A Faster Parameterized Algorithm for Treedepth - Semantic Scholar
A Faster Parameterized Algorithm for Treedepth Felix Reidl, Peter Rossmanith, Fernando S´ anchez Villaamil Somnath Sikdar RWTH Aachen University
July 11, 2014
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Treedepth is a width measure.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
5 3
2 1
7 4
6 8
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
5 3 1
7 4
1
2
3
2
6 6
5
4 7
8
8
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
5 3 1
7 4
1
2
3
2
6 6
5
4 7
8
8
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Definition (Treedepth decomposition) A treedepth decomposition of a graph G is a rooted forest F such that V (G ) ⊆ V (F ) and E (G ) ⊆ E (clos(F )).
Definition (Treedepth) The treedepth td(G ) of a graph G is the minimum height of any treedepth decomposition of G .
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
A strange width measure... “So many choices” —Dr. Dre
A graph G has treedepth at most t if G is a subgraph the closure of a tree (forest) of height ≤ t G has a centered coloring with t colors G has a ranked coloring with t colors G is the subgraph of a trivially perfect graph with clique size ≤ t
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Centered Coloring
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Centered Coloring
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Centered Coloring
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Ranked Coloring
1 2 1
5 4 3
Fernando S´ anchez Villaamil (RWTH)
1 2 1
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Ranked Coloring
1 2 1
5 4 3
Fernando S´ anchez Villaamil (RWTH)
1 2 1
5 4 2
3
2
1
1 1
1
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Trivially Perfect Graphs G is the subgraph of a trivially perfect graph with clique size at most t.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
10 / 51
Treedepth
Trivially Perfect Graphs G is the subgraph of a trivially perfect graph with clique size at most t.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
10 / 51
Treedepth
Arises again and again Introduced as... minimum elimination tree by Pothen [1988] ordered coloring by Katchalski et al. [1995] vertex ranking by Bodlaender et al. [1998] again as treedepth by Neˇsetˇril and Ossona de Mendez [2008]
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
11 / 51
Treedepth
Arises again and again Introduced as... minimum elimination tree by Pothen [1988] ordered coloring by Katchalski et al. [1995] vertex ranking by Bodlaender et al. [1998] again as treedepth by Neˇsetˇril and Ossona de Mendez [2008] Related to... layouting of VLSI chips star height of regular languages characterizing bounded expansion graph classes counting subgraphs [New results coming]
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
11 / 51
Treedepth
Arises again and again Introduced as... minimum elimination tree by Pothen [1988] ordered coloring by Katchalski et al. [1995] vertex ranking by Bodlaender et al. [1998] again as treedepth by Neˇsetˇril and Ossona de Mendez [2008] Related to... layouting of VLSI chips star height of regular languages characterizing bounded expansion graph classes counting subgraphs [New results coming] Personal opinion: Treedepth is the most useful definition.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
11 / 51
Treedepth
1
2
3
4
4
5
3
7
6
5
1
2
7
6
Treedepth t → Maximal path length 2t − 1.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Treedepth t → Maximal path length 2t − 1.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
13 / 51
Treedepth
Treedepth t → Maximal path length 2t − 1.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
13 / 51
Treedepth
Basic results
A DFS is a Treedepth decomposition
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
14 / 51
Treedepth
Basic results
A DFS is a Treedepth decomposition
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
14 / 51
Treedepth
Basic results
A DFS is a Treedepth decomposition
Treedepth t ⇒ Maximal path length 2t − 1 ⇒ 2t -approximation
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Basic results
Treedepth to pathwidth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Basic results
Treedepth to pathwidth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
15 / 51
Treedepth
Basic results
Treedepth to pathwidth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
15 / 51
Treedepth
Basic results
Treedepth to pathwidth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
15 / 51
Treedepth
Basic results
Treedepth to pathwidth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
15 / 51
Treedepth
Basic results
Treedepth to pathwidth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
15 / 51
Treedepth
Basic results
Treedepth to pathwidth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
15 / 51
Treedepth
Basic results
Treedepth to pathwidth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
15 / 51
