Moving Vertices to Make Drawings Plane Xavier Goaoc INRIA Lorraine - - PowerPoint PPT Presentation

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Moving Vertices to Make Drawings Plane Xavier Goaoc INRIA Lorraine - - PowerPoint PPT Presentation

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Moving Vertices to Make Drawings Plane Xavier Goaoc INRIA Lorraine FR Jan Kratochv l Charles U CZ Yoshio Okamoto Toyohashi U Tech JP Chan-Su Shin Hankuk U Foreign Studies KR Alexander Wolff TU Eindhoven NL September 24, 2007 @

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Moving Vertices to Make Drawings Plane

Xavier Goaoc

INRIA Lorraine FR

Jan Kratochv´ ıl

Charles U CZ

Yoshio Okamoto

Toyohashi U Tech JP

Chan-Su Shin

Hankuk U Foreign Studies KR

Alexander Wolff

TU Eindhoven NL

September 24, 2007 @ 15th International Conference on Graph Drawing Swiss-Grand Resort & Spa Bondi Beach, Sydney, Australia

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p Game “Planarity” by John Tantalo

http://www.planarity.net/ . Given: Given: a straight-line drawing of a planar graph G

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p Game “Planarity” by John Tantalo

http://www.planarity.net/ . Given: Given: a straight-line drawing of a planar graph G . Task: Task: to make it non-crossing (i.e., plane)

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p Game “Planarity” by John Tantalo

http://www.planarity.net/ . Given: Given: a straight-line drawing of a planar graph G . Task: Task: to make it non-crossing (i.e., plane) . Operation: Operation: move vertices on the plane (by dragging)

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p Game “Planarity” by John Tantalo

http://www.planarity.net/ . Given: Given: a straight-line drawing of a planar graph G . Task: Task: to make it non-crossing (i.e., plane) . Operation: Operation: move vertices on the plane (by dragging)

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p Game “Planarity” by John Tantalo

http://www.planarity.net/ . Given: Given: a straight-line drawing of a planar graph G . Task: Task: to make it non-crossing (i.e., plane) . Operation: Operation: move vertices on the plane (by dragging)

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p Game “Planarity” by John Tantalo

http://www.planarity.net/ . Given: Given: a straight-line drawing of a planar graph G . Task: Task: to make it non-crossing (i.e., plane) . Operation: Operation: move vertices on the plane (by dragging)

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p Game “Planarity” by John Tantalo

http://www.planarity.net/ . Given: Given: a straight-line drawing of a planar graph G . Task: Task: to make it non-crossing (i.e., plane) . Operation: Operation: move vertices on the plane (by dragging)

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p As a theoretician...

. Questions Questions

  • (computational question)
  • (combinatorial question)

− →

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p As a theoretician...

. Questions Questions

  • (computational question)

How hard to find a min # of vertices to move?

  • (combinatorial question)

− →

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p As a theoretician...

. Questions Questions

  • (computational question)

How hard to find a min # of vertices to move?

  • (combinatorial question)

How many vertices must be moved in worst case? − →

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p Complexity results

. Theorem Theorem

  • It is NP-hard to compute the min # of vertices to move

in order to make a given drawing plane

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p Complexity results

. Theorem Theorem

  • It is NP-hard to compute the min # of vertices to move

in order to make a given drawing plane

  • It is NP-hard to approximate (1 + min #)

within a factor of n1−ε (for any fixed ε ∈ (0, 1]) where n = # of vertices

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p Combinatorial results

We switch to the max # of vertices that we can keep fixed

  • For n-vertex cycles

(Pach & Tardos (GD ’01, DCG ’02))

we can always keep ⌊√n⌋ vertices we can’t keep O((n log n)2/3) vertices in some cases . Theorem Theorem

  • For n-vertex trees
  • For n-vertex planar graphs
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p Combinatorial results

We switch to the max # of vertices that we can keep fixed

  • For n-vertex cycles

(Pach & Tardos (GD ’01, DCG ’02))

we can always keep ⌊√n⌋ vertices we can’t keep O((n log n)2/3) vertices in some cases . Theorem Theorem

