashi and Kreutzer to improve this XP algorithm into an FPT algorithm. Towards 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. The second algorithm finds, in a digraph of large directed tree-width, a well-linked set of order k with some additional properties. 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.

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Sándor Kisfaludi-Bak

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