Stick Graphs with Length Constraints Steven Chaplick, Philipp - - PowerPoint PPT Presentation

Stick Graphs with Length Constraints

Steven Chaplick, Philipp Kindermann, Andre L¨

• ffler,

Florian Thiele, Alexander Wolff, Alexander Zaft, and Johannes Zink

Introduction

• Given a collection S of geometric objects

2

Introduction

• Given a collection S of geometric objects
• The intersection graph G of S has

– S as its vertex set – an edge for each two elements of S that intersect or touch each other.

2

Introduction

• S: line segments

⇒ G: segment graph

• Given a collection S of geometric objects
• The intersection graph G of S has

– S as its vertex set – an edge for each two elements of S that intersect or touch each other.

2

Introduction

• S: line segments

⇒ G: segment graph

• Given a collection S of geometric objects
• The intersection graph G of S has

– S as its vertex set – an edge for each two elements of S that intersect or touch each other.

segment graphs

2

Introduction

• S: line segments

⇒ G: segment graph

• Given a collection S of geometric objects
• The intersection graph G of S has

– S as its vertex set – an edge for each two elements of S that intersect or touch each other.

• S: horizontal & vertical segments

⇒ G: grid intersection graph

segment graphs

2

Introduction

• S: line segments

⇒ G: segment graph

• Given a collection S of geometric objects
• The intersection graph G of S has

– S as its vertex set – an edge for each two elements of S that intersect or touch each other.

• S: horizontal & vertical segments

⇒ G: grid intersection graph

segment graphs grid intersec. graphs

2

Introduction

• S: line segments

⇒ G: segment graph

• Given a collection S of geometric objects
• The intersection graph G of S has

– S as its vertex set – an edge for each two elements of S that intersect or touch each other.

• S: horizontal & vertical segments

⇒ G: grid intersection graph

• S: horizontal & vertical segments grounded
• n a line of slope −1

⇒ G: stick graph

segment graphs grid intersec. graphs

2

Introduction

• S: line segments

⇒ G: segment graph

• Given a collection S of geometric objects
• The intersection graph G of S has

– S as its vertex set – an edge for each two elements of S that intersect or touch each other.

• S: horizontal & vertical segments

⇒ G: grid intersection graph

• S: horizontal & vertical segments grounded
• n a line of slope −1

⇒ G: stick graph

segment graphs stick graphs grid intersec. graphs

2

Introduction

• S: line segments

⇒ G: segment graph

• S: horizontal & vertical segments grounded on

two parallel lines ⇒ G: bipartite permutation graph

• Given a collection S of geometric objects
• The intersection graph G of S has

– S as its vertex set – an edge for each two elements of S that intersect or touch each other.

• S: horizontal & vertical segments

⇒ G: grid intersection graph

• S: horizontal & vertical segments grounded
• n a line of slope −1

⇒ G: stick graph

segment graphs stick graphs grid intersec. graphs

2

Introduction

• S: line segments

⇒ G: segment graph

• S: horizontal & vertical segments grounded on

two parallel lines ⇒ G: bipartite permutation graph

• Given a collection S of geometric objects
• The intersection graph G of S has

– S as its vertex set – an edge for each two elements of S that intersect or touch each other.

• S: horizontal & vertical segments

⇒ G: grid intersection graph

• S: horizontal & vertical segments grounded
• n a line of slope −1

⇒ G: stick graph

segment graphs bip. permu. graphs stick graphs grid intersec. graphs

2

Computational Complexity

Recognition problem: Decide whether a given graph is an intersection graph.

3

Computational Complexity

segment graphs bip. permu. graphs stick graphs grid intersec. graphs Recognition problem: Decide whether a given graph is an intersection graph. How hard for each class?