Treedepth
Basic results
Treedepth to pathwidth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
15 / 51
Treedepth
Basic results
Treedepth to pathwidth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
15 / 51
Treedepth
Basic results
Treedepth to pathwidth
tw(G ) ≤ pw(G ) ≤ td(G ) − 1 Treedepth t ⇒ Path decomposition of width 2t − 2 Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Fernando S´ anchez Villaamil (RWTH)
Basic results
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Fernando S´ anchez Villaamil (RWTH)
Basic results
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Fernando S´ anchez Villaamil (RWTH)
Basic results
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Basic results
Treedepth by bruteforce
G
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
17 / 51
Treedepth
Basic results
Treedepth by bruteforce
G S
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
17 / 51
Treedepth
Basic results
Treedepth by bruteforce S
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
17 / 51
Treedepth
Basic results
Treedepth by bruteforce
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
17 / 51
Treedepth
Basic results
Treedepth by bruteforce
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
17 / 51
Treedepth
Basic results
Parameterized algorithms
Open problem by Neˇsetˇril and Ossona de Mendez [2012] Is there a simple linear time algorithm to check td(G ) ≤ t for fixed t?
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
18 / 51
Treedepth
Basic results
Parameterized algorithms
Open problem by Neˇsetˇril and Ossona de Mendez [2012] Is there a simple linear time algorithm to check td(G ) ≤ t for fixed t? In f (t) · n3 time by Robertson and Seymour. tw(G ) ≤ td(G ) − 1 ⇒ By Courcelle’s Theorem
2 22
..
.t
· n.
Algorithm by Bodlaender et. al. with running time 2O(w
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
2 t)
· n2 .
July 11, 2014
18 / 51
Treedepth
Basic results
Our results: O(t)
A (relatively) simple direct algorithm in time 22 A fast algorithm in time
Fernando S´ anchez Villaamil (RWTH)
2 2O(t )
· n.
· n.
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Basic results
Our results: O(t)
A (relatively) simple direct algorithm in time 22 A fast algorithm in time
2 2O(t )
· n.
· n.
Both results follow from an algorithm on tree decompositions which runs in time 2O(wt) · n.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Where could the introduced node u be?
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
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The algorithm
Definition (Nice treedepth decomposition) We say that T is nice if for every vertex x ∈ V (T ), the subgraph of G induced by the vertices in Tx is connected.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
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The algorithm
Lemma For any graph there exists a treedepth decomposition of minimal depth which is nice.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
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The algorithm
Where could the introduced node u be?
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Theorem Given a graph G with n nodes and a tree decomposition of G of width w the treedepth t of G can be decided in time 2O(wt) · n.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Simple algorithm
Simple algorithm
1
Depth-first-search to construct treedepth decomposition T .
2
If depth greater than 2t − 1 say NO.
3
Construct path decomposition P from T of width 2t .
4
Run algorithm on P.
Theorem There is a (simple) algorithm to decide if a graph G with n nodes has O(t) treedepth t which runs in time 22 · n.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Fast algorithm
Fast algorithm
1
Use single exponential 5-approximation for treewidth1 .
2
Remember tw(G ) ≤ pw(G ) ≤ td(G ) − 1.
3
If width is greater than 5t say NO.
4
Else run algorithm on tree decomposition.
Theorem There is a algorithm to decide if a graph G with n nodes has treedepth t 2 which runs in time 2O(t ) · n.
1 Very useful result by Hans Bodlaender, P˚ al G. Drange, Markus S. Dregi, Fedor V. Fomin, Daniel Lokshtanov and Michal Pilipczuk Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
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The End
Thank you for listening. Questions?
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
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July 11, 2014
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Treedepth is a width measure.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
5 3
2 1
7 4
6 8
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
5 3 1
7 4
1
2
3
2
6 6
5
4 7
8
8
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
5 3 1
7 4
1
2
3
2
6 6
5
4 7
8
8
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Definition (Treedepth decomposition) A treedepth decomposition of a graph G is a rooted forest F such that V (G ) ⊆ V (F ) and E (G ) ⊆ E (clos(F )).