  • For n-vertex trees

we can always keep ⌊√n/3⌋ vertices we can’t keep ⌈n/3⌉+4 vertices in some cases

  • For n-vertex planar graphs
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p Combinatorial results

We switch to the max # of vertices that we can keep fixed

  • For n-vertex cycles

(Pach & Tardos (GD ’01, DCG ’02))

we can always keep ⌊√n⌋ vertices we can’t keep O((n log n)2/3) vertices in some cases . Theorem Theorem

  • For n-vertex trees

we can always keep ⌊√n/3⌋ vertices we can’t keep ⌈n/3⌉+4 vertices in some cases

  • For n-vertex planar graphs

we can always keep 3 vertices we can’t keep ⌈ √ n−2⌉+1 vertices in some cases

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p Combinatorial results

We switch to the max # of vertices that we can keep fixed . Theorem Theorem Lower Bound Upper Bound Cycles ⌊√n⌋ O((n log n)2/3) Trees ⌊√n/3⌋ ⌈n/3⌉ + 4 General 3 ⌈ √ n−2⌉+1

Pach & Tardos (GD ’01, DCG ’02)

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p History

  • Aug 06: this work started
  • Jun 07: submitted to GD
  • Jul 07: accepted for GD
  • Today: presented at GD
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p History

  • Aug 06: this work started
  • May 07: Verbitsky @ arXiv
  • Jun 07: submitted to GD
  • Jul 07: accepted for GD
  • Today: presented at GD
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p History

  • Aug 06: this work started
  • May 07: Verbitsky @ arXiv
  • Jun 07: submitted to GD
  • Jun 07: aware of Pach & Tardos (DCG ’02)
  • Jul 07: accepted for GD
  • Today: presented at GD
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p History

  • Aug 06: this work started
  • May 07: Verbitsky @ arXiv
  • Jun 07: submitted to GD
  • Jun 07: aware of Pach & Tardos (DCG ’02)
  • Jul 07: accepted for GD
  • Jul 07: Kang, Schacht & Verbitsky @ arXiv
  • Today: presented at GD
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p History

  • Aug 06: this work started
  • May 07: Verbitsky @ arXiv
  • Jun 07: submitted to GD
  • Jun 07: aware of Pach & Tardos (DCG ’02)
  • Jul 07: accepted for GD
  • Jul 07: Kang, Schacht & Verbitsky @ arXiv
  • Sep 07: Spillner & Wolff @ arXiv
  • Today: presented at GD
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p Contents

We concentrate on the complexity result

  • Problem statement (more formally)
  • NP-hardness proof
  • Inapproximability (briefly)
  • Connection to the one-bend embeddability problem
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p Def.: Straight-Line Drawing

. Setup: Setup: G = (V, E) an undirected graph (w/o loop or parallel edges)

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p Def.: Straight-Line Drawing

. Setup: Setup: G = (V, E) an undirected graph (w/o loop or parallel edges) . Def: Def: A (straight-line) drawing of G is an injective map δ: V → I R2, image of {u, v} ∈ E is a line segment δ(u)δ(v)

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p Def.: Plane Drawing

. Setup: Setup: G = (V, E) an undirected graph (w/o loop or parallel edges) . Def: Def: A drawing δ of G is plane if (the images under δ of) two edges are

  • nly allowed to share a common endpoint
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p Def.: Plane Drawing

. Setup: Setup: G = (V, E) an undirected graph (w/o loop or parallel edges) . Def: Def: A drawing δ of G is plane if (the images under δ of) two edges are

  • nly allowed to share a common endpoint
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p Def.: Planar Graph

. Setup: Setup: G = (V, E) an undirected graph (w/o loop or parallel edges) . Def: Def: A graph G is planar if ∃ a plane drawing of G

(characterization due to F´ ary ’48)

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p Def.: Planar Graph

. Setup: Setup: G = (V, E) an undirected graph (w/o loop or parallel edges) . Def: Def: A graph G is planar if ∃ a plane drawing of G

(characterization due to F´ ary ’48)

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p Def.: Distance of Drawings

