the directed grid...
TRANSCRIPT
![Page 1: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/1.jpg)
AlgorithmicDigraphStructureTheory
TheDirectedGridTheorem
Stephan Kreutzer
Technical University Berlin
Joint work with Ken-ichi Kawarabayashi, NII, Tokyo
StephanKreutzer–TUBerlin
![Page 2: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/2.jpg)
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
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The Directed Grid TheoremStephan Kreutzer – TU Berlin
planar
apexfree/lin.localtw
excludedminors
trees treewidth
allgraphclasses
bidimensionalityirrelevantvertices
…
structural decomposition
…
StructureTheoryforUndirectedGraphs
3
dynamicprogramming
AlgorithmicTechniquesStructuralResults
nowheredense
![Page 4: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/4.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
planar
apexfree/lin.localtw
excludedminors
trees treewidth
allgraphclasses
bidimensionalityirrelevantvertices
…
structural decomposition
…
StructureTheoryforUndirectedGraphs
3
dynamicprogramming
AlgorithmicTechniquesStructuralResults
nowheredense
![Page 5: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/5.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
StructureTheoryforDirectedGraphs
4
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The Directed Grid TheoremStephan Kreutzer – TU Berlin
UndirectedTreeWidth
5
1 2
3
54
6 7
98
10 11
![Page 7: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/7.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
1 2
3
UndirectedTreeWidth
5
1 2
3
54
6 7
98
10 11
![Page 8: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/8.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
1 2
3
UndirectedTreeWidth
5
1 2
3
54
6 7
98
10 11
![Page 9: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/9.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
1 2
3
UndirectedTreeWidth
5
1 2
3
54
6 7
98
10 11
![Page 10: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/10.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
1 2
3
UndirectedTreeWidth
5
1 2
3
54
6 7
98
10 11
![Page 11: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/11.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
1
3
5
1 2
3
UndirectedTreeWidth
5
1 2
3
54
6 7
98
10 11
![Page 12: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/12.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
1
3
5
1 2
3
UndirectedTreeWidth
5
1 2
3
54
6 7
98
10 11
![Page 13: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/13.jpg)
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
11
![Page 14: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/14.jpg)
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
![Page 15: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/15.jpg)
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
![Page 16: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/16.jpg)
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
![Page 17: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/17.jpg)
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
![Page 18: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/18.jpg)
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
![Page 19: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/19.jpg)
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
![Page 20: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/20.jpg)
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
![Page 21: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/21.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
DirectedTreeWidth
7
![Page 22: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/22.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
DirectedTreeWidth
7
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The Directed Grid TheoremStephan Kreutzer – TU Berlin
DirectedTreeWidth
7
![Page 24: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/24.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
DirectedTreeWidth
8
![Page 25: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/25.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
StructureTheoryforDirectedGraphs
9
s1s2
s3
t1 t2
t3
![Page 26: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/26.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
ApplicationsofDirectedTreeWidth
10
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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
![Page 28: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/28.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
Graphsvs.Digraphs
11
planar
Apexfree/lin.localtw
excludedminors
trees treewidth
allgraphclasses
UndirectedGraphs
…
DirectedGraphs
directedtreewidth
TheGridTheorem
![Page 29: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/29.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
TheDirectedGridConjecture
12
![Page 30: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/30.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
TheDirectedGridConjecture
12
Butterfly minor: contract an edge if only outgoing of tail or only incoming of head.
![Page 31: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/31.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
TheDirectedGridConjecture
13
![Page 32: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/32.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
TheDirectedGridTheorem
14
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The Directed Grid TheoremStephan Kreutzer – TU Berlin
ProofoftheDirectedGridTheorem
15
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The Directed Grid TheoremStephan Kreutzer – TU Berlin
ProofoftheDirectedGridTheorem
16
High Directed Tree Width
![Page 35: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/35.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
ProofoftheDirectedGridTheorem
16
High Directed Tree WidthDirected Bramble
![Page 36: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/36.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
ProofoftheDirectedGridTheorem
16
High Directed Tree Width Well-Linked Path SystemDirected Bramble
![Page 37: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/37.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
Well-LinkedPathSystems
17
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The Directed Grid TheoremStephan Kreutzer – TU Berlin
Well-LinkedPathSystems
17
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The Directed Grid TheoremStephan Kreutzer – TU Berlin
Well-LinkedPathSystems
17
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The Directed Grid TheoremStephan Kreutzer – TU Berlin
Well-LinkedPathSystems
17
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The Directed Grid TheoremStephan Kreutzer – TU Berlin
AWeb
18
Q
P
![Page 42: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/42.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
AWeb
18
Q
P
![Page 43: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/43.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
AWeb
19
Q
P
![Page 44: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/44.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
AWeb
19
Q
P
![Page 45: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/45.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
AWeb
19
Q
P
![Page 46: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/46.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
AWeb
20
Q
P
![Page 47: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/47.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
Well-LinkedFences
21
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The Directed Grid TheoremStephan Kreutzer – TU Berlin
Bottom-UpLinkages
22
R
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The Directed Grid TheoremStephan Kreutzer – TU Berlin
AvoidingaSub-Fence
23
R
![Page 50: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/50.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
AvoidingaSub-Fence
23
R
![Page 51: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/51.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
GoingupRow-By-Row
24
![Page 52: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/52.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
TheDirectedGridTheorem
25
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The Directed Grid TheoremStephan Kreutzer – TU Berlin
Erdős-PósaPropertyforDigraphs
26
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The Directed Grid TheoremStephan Kreutzer – TU Berlin
Erdős-PósaPropertyforDigraphs
26
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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.
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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.
![Page 57: The Directed Grid Theorem2016.highlightsofalgorithms.org/wp-content/uploads/2015/12/Stephan-Kreutzer.pdfAlgorithmic Digraph Structure Theory The Directed Grid Theorem Stephan Kreutzer](https://reader033.vdocuments.us/reader033/viewer/2022042313/5edc9f1cad6a402d66675dc6/html5/thumbnails/57.jpg)
The Directed Grid TheoremStephan Kreutzer – TU Berlin
Semi-IntegralDisjointPathsProblem
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•
s1s2
s3
t1 t2
t3
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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).
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The Directed Grid TheoremStephan Kreutzer – TU Berlin 29
Conclusion
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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
30
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The Directed Grid TheoremStephan Kreutzer – TU Berlin
planar
Apexfree/lin.localtw
excludedminors
trees treewidth
allgraphclasses
…
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nowheredense
Directedtreewidth
StructureTheoryforUndirectedGraphs
nowherecrownful