A Faster Parameterized Algorithm for Treedepth Felix Reidl, Peter - - PowerPoint PPT Presentation

▶
a faster parameterized algorithm for treedepth
SMART_READER_LITE
LIVE PREVIEW

A Faster Parameterized Algorithm for Treedepth Felix Reidl, Peter - - PowerPoint PPT Presentation

May 18, 2023 48 likes •770 views

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

slide-1
SLIDE 1

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 1 / 51

slide-2
SLIDE 2

Treedepth

Treedepth is a width measure.

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 2 / 51

slide-3
SLIDE 3

Treedepth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 3 / 51

slide-4
SLIDE 4

Treedepth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 3 / 51

slide-5
SLIDE 5

Treedepth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 3 / 51

slide-6
SLIDE 6

Treedepth

7 3 5 2 6 1 4 8

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 4 / 51

slide-7
SLIDE 7

Treedepth

7 3 5 2 6 1 4 8 7 3 5 2 6 1 4 8

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 4 / 51

slide-8
SLIDE 8

Treedepth

7 3 5 2 6 1 4 8 7 3 5 2 6 1 4 8

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 4 / 51

slide-9
SLIDE 9

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

slide-10
SLIDE 10

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

slide-11
SLIDE 11

Treedepth

Centered Coloring

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 7 / 51

slide-12
SLIDE 12

Treedepth

Centered Coloring

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 8 / 51

slide-13
SLIDE 13

Treedepth

Centered Coloring

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 8 / 51

slide-14
SLIDE 14

Treedepth

Ranked Coloring

1 1 1 1 2 2 3 4 5

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 9 / 51

slide-15
SLIDE 15

Treedepth

Ranked Coloring

1 1 1 1 2 2 3 4 5

1 1 1 1 2 2 3 4 5

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 9 / 51

slide-16
SLIDE 16

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

slide-17
SLIDE 17

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

slide-18
SLIDE 18

Treedepth

Arises again and again

Introduced as... minimum elimination tree by Pothen [1988]

  • rdered coloring by Katchalski et al. [1995]

vertex ranking by Bodlaender et al. [1998] again as treedepth by Neˇ setˇ ril and Ossona de Mendez [2008]

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 11 / 51

slide-19
SLIDE 19

Treedepth

Arises again and again

Introduced as... minimum elimination tree by Pothen [1988]

  • rdered coloring by Katchalski et al. [1995]

vertex ranking by Bodlaender et al. [1998] again as treedepth by Neˇ setˇ ril 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

slide-20
SLIDE 20

Treedepth

Arises again and again

Introduced as... minimum elimination tree by Pothen [1988]

  • rdered coloring by Katchalski et al. [1995]

vertex ranking by Bodlaender et al. [1998] again as treedepth by Neˇ setˇ ril 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

slide-21
SLIDE 21

Treedepth

7 3 5 2 6 1 4

7 3 5 2 6 1 4 Treedepth t → Maximal path length 2t − 1.

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 12 / 51

slide-22
SLIDE 22

Treedepth

Treedepth t → Maximal path length 2t − 1.

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 13 / 51

slide-23
SLIDE 23

Treedepth

Treedepth t → Maximal path length 2t − 1.