. Setup: Setup: G = (V, E) an undirected graph (w/o loop or parallel edges) . Def: Def: The distance of two drawings δ, δ′ of G is d(δ, δ′) = |{v ∈ V | δ(v) = δ′(v)}|

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p Def.: MMV (Min Moved Vertices)

. Setup: Setup: G = (V, E) an undirected graph (w/o loop or parallel edges) δ a drawing of G . Def: Def: MMV(G, δ) = min

δ ′plane d(δ, δ′)

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p Def.: MKV (Max Kept Vertices)

. Setup: Setup: G = (V, E) an n-vertex undirected graph (w/o loop or parallel edges) δ a drawing of G . Def: Def: MKV(G, δ) = n − MMV(G, δ)

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p Contents

  • Problem statement (more formally)
  • NP-hardness proof
  • Inapproximability (briefly)
  • Connection to the one-bend embeddability problem
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p Complexity results

. Theorem Theorem

  • For a given planar graph G and a drawing δ of G,

it is NP-hard to compute MMV(G, δ)

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p Complexity results

. Theorem Theorem

  • For a given planar graph G and a drawing δ of G,

it is NP-hard to compute MMV(G, δ) . Proof Proof

  • Reduction from Planar 3SAT

.

NP-complete (Lichtenstein (SICOMP ’82))

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p Complexity results

. Theorem Theorem

  • For a given planar graph G and a drawing δ of G,

it is NP-hard to compute MMV(G, δ) . Proof Proof

  • Reduction from Planar 3SAT

.

NP-complete (Lichtenstein (SICOMP ’82))

. Note Note

  • More direct proof by Verbitsky (arXiv ’07),

but does not generalize to inapproximability

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p Planar 3SAT

. Input Input : 3CNF formula ϕ with a planar variable-clause graph . Question Question : Is ϕ satisfiable? Note: such a graph can be embedded as below

(Knuth & Ragunathan (SIAMDM ’92))

x1 x2 x3 x4 x5 x6

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p Basic strategy

Given a planar 3CNF formula ϕ

  • Construct a planar graph Gϕ and a drawing δϕ s.t.
  • ϕ is satisfiable ⇔

δϕ can be made plane by moving ≤ K vertices

  • Vertices: two types
  • Mobile vertices (those that may move)
  • Immobile vertices (those that are meant not to move)
  • Edges: each contributes to ≤ 1 crossing
  • Gadgets: two types
  • Variable gadgets
  • Clause gadgets
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p Gadget: block

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p Gadget: block

immobile vertex mobile vertex

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p Gadget: block

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p Gadget: block

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p Gadget: block

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p Important properties of gadgets

  • Each mobile vertex has exactly two incident edges
  • These two edges have crossings
  • Mobile vertices are not adjacent
  • Movement of a mobile vertex can get rid of two crossings
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p Gadget: variable

# blocks = # occurences

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p Gadget: variable

# blocks = # occurences

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p Gadget: variable

# blocks = # occurences

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p Gadget: variable

# blocks = # occurences

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p Gadget: variable

# blocks = # occurences

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p Gadget: variable

# blocks = # occurences

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p Gadget: variable

# blocks = # occurences

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p Gadget: variable

# blocks = # occurences

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p Gadgets: connection to clause gadgets

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p Gadgets: connection to clause gadgets

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p Gadgets: connection to clause gadgets

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p Gadgets: connection to clause gadgets

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p Gadgets: connection to clause gadgets

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p Gadgets: connection to clause gadgets

a clause gadget Interfere!

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p Gadgets: connection to clause gadgets

a clause gadget Interfere!

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p Gadgets: connection to clause gadgets

a clause gadget Interfere iff the blue choice is taken

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p Gadgets: connection to clause gadgets

a clause gadget is taken the green choice Interfere iff

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p Gadgets: clause

connections connections

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p Gadgets: clause

connections connections

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p Gadgets: clause

connections connections

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p Gadgets: clause

connections connections

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p Gadgets: clause

connections connections

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p Gadgets: clause

connections connections

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p Gadgets: clause

connections connections

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p Gadgets: clause

connections connections

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p Gadgets: clause

connections connections interfere!