3

Computational Complexity

segment graphs bip. permu. graphs stick graphs grid intersec. graphs Recognition problem: Decide whether a given graph is an intersection graph. How hard for each class? segment graphs: ∃R-complete [Kratochv´ ıl, Matouˇ sek ’94; Matouˇ sek ’14]

3

Computational Complexity

segment graphs bip. permu. graphs stick graphs grid intersec. graphs Recognition problem: Decide whether a given graph is an intersection graph. How hard for each class? segment graphs: ∃R-complete [Kratochv´ ıl, Matouˇ sek ’94; Matouˇ sek ’14] grid intersection graphs: NP-complete [Kratochv´ ıl ’94]

3

Computational Complexity

segment graphs bip. permu. graphs stick graphs grid intersec. graphs Recognition problem: Decide whether a given graph is an intersection graph. How hard for each class? segment graphs: ∃R-complete [Kratochv´ ıl, Matouˇ sek ’94; Matouˇ sek ’14] grid intersection graphs: NP-complete [Kratochv´ ıl ’94] (bipart.) permutation graphs: linear time [Spinrad et al. ’87; Kratsch et al. ’06]

3

Computational Complexity

segment graphs bip. permu. graphs stick graphs grid intersec. graphs Recognition problem: Decide whether a given graph is an intersection graph. How hard for each class? segment graphs: ∃R-complete [Kratochv´ ıl, Matouˇ sek ’94; Matouˇ sek ’14] grid intersection graphs: NP-complete [Kratochv´ ıl ’94] (bipart.) permutation graphs: linear time [Spinrad et al. ’87; Kratsch et al. ’06] stick graphs: ???

3

Computational Complexity

segment graphs bip. permu. graphs stick graphs grid intersec. graphs Recognition problem: Decide whether a given graph is an intersection graph. How hard for each class? segment graphs: ∃R-complete [Kratochv´ ıl, Matouˇ sek ’94; Matouˇ sek ’14] grid intersection graphs: NP-complete [Kratochv´ ıl ’94] (bipart.) permutation graphs: linear time [Spinrad et al. ’87; Kratsch et al. ’06] stick graphs: ???

remains open. . .

3

Versions of Stick Graph Recognition

4

Versions of Stick Graph Recognition

Given a bipartite Graph G = (A ˙ ∪B, E), does G admit a stick representation . . .

4

Versions of Stick Graph Recognition

Given a bipartite Graph G = (A ˙ ∪B, E), does G admit a stick representation . . .

• STICK: . . . ?

4

Versions of Stick Graph Recognition

Given a bipartite Graph G = (A ˙ ∪B, E), does G admit a stick representation . . .

• STICK: . . . ?

4

Versions of Stick Graph Recognition

Given a bipartite Graph G = (A ˙ ∪B, E), does G admit a stick representation . . .

• STICK: . . . ?
• STICKA:

. . . if a permutation of the vertices in A is given?

4

Versions of Stick Graph Recognition

Given a bipartite Graph G = (A ˙ ∪B, E), does G admit a stick representation . . .

• STICK: . . . ?
• STICKA:

. . . if a permutation of the vertices in A is given?

4

Versions of Stick Graph Recognition

Given a bipartite Graph G = (A ˙ ∪B, E), does G admit a stick representation . . .

• STICK: . . . ?
• STICKA:

. . . if a permutation of the vertices in A is given?

• STICKAB:

. . . if a permutation of the vertices in A and a permutation of the vertices in B is given?

4

Versions of Stick Graph Recognition

Given a bipartite Graph G = (A ˙ ∪B, E), does G admit a stick representation . . .

• STICK: . . . ?
• STICKA:

. . . if a permutation of the vertices in A is given?

• STICKAB:

. . . if a permutation of the vertices in A and a permutation of the vertices in B is given?

4

Versions of Stick Graph Recognition

Given a bipartite Graph G = (A ˙ ∪B, E), does G admit a stick representation . . .

• STICK: . . . ?
• STICKA:

. . . if a permutation of the vertices in A is given?

• STICKAB:

. . . if a permutation of the vertices in A and a permutation of the vertices in B is given?

• STICKfix

⋆ :

. . . if a stick length for each vertex (and possibly permutations of A, or A and B) is given?

4

Versions of Stick Graph Recognition

Given a bipartite Graph G = (A ˙ ∪B, E), does G admit a stick representation . . .

• STICK: . . . ?
• STICKA:

. . . if a permutation of the vertices in A is given?

• STICKAB:

. . . if a permutation of the vertices in A and a permutation of the vertices in B is given?

