Adapting the Directed Grid Theorem into an FPT algorithm

March 28 2022. Room B115 at 14:00

(Hybrid) Cliquez ici pour participer à la réunion

Speaker: Raul Wayne Teixeira Lopes

Abstract: 

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]. 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. Their constructive 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 talk, we present the ideas used in our adaptation of the Directed Grid Theorem into an FPT algorithm. We provide 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. [JCTB, 2001]. The second one uses a bramble B that naturally occurs in digraphs of large directed tree-width to find a well-linked set of order k whose vertices appear in a path hitting all elements of B. 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|.

Joint work with Victor Campos, Ana Karolinna Maia, and Ignasi Sau.