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p Gadgets: clause

connections connections

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p Gadgets: clause

connections connections interfere!

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p Gadgets: clause

connections connections

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p Gadgets: clause

connections connections interefere!

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p Gadgets: clause

connections connections interefere!

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p Gadgets: clause

connections connections

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p Gadgets: clause

connections connections

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p Gadgets: clause

connections connections

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p Gadgets: clause

connections connections

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p Gadgets: clause

connections connections

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p Gadgets: clause

connections connections

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p Summing up...

  • ϕ satisfiable ⇒

Movement of blue vertices suffices # moved vertices = # initial crossings/2

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p Summing up...

  • ϕ satisfiable ⇒

Movement of blue vertices suffices # moved vertices = # initial crossings/2

  • ϕ unsatisfiable ⇒

Movement of blue vertices doesn’t suffice ∴ at least one crossing requires both endpoints to move # moved vertices ≥ # initial crossings/2 + 1

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p Summing up...

  • ϕ satisfiable ⇒

Movement of blue vertices suffices # moved vertices = # initial crossings/2

  • ϕ unsatisfiable ⇒

Movement of blue vertices doesn’t suffice ∴ at least one crossing requires both endpoints to move # moved vertices ≥ # initial crossings/2 + 1 Conclude the reduction

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p Contents

  • Problem statement (more formally)
  • NP-hardness proof
  • Inapproximability (briefly)
  • Connection to the one-bend embeddability problem
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p Inapproximability result

. Theorem Theorem

  • For a given planar graph G and a drawing δ of G,

it is NP-hard to approximate 1+MMV(G, δ) within a factor of n1−ε (∀ fixed ε ∈ (0, 1]) Remark

  • Since MMV(G, δ) could be zero,

we modify the objective value by adding one for the approximation to make sense.

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p Proof idea

  • Use the same reduction as the NP-hardness proof
  • Replace every immobile vertex with an immobile star
  • Immobile stars give us a large gap

∴ Calculation shows our inapproximability

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p Contents

  • Problem statement (more formally)
  • NP-hardness proof
  • Inapproximability (briefly)
  • Connection to the one-bend embeddability problem
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p Def.: k-bend embedding

. Setup: Setup: G = (V, E) a planar graph . Def: Def: A k-bend embedding of G is an embedding of G into a plane s.t. every edge is drawn as a non-crossing polygonal chain with k bends

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p Def.: k-bend embedding

. Setup: Setup: G = (V, E) a planar graph . Def: Def: A k-bend embedding of G is an embedding of G into a plane s.t. every edge is drawn as a non-crossing polygonal chain with k bends

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p Def.: k-bend point-set embeddability

. Setup: Setup: G = (V, E) a planar graph . Def: Def: G is k-bend (point-set) embeddable if ∀ S ⊂ I R2 with |S| = |V| ∃ a bijection δ: V → S s.t. G can be k-bend embedded while each v ∈ V is placed at δ(v) ∈ S

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p Known results

(Kaufmann & Wiese (GD ’99, JGAA ’02))

  • G 4-connected planar ⇒ G 1-bend embeddable
  • G planar ⇒ G 2-bend embeddable
  • It is NP-complete to decide if

for a given planar G = (V, E) and a point set S ∃ a bijection δ: V → S that makes it possible to 1-bend embed G on S

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p New result

. Theorem Theorem

  • For a given planar graph G = (V, E), a point set S and

a bijection δ: V → S it is NP-hard to decide if δ makes it possible to 1-bend embed G on S Reminder . Kaufmann–Wiese ’02 Kaufmann–Wiese ’02

  • For a given planar graph G = (V, E) and a point set S

it is NP-hard to decide if ∃ a bijection δ: V → S that makes it possible to 1-bend embed G on S

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p Proof idea

  • Use the same reduction as the NP-hardness of MMV
  • But contract the mobile vertices
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p Proof idea

  • Use the same reduction as the NP-hardness of MMV
  • But contract the mobile vertices

