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

The Directed Grid TheoremStephan Kreutzer – TU Berlin

planar

apexfree/lin.localtw

excludedminors

trees treewidth

allgraphclasses

bidimensionalityirrelevantvertices

…

structural decomposition

…

StructureTheoryforUndirectedGraphs

3

dynamicprogramming

AlgorithmicTechniquesStructuralResults

nowheredense

The Directed Grid TheoremStephan Kreutzer – TU Berlin

planar

apexfree/lin.localtw

excludedminors

trees treewidth

allgraphclasses

bidimensionalityirrelevantvertices

…

structural decomposition

…

StructureTheoryforUndirectedGraphs

3

dynamicprogramming

AlgorithmicTechniquesStructuralResults

nowheredense

The Directed Grid TheoremStephan Kreutzer – TU Berlin

StructureTheoryforDirectedGraphs

4

The Directed Grid TheoremStephan Kreutzer – TU Berlin

UndirectedTreeWidth

5

1 2

3

54

6 7

98

10 11

The Directed Grid TheoremStephan Kreutzer – TU Berlin

1 2

3

UndirectedTreeWidth

5

1 2

3

54

6 7

98

10 11

The Directed Grid TheoremStephan Kreutzer – TU Berlin

1 2

3

UndirectedTreeWidth

5

1 2

3

54

6 7

98

10 11

The Directed Grid TheoremStephan Kreutzer – TU Berlin

1 2

3

UndirectedTreeWidth

5

1 2

3

54

6 7

98

10 11

The Directed Grid TheoremStephan Kreutzer – TU Berlin

1 2

3

UndirectedTreeWidth

5

1 2

3

54

6 7

98

10 11

The Directed Grid TheoremStephan Kreutzer – TU Berlin

1

3

5

1 2

3

UndirectedTreeWidth

5

1 2

3

54

6 7

98

10 11

The Directed Grid TheoremStephan Kreutzer – TU Berlin

1

3

5

1 2

3

UndirectedTreeWidth

5

1 2

3

54

6 7

98

10 11

The Directed Grid TheoremStephan Kreutzer – TU Berlin

Defined analogous to undirected tree width but based on strong separators

DirectedTreeWidth

6

1 2

3

54

6 7

98

10

8

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The Directed Grid TheoremStephan Kreutzer – TU Berlin

Defined analogous to undirected tree width but based on strong separators

1 2

3

DirectedTreeWidth

6

1 2

3

54

6 7

98

10

8

11

The Directed Grid TheoremStephan Kreutzer – TU Berlin

Defined analogous to undirected tree width but based on strong separators

1 2

3

DirectedTreeWidth

6

1 2

3

54

6 7

98

10

8

11

The Directed Grid TheoremStephan Kreutzer – TU Berlin

Defined analogous to undirected tree width but based on strong separators

1 2

3

DirectedTreeWidth

6

1 2

3

54

6 7

98

10

8

11

The Directed Grid TheoremStephan Kreutzer – TU Berlin

Defined analogous to undirected tree width but based on strong separators

1 2

3

DirectedTreeWidth

6

1 2

3

54

6 7

98

10

8

11

The Directed Grid TheoremStephan Kreutzer – TU Berlin

Defined analogous to undirected tree width but based on strong separators

5

1 2

3

DirectedTreeWidth

6

1 2

3

54

6 7

98

10

8

11

The Directed Grid TheoremStephan Kreutzer – TU Berlin

Defined analogous to undirected tree width but based on strong separators

5

1 2

3

DirectedTreeWidth

6

1 2

3

54

6 7

98

10

8

11

The Directed Grid TheoremStephan Kreutzer – TU Berlin

Defined analogous to undirected tree width but based on strong separators

5

1 2

3

DirectedTreeWidth

6

1 2

3

54

6 7

98

10

8

11

4 6 7

58 58 58

The Directed Grid TheoremStephan Kreutzer – TU Berlin

DirectedTreeWidth

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The Directed Grid TheoremStephan Kreutzer – TU Berlin

DirectedTreeWidth

7

The Directed Grid TheoremStephan Kreutzer – TU Berlin

DirectedTreeWidth

7

The Directed Grid TheoremStephan Kreutzer – TU Berlin

DirectedTreeWidth

8

The Directed Grid TheoremStephan Kreutzer – TU Berlin

StructureTheoryforDirectedGraphs

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s1s2

s3

t1 t2

t3

The Directed Grid TheoremStephan Kreutzer – TU Berlin

ApplicationsofDirectedTreeWidth

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The Directed Grid TheoremStephan Kreutzer – TU Berlin