• STICKfix

⋆ :

. . . if a stick length for each vertex (and possibly permutations of A, or A and B) is given?

4

Versions of Stick Graph Recognition

Given a bipartite Graph G = (A ˙ ∪B, E), does G admit a stick representation . . .

• STICK: . . . ?
• STICKA:

. . . if a permutation of the vertices in A is given?

• STICKAB:

. . . if a permutation of the vertices in A and a permutation of the vertices in B is given?

• STICKfix

⋆ :

. . . if a stick length for each vertex (and possibly permutations of A, or A and B) is given? First investigated by De Luca et al. [GD’18]               

4

Versions of Stick Graph Recognition

Given a bipartite Graph G = (A ˙ ∪B, E), does G admit a stick representation . . .

• STICK: . . . ?
• STICKA:

. . . if a permutation of the vertices in A is given?

• STICKAB:

. . . if a permutation of the vertices in A and a permutation of the vertices in B is given?

• STICKfix

⋆ :

. . . if a stick length for each vertex (and possibly permutations of A, or A and B) is given? First investigated by De Luca et al. [GD’18]               

new

  

4

Complexity of Recognition STICKfix

⋆

STICK⋆ ⋆ A AB

for a bipartite graph G = (A ∪ B, E)

5

Complexity of Recognition STICKfix

⋆

STICK⋆ ⋆ A AB ? ? ? ? ?1

1an O(|A|3|B|3) time algorithm proposed by De Luca et al. turned out to be wrong

O(|A||B|)

[De Luca et al. GD’18] for a bipartite graph G = (A ∪ B, E)

5

Complexity of Recognition STICKfix

⋆

STICK⋆ ⋆ A AB ? ? ? ? ?1

1an O(|A|3|B|3) time algorithm proposed by De Luca et al. turned out to be wrong

O(|A||B|)

[De Luca et al. GD’18]

• ur results

for a bipartite graph G = (A ∪ B, E)

5

Complexity of Recognition STICKfix

⋆

STICK⋆ ⋆ A AB ? ? ? ? ?1

1an O(|A|3|B|3) time algorithm proposed by De Luca et al. turned out to be wrong

O(|A||B|)

[De Luca et al. GD’18]

• ur results

O(|A||B|)

for a bipartite graph G = (A ∪ B, E)

5

Complexity of Recognition STICKfix

⋆

STICK⋆ ⋆ A AB ? ? ? ? ?1

1an O(|A|3|B|3) time algorithm proposed by De Luca et al. turned out to be wrong

O(|A||B|)

[De Luca et al. GD’18]

• ur results

O(|A||B|) O(|E|)

for a bipartite graph G = (A ∪ B, E)

5

Complexity of Recognition STICKfix

⋆

STICK⋆ ⋆ A AB ? ? ? ? ?1

1an O(|A|3|B|3) time algorithm proposed by De Luca et al. turned out to be wrong

O(|A||B|)

[De Luca et al. GD’18]

• ur results

O(|A||B|) O(|E|) NP-complete

for a bipartite graph G = (A ∪ B, E)

5

Complexity of Recognition STICKfix

⋆

STICK⋆ ⋆ A AB ? ? ? ? ?1

1an O(|A|3|B|3) time algorithm proposed by De Luca et al. turned out to be wrong

O(|A||B|)

[De Luca et al. GD’18]

• ur results

O(|A||B|) O(|E|) NP-complete NP-complete

for a bipartite graph G = (A ∪ B, E)

5

Complexity of Recognition STICKfix

⋆

STICK⋆ ⋆ A AB ? ? ? ? ?1

1an O(|A|3|B|3) time algorithm proposed by De Luca et al. turned out to be wrong

O(|A||B|)

[De Luca et al. GD’18]

• ur results

O(|A||B|) O(|E|) NP-complete NP-complete NP-complete

in general:

for a bipartite graph G = (A ∪ B, E)

5

Complexity of Recognition STICKfix

⋆

STICK⋆ ⋆ A AB ? ? ? ? ?1

1an O(|A|3|B|3) time algorithm proposed by De Luca et al. turned out to be wrong

O(|A||B|)

[De Luca et al. GD’18]