Definition (Treedepth) The treedepth td(G ) of a graph G is the minimum height of any treedepth decomposition of G .
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
5 / 51
Treedepth
A strange width measure... “So many choices” —Dr. Dre
A graph G has treedepth at most t if G is a subgraph the closure of a tree (forest) of height ≤ t G has a centered coloring with t colors G has a ranked coloring with t colors G is the subgraph of a trivially perfect graph with clique size ≤ t
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
6 / 51
Treedepth
Centered Coloring
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
7 / 51
Treedepth
Centered Coloring
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
8 / 51
Treedepth
Centered Coloring
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
8 / 51
Treedepth
Ranked Coloring
1 2 1
5 4 3
Fernando S´ anchez Villaamil (RWTH)
1 2 1
Parameterized Algorithm for Treedepth
July 11, 2014
9 / 51
Treedepth
Ranked Coloring
1 2 1
5 4 3
Fernando S´ anchez Villaamil (RWTH)
1 2 1
5 4 2
3
2
1
1 1
1
Parameterized Algorithm for Treedepth
July 11, 2014
9 / 51
Treedepth
Trivially Perfect Graphs G is the subgraph of a trivially perfect graph with clique size at most t.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
10 / 51
Treedepth
Trivially Perfect Graphs G is the subgraph of a trivially perfect graph with clique size at most t.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
10 / 51
Treedepth
Arises again and again Introduced as... minimum elimination tree by Pothen [1988] ordered coloring by Katchalski et al. [1995] vertex ranking by Bodlaender et al. [1998] again as treedepth by Neˇsetˇril and Ossona de Mendez [2008]
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
11 / 51
Treedepth
Arises again and again Introduced as... minimum elimination tree by Pothen [1988] ordered coloring by Katchalski et al. [1995] vertex ranking by Bodlaender et al. [1998] again as treedepth by Neˇsetˇril and Ossona de Mendez [2008] Related to... layouting of VLSI chips star height of regular languages characterizing bounded expansion graph classes counting subgraphs [New results coming]
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
11 / 51
Treedepth
Arises again and again Introduced as... minimum elimination tree by Pothen [1988] ordered coloring by Katchalski et al. [1995] vertex ranking by Bodlaender et al. [1998] again as treedepth by Neˇsetˇril and Ossona de Mendez [2008] Related to... layouting of VLSI chips star height of regular languages characterizing bounded expansion graph classes counting subgraphs [New results coming] Personal opinion: Treedepth is the most useful definition.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
11 / 51
Treedepth
1
2
3
4
4
5
3
7
6
5
1
2
7
6
Treedepth t → Maximal path length 2t − 1.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
12 / 51
Treedepth
Treedepth t → Maximal path length 2t − 1.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
13 / 51
Treedepth
Treedepth t → Maximal path length 2t − 1.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
13 / 51
Treedepth
Basic results
A DFS is a Treedepth decomposition
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
14 / 51
Treedepth
Basic results
A DFS is a Treedepth decomposition
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
14 / 51
Treedepth
Basic results
A DFS is a Treedepth decomposition
Treedepth t ⇒ Maximal path length 2t − 1 ⇒ 2t -approximation
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
14 / 51
Treedepth
Basic results
Treedepth to pathwidth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
15 / 51
Treedepth
Basic results
Treedepth to pathwidth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
15 / 51
Treedepth
Basic results
Treedepth to pathwidth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
15 / 51
Treedepth
Basic results
Treedepth to pathwidth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
15 / 51
Treedepth
Basic results
Treedepth to pathwidth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
15 / 51
Treedepth
Basic results
Treedepth to pathwidth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
15 / 51
Treedepth
Basic results
Treedepth to pathwidth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
15 / 51
Treedepth
Basic results
Treedepth to pathwidth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
15 / 51
Treedepth
Basic results
Treedepth to pathwidth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
15 / 51
Treedepth
Basic results