Remark: a similar inapproximability holds . Theorem Theorem

  • For a given planar graph G = (V, E), a point set S and

a bijection δ: V → S it is NP-hard to approximate min # total bends (+1) when G is embedded on S with the correspondence δ within a factor of n1−ε (∀ fixed ε ∈ (0, 1])

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p Contents

  • Problem statement (more formally)
  • NP-hardness proof
  • Inapproximability (briefly)
  • Connection to the one-bend embeddability problem
  • Concluding remarks
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p Combinatorial results

max # of vertices that we can keep fixed Lower Bound Upper Bound Cycles ⌊√n⌋ O((n log n)2/3) Trees ⌊√n/3⌋ ⌈n/3⌉ + 4 General 3 ⌈ √ n−2⌉+1

Pach & Tardos (GD ’01, DCG ’02)

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p Combinatorial results

max # of vertices that we can keep fixed Lower Bound Upper Bound Cycles ⌊√n⌋ O((n log n)2/3) Trees ⌊√n/3⌋ ⌈n/3⌉ + 4 Outerplanar √ n−1/3 2 √ n−1+1 General 3 ⌈ √ n−2⌉+1 Ω(

  • log n/ log log n)

Pach & Tardos (GD ’01, DCG ’02) Spillner & Wolff (arXiv Sept ’07)

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p Summary and open problems: computational

Results

  • Minimizing the number of moved vertices is
  • NP-hard to compute precisely
  • NP-hard to compute approximately with factor n1−ε
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p Summary and open problems: computational

Results

  • Minimizing the number of moved vertices is
  • NP-hard to compute precisely
  • NP-hard to compute approximately with factor n1−ε

Open Problems

  • Minimizing the number of moved vertices is
  • hard in the parameterized sense?
  • Maximizing the number of kept vertices is
  • NP-hard to compute approximately??
  • hard in the parameterized sense?
  • How about when we restrict a graph class?
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p Summary and open problems: computational

Results

  • Minimizing the number of moved vertices is
  • NP-hard to compute precisely
  • NP-hard to compute approximately with factor n1−ε

Open Problems

  • Minimizing the number of moved vertices is
  • hard in the parameterized sense?
  • Maximizing the number of kept vertices is
  • NP-hard to compute approximately??
  • hard in the parameterized sense?
  • How about when we restrict a graph class?

[End of Talk]

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p

Supplementary slides

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p Min Vertex Cover of line segments

Vertex cover = set of vertices that doesn’t miss any edge.

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p Min Vertex Cover of line segments

Vertex cover = set of vertices that doesn’t miss any edge.

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p Reduction

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p Reduction

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p Cycles: Lower Bound

Recall: MKV(G, δ) = max # vertices we can keep fixed to make δ plane . Theorem Theorem (Pach & Tardos GD ’01, DCG ’02) For any drawing δ of an n-cycle Cn MKV(Cn, δ) ≥ ⌊√n⌋

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p Cycles: Lower Bound

Recall: MKV(G, δ) = max # vertices we can keep fixed to make δ plane . Theorem Theorem (Pach & Tardos GD ’01, DCG ’02) For any drawing δ of an n-cycle Cn MKV(Cn, δ) ≥ ⌊√n⌋ . Proof Proof Use the Erd˝

  • s–Szekeres theorem
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p Erd˝

  • s–Szekeres for a monotone subsequence

. Lemma Lemma

(Erd˝

  • s and Szekeres ’35)

A sequence of n different real numbers contains a monotone subsequence of length (at least) ⌊√n⌋ 3 9 12 16 7 6 13 1 10 11 4 8 2 15 5 14 n = 16

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p Erd˝

  • s–Szekeres for a monotone subsequence

. Lemma Lemma

(Erd˝

  • s and Szekeres ’35)

A sequence of n different real numbers contains a monotone subsequence of length (at least) ⌊√n⌋ 3 9 12 16 7 6 13 1 10 11 4 8 2 15 5 14 n = 16

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p Erd˝

  • s–Szekeres for a monotone subsequence

. Lemma Lemma

(Erd˝

  • s and Szekeres ’35)

A sequence of n different real numbers contains a monotone subsequence of length (at least) ⌊√n⌋ 3 9 12 16 7 6 13 1 10 11 4 8 2 15 5 14 n = 16

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p From drawings to sequences 1 3 4 5 6 7 2 9 8

Given a drawing...