• ur results

O(|A||B|) O(|E|) NP-complete NP-complete NP-complete

O((|A| + |B|)2)

in general: w/o isolated vtc.:

for a bipartite graph G = (A ∪ B, E)

5

Complexity of Recognition STICKfix

⋆

STICK⋆ ⋆ A AB ? ? ? ? ?1

1an O(|A|3|B|3) time algorithm proposed by De Luca et al. turned out to be wrong

O(|A||B|)

[De Luca et al. GD’18]

• ur results

O(|A||B|) O(|E|) NP-complete NP-complete NP-complete

O((|A| + |B|)2)

in general: w/o isolated vtc.:

for a bipartite graph G = (A ∪ B, E)

next

5

Complexity of Recognition STICKfix

⋆

STICK⋆ ⋆ A AB ? ? ? ? ?1

1an O(|A|3|B|3) time algorithm proposed by De Luca et al. turned out to be wrong

O(|A||B|)

[De Luca et al. GD’18]

• ur results

O(|A||B|) O(|E|) NP-complete NP-complete NP-complete

O((|A| + |B|)2)

in general: w/o isolated vtc.:

for a bipartite graph G = (A ∪ B, E)

next afterwards

5

Algorithm for STICKA O(|A||B|) O(|E|) NP-complete NP-complete NP-complete

O((|A| + |B|)2)

in general: w/o isolated vtc.:

STICKfix

⋆

STICK⋆ ⋆ A AB ?

6

Algorithm for STICKA

• Sweep-line along the ordered vertical sticks in A:

enter event (i) and exit event (i ) for each ai ∈ A

7

Algorithm for STICKA

• Sweep-line along the ordered vertical sticks in A:

enter event (i) and exit event (i ) for each ai ∈ A

• Let p ∈ { i, i

}, G i := subgraph induced by a1, . . . , ai and their neighbors, Bp := vertices in B that intersect the sweep-line at event p

7

Algorithm for STICKA

• Sweep-line along the ordered vertical sticks in A:

enter event (i) and exit event (i ) for each ai ∈ A

• Let p ∈ { i, i

}, G i := subgraph induced by a1, . . . , ai and their neighbors, Bp := vertices in B that intersect the sweep-line at event p

• (Rooted) tree data structure T p:

7

Algorithm for STICKA

• Sweep-line along the ordered vertical sticks in A:

enter event (i) and exit event (i ) for each ai ∈ A

• Let p ∈ { i, i

}, G i := subgraph induced by a1, . . . , ai and their neighbors, Bp := vertices in B that intersect the sweep-line at event p

• (Rooted) tree data structure T p:

– contains two types of nodes: leaves and non-leaves

7

Algorithm for STICKA

• Sweep-line along the ordered vertical sticks in A:

enter event (i) and exit event (i ) for each ai ∈ A

• Let p ∈ { i, i

}, G i := subgraph induced by a1, . . . , ai and their neighbors, Bp := vertices in B that intersect the sweep-line at event p

• (Rooted) tree data structure T p:

– contains two types of nodes: leaves and non-leaves – each leaf corresponds to a vertex in Bp

7

Algorithm for STICKA

• Sweep-line along the ordered vertical sticks in A:

enter event (i) and exit event (i ) for each ai ∈ A

• Let p ∈ { i, i

}, G i := subgraph induced by a1, . . . , ai and their neighbors, Bp := vertices in B that intersect the sweep-line at event p

• (Rooted) tree data structure T p:

– contains two types of nodes: leaves and non-leaves – each leaf corresponds to a vertex in Bp – the order of leaves is free; the order of non-leaves is fixed

7

Algorithm for STICKA

• Sweep-line along the ordered vertical sticks in A:

enter event (i) and exit event (i ) for each ai ∈ A

• Let p ∈ { i, i

}, G i := subgraph induced by a1, . . . , ai and their neighbors, Bp := vertices in B that intersect the sweep-line at event p

• (Rooted) tree data structure T p:

– contains two types of nodes: leaves and non-leaves – each leaf corresponds to a vertex in Bp – the order of leaves is free; the order of non-leaves is fixed – encodes all realizable permutations of Bp

