DeepAI

# Adapting the Directed Grid Theorem into an FPT Algorithm

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.

• 12 publications
• 6 publications
• 3 publications
• 26 publications
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...
05/21/2018

### Finding forbidden minors in sublinear time: a n^1/2+o(1)-query one-sided tester for minor closed properties on bounded degree graphs

Let G be an undirected, bounded degree graph with n vertices. Fix a fini...
07/15/2022

### Fixed-parameter tractability of Directed Multicut with three terminal pairs parameterized by the size of the cutset: twin-width meets flow-augmentation

We show fixed-parameter tractability of the Directed Multicut problem wi...
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...