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
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
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
7
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
9
s1s2
s3
t1 t2
t3
The Directed Grid TheoremStephan Kreutzer – TU Berlin
ApplicationsofDirectedTreeWidth
10
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
12
The Directed Grid TheoremStephan Kreutzer – TU Berlin
TheDirectedGridConjecture
12
Butterfly minor: contract an edge if only outgoing of tail or only incoming of head.
The Directed Grid TheoremStephan Kreutzer – TU Berlin
TheDirectedGridConjecture
13
The Directed Grid TheoremStephan Kreutzer – TU Berlin
TheDirectedGridTheorem
14
The Directed Grid TheoremStephan Kreutzer – TU Berlin
ProofoftheDirectedGridTheorem
15
The Directed Grid TheoremStephan Kreutzer – TU Berlin
ProofoftheDirectedGridTheorem
16
High Directed Tree Width
The Directed Grid TheoremStephan Kreutzer – TU Berlin
ProofoftheDirectedGridTheorem
16
High Directed Tree WidthDirected Bramble
The Directed Grid TheoremStephan Kreutzer – TU Berlin
ProofoftheDirectedGridTheorem
16
High Directed Tree Width Well-Linked Path SystemDirected Bramble
The Directed Grid TheoremStephan Kreutzer – TU Berlin
Well-LinkedPathSystems
17
The Directed Grid TheoremStephan Kreutzer – TU Berlin
Well-LinkedPathSystems
17
The Directed Grid TheoremStephan Kreutzer – TU Berlin
Well-LinkedPathSystems
17
The Directed Grid TheoremStephan Kreutzer – TU Berlin
Well-LinkedPathSystems
17
The Directed Grid TheoremStephan Kreutzer – TU Berlin
AWeb
18
Q
P
The Directed Grid TheoremStephan Kreutzer – TU Berlin
AWeb
18
Q
P
The Directed Grid TheoremStephan Kreutzer – TU Berlin
AWeb
19
Q
P
The Directed Grid TheoremStephan Kreutzer – TU Berlin
AWeb
19
Q
P
The Directed Grid TheoremStephan Kreutzer – TU Berlin
AWeb
19
Q
P
The Directed Grid TheoremStephan Kreutzer – TU Berlin
AWeb
20
Q
P
The Directed Grid TheoremStephan Kreutzer – TU Berlin
Well-LinkedFences
21
The Directed Grid TheoremStephan Kreutzer – TU Berlin
Bottom-UpLinkages
22
R
The Directed Grid TheoremStephan Kreutzer – TU Berlin
AvoidingaSub-Fence
23
R
The Directed Grid TheoremStephan Kreutzer – TU Berlin
AvoidingaSub-Fence
23
R
The Directed Grid TheoremStephan Kreutzer – TU Berlin
GoingupRow-By-Row
24
The Directed Grid TheoremStephan Kreutzer – TU Berlin
TheDirectedGridTheorem
25
The Directed Grid TheoremStephan Kreutzer – TU Berlin
Erdős-PósaPropertyforDigraphs
26
The Directed Grid TheoremStephan Kreutzer – TU Berlin
Erdős-PósaPropertyforDigraphs
26
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
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
Semi-IntegralDisjointPathsProblem
28
•
s1s2
s3
t1 t2
t3
The Directed Grid TheoremStephan Kreutzer – TU Berlin
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 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
The Directed Grid TheoremStephan Kreutzer – TU Berlin
planar
Apexfree/lin.localtw
excludedminors
trees treewidth
allgraphclasses
…
31
nowheredense
Directedtreewidth
StructureTheoryforUndirectedGraphs
nowherecrownful