7

Algorithm for STICKA

• Sweep-line along the ordered vertical sticks in A:

enter event (i) and exit event (i ) for each ai ∈ A

• Let p ∈ { i, i

}, G i := subgraph induced by a1, . . . , ai and their neighbors, Bp := vertices in B that intersect the sweep-line at event p

• (Rooted) tree data structure T p:

– contains two types of nodes: leaves and non-leaves – each leaf corresponds to a vertex in Bp – the order of leaves is free; the order of non-leaves is fixed – encodes all realizable permutations of Bp

7

Algorithm for STICKA

• Sweep-line along the ordered vertical sticks in A:

enter event (i) and exit event (i ) for each ai ∈ A

• Let p ∈ { i, i

}, G i := subgraph induced by a1, . . . , ai and their neighbors, Bp := vertices in B that intersect the sweep-line at event p

• (Rooted) tree data structure T p:

– contains two types of nodes: leaves and non-leaves – each leaf corresponds to a vertex in Bp – the order of leaves is free; the order of non-leaves is fixed – encodes all realizable permutations of Bp

7

Example for STICKA

G:

a1 a2 a3

G 0 : i = 0 B0 = ∅ Event: Start T 0 :

b1 b3 b5 b2 b4 b6 fixed order free order

8

Example for STICKA

G:

a1 a2 a3

G 1 : i = 1 B1 = {b1, b2, b4} Event: 1 T 1 :

b1 b3 b5 b2 b4 b6 a1 b1 b2 b4 b1 b2 b4 b1 b2 b4 fixed order free order a1

8

Example for STICKA

G:

a1 a2 a3 b1 b3 b5 b2 b4 b6

G 1 : i = 1 B1 = {b1, b2, b4} Event: 1 T 1 :

a1 b1 b2 b4 b1 b2 b4 a1 b1 b2 b4 fixed order free order

8

Example for STICKA

a2 b5 b3

G:

a1 a2 a3 b1 b3 b5 b2 b4 b6

G 2 : i = 2 B2 = {b1, b2, b3,b4, b5} Event: 2 T 2 :

a1 b1 b2 b4 b1 b2 b3 b5 b4 b4 b1 b2 b3 b5 a1 fixed order free order a2

8

Example for STICKA

G:

a1 a2 a3 b1 b3 b5 b2 b4 b6

G 2 :

a1 b1 b2 b4 a2 b5 b3 b4 b2 b5 a1 a2

i = 2 B2 = {b1,b2, b3,b4, b5} Event: 2 T 2 :

b1 b2 b3 b5 b4 fixed order free order b1 b3

8

Example for STICKA

a3 b6

G:

a1 a2 a3 b1 b3 b5 b2 b4 b6

G 3 :

b4 b2 b5 a1 a2 b1 b3

i = 3 B3 = {b2, b4, b5, b6} Event: 3 T 3 :

a1 a2 b1 b3 b5 b2 b4 b2 b5 b4 b6 b6 fixed order free order a3

8

Example for STICKA

G:

a1 a2 a3 b1 b3 b5 b2 b4 b6

G 3 :

a1 a2 b1 b3 a1 a2 a3 b1 b3 b5 b2 b4 b6 b2 b5 b4 b6 a3

i = 3 B3 = {b2, b4, b5, b6} Event: 3 T 3 :

fixed order free order b4 b2 b5 b6

8

Example for STICKA

G:

a1 a2 a3 b1 b3 b5 b2 b4 b6 a1 a2 b1 b3 a3

i = 3 Event: End

fixed order free order b4 b2 b5 b6

8

Example for STICKA

G:

a1 a2 a3 b1 b3 b5 b2 b4 b6 a1 a2 b1 b3 a3

i = 3 Event: End

fixed order free order b4 b2 b5 b6

Runtime in O(|A| · |B|)

8

STICKfix

AB with isolated vertices

O(|A||B|) O(|E|) NP-complete NP-complete NP-complete

O((|A| + |B|)2)

in general: w/o isolated vtc.:

STICKfix

⋆

STICK⋆ ⋆ A AB ?

