DeepAI
Log In Sign Up

Adapting the Directed Grid Theorem into an FPT Algorithm

07/15/2020
by   Victor Campos, et al.
0

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 size 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 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 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.

READ FULL TEXT

page 1

page 2

page 3

page 4

12/02/2017

Non-branching tree-decompositions

We prove that if a graph has a tree-decomposition of width at most w, th...
02/04/2019

Cyclewidth and the Grid Theorem for Perfect Matching Width of Bipartite Graphs

A connected graph G is called matching covered if every edge of G is con...
03/15/2021

Constant congestion brambles in directed graphs

The Directed Grid Theorem, stating that there is a function f such that ...
09/28/2020

The canonical directed tree decomposition and its applications to the directed disjoint paths problem

The canonical tree-decomposition theorem, given by Robertson and Seymour...
04/05/2019

Reducing Topological Minor Containment to the Unique Linkage Theorem

In the Topological Minor Containment problem (TMC) problem two undirecte...