Graphsvs.Digraphs

11

planar

Apexfree/lin.localtw

excludedminors

trees treewidth

allgraphclasses

UndirectedGraphs

…

DirectedGraphs

directedtreewidth

The Directed Grid TheoremStephan Kreutzer – TU Berlin

Graphsvs.Digraphs

11

planar

Apexfree/lin.localtw

excludedminors

trees treewidth

allgraphclasses

UndirectedGraphs

…

DirectedGraphs

directedtreewidth

TheGridTheorem

The Directed Grid TheoremStephan Kreutzer – TU Berlin

TheDirectedGridConjecture

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The Directed Grid TheoremStephan Kreutzer – TU Berlin

TheDirectedGridConjecture

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

The Directed Grid TheoremStephan Kreutzer – TU Berlin

TheDirectedGridConjecture

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The Directed Grid TheoremStephan Kreutzer – TU Berlin

TheDirectedGridTheorem

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The Directed Grid TheoremStephan Kreutzer – TU Berlin

ProofoftheDirectedGridTheorem

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The Directed Grid TheoremStephan Kreutzer – TU Berlin

ProofoftheDirectedGridTheorem

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High Directed Tree Width

The Directed Grid TheoremStephan Kreutzer – TU Berlin

ProofoftheDirectedGridTheorem

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High Directed Tree WidthDirected Bramble

The Directed Grid TheoremStephan Kreutzer – TU Berlin

ProofoftheDirectedGridTheorem

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High Directed Tree Width Well-Linked Path SystemDirected Bramble

The Directed Grid TheoremStephan Kreutzer – TU Berlin

Well-LinkedPathSystems

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The Directed Grid TheoremStephan Kreutzer – TU Berlin

Well-LinkedPathSystems

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The Directed Grid TheoremStephan Kreutzer – TU Berlin

Well-LinkedPathSystems

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The Directed Grid TheoremStephan Kreutzer – TU Berlin

Well-LinkedPathSystems

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The Directed Grid TheoremStephan Kreutzer – TU Berlin

AWeb

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Q

P

The Directed Grid TheoremStephan Kreutzer – TU Berlin

AWeb

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Q

P

The Directed Grid TheoremStephan Kreutzer – TU Berlin

AWeb

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Q

P

The Directed Grid TheoremStephan Kreutzer – TU Berlin

AWeb

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Q

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The Directed Grid TheoremStephan Kreutzer – TU Berlin

AWeb

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Q

P

The Directed Grid TheoremStephan Kreutzer – TU Berlin

AWeb

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Q

P

The Directed Grid TheoremStephan Kreutzer – TU Berlin

Well-LinkedFences

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The Directed Grid TheoremStephan Kreutzer – TU Berlin

Bottom-UpLinkages

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R

The Directed Grid TheoremStephan Kreutzer – TU Berlin

AvoidingaSub-Fence

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The Directed Grid TheoremStephan Kreutzer – TU Berlin

AvoidingaSub-Fence

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The Directed Grid TheoremStephan Kreutzer – TU Berlin

GoingupRow-By-Row

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The Directed Grid TheoremStephan Kreutzer – TU Berlin

TheDirectedGridTheorem

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The Directed Grid TheoremStephan Kreutzer – TU Berlin

Erdős-PósaPropertyforDigraphs

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The Directed Grid TheoremStephan Kreutzer – TU Berlin

Erdős-PósaPropertyforDigraphs

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The Directed Grid TheoremStephan Kreutzer – TU Berlin

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.

The Directed Grid TheoremStephan Kreutzer – TU Berlin

Erdős-PósaPropertyforDigraphs

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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.

The Directed Grid TheoremStephan Kreutzer – TU Berlin

Semi-IntegralDisjointPathsProblem

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•

s1s2

s3

t1 t2

t3

The Directed Grid TheoremStephan Kreutzer – TU Berlin

Semi-IntegralDisjointPathsProblem

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•

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 TheoremStephan Kreutzer – TU Berlin

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

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The Directed Grid TheoremStephan Kreutzer – TU Berlin

planar

Apexfree/lin.localtw

excludedminors

trees treewidth

allgraphclasses

…

31

nowheredense

Directedtreewidth

StructureTheoryforUndirectedGraphs

nowherecrownful