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 13 / 51

slide-24
SLIDE 24

Treedepth Basic results

A DFS is a Treedepth decomposition

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 14 / 51

slide-25
SLIDE 25

Treedepth Basic results

A DFS is a Treedepth decomposition

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 14 / 51

slide-26
SLIDE 26

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

slide-27
SLIDE 27

Treedepth Basic results

Treedepth to pathwidth

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

slide-28
SLIDE 28

Treedepth Basic results

Treedepth to pathwidth

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

slide-29
SLIDE 29

Treedepth Basic results

Treedepth to pathwidth

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

slide-30
SLIDE 30

Treedepth Basic results

Treedepth to pathwidth

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

slide-31
SLIDE 31

Treedepth Basic results

Treedepth to pathwidth

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

slide-32
SLIDE 32

Treedepth Basic results

Treedepth to pathwidth

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

slide-33
SLIDE 33

Treedepth Basic results

Treedepth to pathwidth

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

slide-34
SLIDE 34

Treedepth Basic results

Treedepth to pathwidth

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

slide-35
SLIDE 35

Treedepth Basic results

Treedepth to pathwidth

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

slide-36
SLIDE 36

Treedepth Basic results

Treedepth to pathwidth

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

slide-37
SLIDE 37

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

slide-38
SLIDE 38

Treedepth Basic results Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 16 / 51

slide-39
SLIDE 39

Treedepth Basic results Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 16 / 51

slide-40
SLIDE 40

Treedepth Basic results Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 16 / 51

slide-41
SLIDE 41

Treedepth Basic results

Treedepth by bruteforce

G

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 17 / 51

slide-42
SLIDE 42

Treedepth Basic results

Treedepth by bruteforce

G S

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 17 / 51

slide-43
SLIDE 43

Treedepth Basic results

Treedepth by bruteforce S

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 17 / 51

slide-44
SLIDE 44

Treedepth Basic results

Treedepth by bruteforce

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 17 / 51

slide-45
SLIDE 45

Treedepth Basic results

Treedepth by bruteforce

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 17 / 51

slide-46
SLIDE 46

Treedepth Basic results

Parameterized algorithms

Open problem by Neˇ setˇ ril 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

slide-47
SLIDE 47

Treedepth Basic results

Parameterized algorithms

Open problem by Neˇ setˇ ril 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 222...t · n. Algorithm by Bodlaender et. al. with running time 2O(w2t) · n2.

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 18 / 51

slide-48
SLIDE 48

Treedepth Basic results

Our results: A (relatively) simple direct algorithm in time 22O(t) · n. A fast algorithm in time 2O(t2) · n.

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 19 / 51

slide-49
SLIDE 49

Treedepth Basic results

Our results: A (relatively) simple direct algorithm in time 22O(t) · n. A fast algorithm in time 2O(t2) · 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

slide-50
SLIDE 50

The algorithm Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 20 / 51

slide-51
SLIDE 51

The algorithm Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 20 / 51

slide-52
SLIDE 52

The algorithm Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 20 / 51

slide-53
SLIDE 53

The algorithm Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 20 / 51

slide-54
SLIDE 54

The algorithm

Where could the introduced node u be?

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 20 / 51

slide-55
SLIDE 55

The algorithm Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 21 / 51

slide-56
SLIDE 56

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 21 / 51

slide-57
SLIDE 57

The algorithm Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 22 / 51

slide-58
SLIDE 58

The algorithm Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 22 / 51

slide-59
SLIDE 59

The algorithm Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 22 / 51

slide-60
SLIDE 60

The algorithm Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 22 / 51

slide-61
SLIDE 61

The algorithm Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 22 / 51

slide-62
SLIDE 62

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 23 / 51

slide-63
SLIDE 63

The algorithm

Where could the introduced node u be?

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 24 / 51

slide-64
SLIDE 64

The algorithm Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 25 / 51

slide-65
SLIDE 65

The algorithm Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 26 / 51

slide-66
SLIDE 66

The algorithm Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 26 / 51

slide-67
SLIDE 67

The algorithm Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 26 / 51

slide-68
SLIDE 68

The algorithm Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 27 / 51

slide-69
SLIDE 69

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 28 / 51

slide-70
SLIDE 70

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 treedepth t which runs in time 22O(t) · n.

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 29 / 51

slide-71
SLIDE 71

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 which runs in time 2O(t2) · n.

1Very useful result by Hans Bodlaender, P˚

al G. Drange, Markus S. Dregi, Fedor V. Fomin, Daniel Lokshtanov and Micha l Pilipczuk

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 30 / 51

slide-72
SLIDE 72

The End

Thank you for listening. Questions?

Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 31 / 51