Treedepth to pathwidth
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
15 / 51
Treedepth
Basic results
Treedepth to pathwidth
tw(G ) ≤ pw(G ) ≤ td(G ) − 1 Treedepth t ⇒ Path decomposition of width 2t − 2 Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
15 / 51
Treedepth
Fernando S´ anchez Villaamil (RWTH)
Basic results
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Fernando S´ anchez Villaamil (RWTH)
Basic results
Parameterized Algorithm for Treedepth
July 11, 2014
16 / 51
Treedepth
Fernando S´ anchez Villaamil (RWTH)
Basic results
Parameterized Algorithm for Treedepth
July 11, 2014
16 / 51
Treedepth
Basic results
Treedepth by bruteforce
G
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Basic results
Treedepth by bruteforce
G S
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
17 / 51
Treedepth
Basic results
Treedepth by bruteforce S
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
17 / 51
Treedepth
Basic results
Treedepth by bruteforce
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
17 / 51
Treedepth
Basic results
Treedepth by bruteforce
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
17 / 51
Treedepth
Basic results
Parameterized algorithms
Open problem by Neˇsetˇril and Ossona de Mendez [2012] Is there a simple linear time algorithm to check td(G ) ≤ t for fixed t?
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
18 / 51
Treedepth
Basic results
Parameterized algorithms
Open problem by Neˇsetˇril and Ossona de Mendez [2012] Is there a simple linear time algorithm to check td(G ) ≤ t for fixed t? In f (t) · n3 time by Robertson and Seymour. tw(G ) ≤ td(G ) − 1 ⇒ By Courcelle’s Theorem
2 22
..
.t
· n.
Algorithm by Bodlaender et. al. with running time 2O(w
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
2 t)
· n2 .
July 11, 2014
18 / 51
Treedepth
Basic results
Our results: O(t)
A (relatively) simple direct algorithm in time 22 A fast algorithm in time
Fernando S´ anchez Villaamil (RWTH)
2 2O(t )
· n.
· n.
Parameterized Algorithm for Treedepth
July 11, 2014
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Treedepth
Basic results
Our results: O(t)
A (relatively) simple direct algorithm in time 22 A fast algorithm in time
2 2O(t )
· n.
· n.
Both results follow from an algorithm on tree decompositions which runs in time 2O(wt) · n.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
19 / 51
The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
20 / 51
The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
20 / 51
The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
20 / 51
The algorithm
Where could the introduced node u be?
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
20 / 51
The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Definition (Nice treedepth decomposition) We say that T is nice if for every vertex x ∈ V (T ), the subgraph of G induced by the vertices in Tx is connected.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Lemma For any graph there exists a treedepth decomposition of minimal depth which is nice.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Where could the introduced node u be?
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Theorem Given a graph G with n nodes and a tree decomposition of G of width w the treedepth t of G can be decided in time 2O(wt) · n.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Simple algorithm
Simple algorithm
1
Depth-first-search to construct treedepth decomposition T .
2
If depth greater than 2t − 1 say NO.
3
Construct path decomposition P from T of width 2t .
4
Run algorithm on P.
Theorem There is a (simple) algorithm to decide if a graph G with n nodes has O(t) treedepth t which runs in time 22 · n.
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The algorithm
Fast algorithm
Fast algorithm
1
Use single exponential 5-approximation for treewidth1 .
2
Remember tw(G ) ≤ pw(G ) ≤ td(G ) − 1.
3
If width is greater than 5t say NO.
4
Else run algorithm on tree decomposition.
Theorem There is a algorithm to decide if a graph G with n nodes has treedepth t 2 which runs in time 2O(t ) · n.
1 Very useful result by Hans Bodlaender, P˚ al G. Drange, Markus S. Dregi, Fedor V. Fomin, Daniel Lokshtanov and Michal Pilipczuk Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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The End
Thank you for listening. Questions?
Fernando S´ anchez Villaamil (RWTH)
Parameterized Algorithm for Treedepth
July 11, 2014
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