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p From drawings to sequences 1 3 4 5 6 7 2 9 8

A point in general position

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p From drawings to sequences 1 3 4 5 6 7 2 9 8

Obtain a sequence according to the angular order 5 8 7 4 6 1 3 2 9

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p From drawings to sequences 1 3 4 5 6 7 2 9 8

Obtain a sequence according to the angular order 5 8 7 4 6 1 3 2 9

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p From drawings to sequences 1 3 4 5 6 7 2 9 8

Obtain a sequence according to the angular order 5 8 7 4 6 1 3 2 9

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p From drawings to sequences 1 3 4 5 6 7 2 9 8

Obtain a sequence according to the angular order 5 8 7 4 6 1 3 2 9

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p From drawings to sequences 1 3 4 5 6 7 2 9 8

Obtain a sequence according to the angular order 5 8 7 4 6 1 3 2 9

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p From drawings to sequences 1 3 4 5 6 7 2 9 8

Obtain a sequence according to the angular order 5 8 7 4 6 1 3 2 9

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p From drawings to sequences 1 3 4 5 6 7 2 9 8

Obtain a sequence according to the angular order 5 8 7 4 6 1 3 2 9

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p From drawings to sequences 1 3 4 5 6 7 2 9 8

Obtain a sequence according to the angular order 5 8 7 4 6 1 3 2 9

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p From drawings to sequences 1 3 4 5 6 7 2 9 8

Obtain a sequence according to the angular order 5 8 7 4 6 1 3 2 9

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p From drawings to sequences 1 3 4 5 6 7 2 9 8

Choose a monotone subsequence of length ⌊√n⌋ 5 8 7 4 6 1 3 2 9

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p From drawings to sequences 1 3 4 5 6 7 2 9 8

Choose a monotone subsequence of length ⌊√n⌋ 5 8 7 4 6 1 3 2 9

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p From drawings to sequences 1 3 4 5 6 7 2 9 8

Keep pts in the subseq fixed, and move remaining pts 5 8 7 4 6 1 3 2 9

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p From drawings to sequences 1 3 4 5 6 7 2 9 8 1 4 5 6 7 9 8 2 3

Keep pts in the subseq fixed, and move remaining pts 5 8 7 4 6 1 3 2 9

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p From drawings to sequences 1 3 4 5 6 7 2 9 8 1 4 7 9 8 2 3 5 6

Keep pts in the subseq fixed, and move remaining pts 5 8 7 4 6 1 3 2 9

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p Cycles: Easy Upper Bound

Recall: MKV(G, δ) = max # vertices we can keep fixed to make δ plane . Theorem Theorem ∃ a drawing δ of Cn (n odd) s.t. MKV(Cn, δ) ≤ ⌊n/2⌋

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SLIDE 129

42

p Cycles: Easy Upper Bound

Recall: MKV(G, δ) = max # vertices we can keep fixed to make δ plane . Theorem Theorem ∃ a drawing δ of Cn (n odd) s.t. MKV(Cn, δ) ≤ ⌊n/2⌋ . Proof Proof Use a thrackle:

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p Results of others

Verbitsky (’07) independently obtained the following

  • It is NP-hard to compute MMV(G, δ)
  • For n-vertex planar graphs, MKV ≥ 3
  • MMV(G, δ) ≥ (matching no. of G) − 1
  • For n-vertex planar graphs, δ ≥ 3 and n ≥ 10

we cannot keep (2n + 1)/3 vertices in some cases

the matching no. ≥ (n+2)/3 (Nishizeki & Baybars ’79)

  • For n-vertex planar graphs, 4-connected

we cannot keep (n + 3)/2 vertices in some cases

4-conn. planar graphs are Hamiltonian (Tutte ’56)

  • Also investigate “obfuscation complexity of a graph”

that might be called “max rectilinear crossing number”