the directed grid...

AlgorithmicDigraphStructureTheory

TheDirectedGridTheorem

Stephan Kreutzer

Technical University Berlin

Joint work with Ken-ichi Kawarabayashi, NII, Tokyo

StephanKreutzer–TUBerlin

The Directed Grid TheoremStephan Kreutzer – TU Berlin

Exact algorithms for NP-hard problems on graphs. Structure theory based on tree width, the excluded grid theorem etc. is very successful in dealing with hard computational problems on graphs.

Tree Width. • Width measure introduced by Robertson and Seymour. • Many algorithmic applications on graph classes of bounded tree width. • Plays an important role in graph structure theory based on minors.

AlgorithmicGraphStructureTheory

2


planar

apexfree/lin.localtw

excludedminors

trees treewidth

allgraphclasses

bidimensionalityirrelevantvertices

…

structural decomposition

…

StructureTheoryforUndirectedGraphs

3

dynamicprogramming

AlgorithmicTechniquesStructuralResults

nowheredense


StructureTheoryforDirectedGraphs

4


UndirectedTreeWidth

5

1 2

3

54

6 7

98

10 11


1 2

3

UndirectedTreeWidth

5

1 2

3

54

6 7

98

10 11


1

3

5

1 2

3

UndirectedTreeWidth

5

1 2

3

54

6 7

98

10 11


Defined analogous to undirected tree width but based on strong separators

DirectedTreeWidth

6

1 2

3

54

6 7

98

10

8

11



1 2

3

DirectedTreeWidth

6

1 2

3

54

6 7

98

10

8

11



5

1 2

3

DirectedTreeWidth

6

1 2

3

54

6 7

98

10

8

11



5

1 2

3

DirectedTreeWidth

6

1 2

3

54

6 7

98

10

8

11

4 6 7

58 58 58


DirectedTreeWidth

7


DirectedTreeWidth

8


StructureTheoryforDirectedGraphs

9

s1s2

s3

t1 t2

t3


ApplicationsofDirectedTreeWidth

10


Graphsvs.Digraphs

11

planar

Apexfree/lin.localtw

excludedminors

trees treewidth

allgraphclasses

UndirectedGraphs

…

DirectedGraphs

directedtreewidth


Graphsvs.Digraphs

11

planar


excludedminors

trees treewidth

allgraphclasses

UndirectedGraphs

…

DirectedGraphs

directedtreewidth

TheGridTheorem


TheDirectedGridConjecture

12



12

Butterfly minor: contract an edge if only outgoing of tail or only incoming of head.



13



14


ProofoftheDirectedGridTheorem

15



16

High Directed Tree Width



16

High Directed Tree WidthDirected Bramble



16

High Directed Tree Width Well-Linked Path SystemDirected Bramble


Well-LinkedPathSystems

17


AWeb

18

Q

P


AWeb

19

Q

P


AWeb

20

Q

P


Well-LinkedFences

21


Bottom-UpLinkages

22

R


AvoidingaSub-Fence

23

R


GoingupRow-By-Row

24



25


Erdős-PósaPropertyforDigraphs

26


Erdős-PósaPropertyforDigraphs

27

Theorem. (Amiri, Kawarabayashi, K., Wollan ’14)

A strongly connected digraph H has the Erdős-Pósa property for subdivisions if, and only if, it is a subdivision of a cylindrical grid.

The same is true for butterfly minors.

Proof. Suppose H is a subdivision of a cylindrical grid. Let G be given.

1. If G has large directed tree width, we have a large grid and hence many copies of H.

2. Otherwise, directed tree width is bounded and we can prove it as in the undirected case.


Semi-IntegralDisjointPathsProblem

28

•

s1s2

s3

t1 t2

t3


Semi-IntegralDisjointPathsProblem

28

•

Open Problem. (with Kawarabayashi, K., Wollan)

• As there cannot be such a result for the fully disjoint directed paths problem, the best we can hope for is a half-integral version.

• Can we find a polynomial time algorithm for computing half-integral disjoint paths (if they exist).

The Directed Grid TheoremStephan Kreutzer – TU Berlin 29

Conclusion


The directed grid theorem establishes a duality to directed tree width in terms of a nice planar digraph.

Algorithmic applications. • Is there a polynomial time algorithm for the half-integral disjoint paths

problem? • Simpler proof of integral disjoint paths on planar digraphs, possibly

extending this to digraphs of bounded genus etc. • Algorithmic versions of Erdős-Pósa problems.

Structure theory. • What are directed tree decompositions restricted to planar digraphs? • More generally: combining directed tree width and nowhere crownful

classes of digraphs? • Can we get further structure theorems such as the weak structure theorem?

OpenProblems

30


planar


excludedminors

trees treewidth

allgraphclasses

…

31

nowheredense

Directedtreewidth

StructureTheoryforUndirectedGraphs

nowherecrownful

the directed grid...

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