9

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

false

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

true

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

. . . true a l l c l a u s e g a d g e t s

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

. . . true a l l c l a u s e g a d g e t s

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

false a l l c l a u s e g a d g e t s . . .

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

false a l l c l a u s e g a d g e t s . . .

• Clause gadget:

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

false a l l c l a u s e g a d g e t s . . .

• Clause gadget:

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

false a l l c l a u s e g a d g e t s . . .

• Clause gadget:

≈ 4 ≈ 1

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

false a l l c l a u s e g a d g e t s . . .

• Clause gadget:

x1 = false

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

false a l l c l a u s e g a d g e t s . . .

• Clause gadget:

x1 = true

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

false a l l c l a u s e g a d g e t s . . .

• Clause gadget:

x1 = true x2 = true

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

false a l l c l a u s e g a d g e t s . . .

• Clause gadget:

x1 = true x2 = true x3 = true

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

false a l l c l a u s e g a d g e t s . . .

• Clause gadget:

x1 = true x2 = true x3 = true empty

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

false a l l c l a u s e g a d g e t s . . .

• Clause gadget:

x1 = true x2 = true x3 = true empty

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

false a l l c l a u s e g a d g e t s . . .

• Clause gadget:

x1 = true x2 = true x3 = true empty

≈ 2

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

false a l l c l a u s e g a d g e t s . . .

• Clause gadget:

x1 = true x2 = true empty x3 = false

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

false a l l c l a u s e g a d g e t s . . .

• Clause gadget:

x1 = true x3 = true empty x2 = false

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

false a l l c l a u s e g a d g e t s . . .

• Clause gadget:

x1 = true empty x2 = false x3 = false

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

false a l l c l a u s e g a d g e t s . . .

• Clause gadget:

x1 = false x2 = true x3 = true empty

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

false a l l c l a u s e g a d g e t s . . .

• Clause gadget:

x1 = false x2 = true empty x3 = false

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

false a l l c l a u s e g a d g e t s . . .

• Clause gadget:

x1 = false x3 = true empty x2 = false

10

Hardness of STICKfix

AB

• NP-hardness by reduction from MONOTONE-3-SAT
• Variable gadget:

false a l l c l a u s e g a d g e t s . . .

• Clause gadget:

x1 = false empty x2 = false x3 = false

10

Example

MONOTONE-3-SAT formula: (x1 ∨ x2 ∨ x3)∧ (x2 ∨ x3 ∨ x4)∧ (¬x1 ∨ ¬x2 ∨ ¬x4)

11

Example

MONOTONE-3-SAT formula: (x1 ∨ x2 ∨ x3)∧ (x2 ∨ x3 ∨ x4)∧ (¬x1 ∨ ¬x2 ∨ ¬x4)

                

v a r i a b l e g a d g e t s

11

Example

MONOTONE-3-SAT formula: (x1 ∨ x2 ∨ x3)∧ (x2 ∨ x3 ∨ x4)∧ (¬x1 ∨ ¬x2 ∨ ¬x4)

                

v a r i a b l e g a d g e t s

x4 x1 ¬x2 x3

11

Example

MONOTONE-3-SAT formula: (x1 ∨ x2 ∨ x3)∧ (x2 ∨ x3 ∨ x4)∧ (¬x1 ∨ ¬x2 ∨ ¬x4)

                                                                    

c l a u s e g a d g e t s

                

v a r i a b l e g a d g e t s

x4 x1 ¬x2 x3

11

Example

MONOTONE-3-SAT formula: (x1 ∨ x2 ∨ x3)∧ (x2 ∨ x3 ∨ x4)∧ (¬x1 ∨ ¬x2 ∨ ¬x4)

                                                                    

c l a u s e g a d g e t s

x

1 ∨

x

2 ∨

x

3

                

v a r i a b l e g a d g e t s

x4 x1 ¬x2 x3

11

Example

MONOTONE-3-SAT formula: (x1 ∨ x2 ∨ x3)∧ (x2 ∨ x3 ∨ x4)∧ (¬x1 ∨ ¬x2 ∨ ¬x4)

x

2 ∨

x

3 ∨

x

4

                                                                    

c l a u s e g a d g e t s

x

1 ∨

x

2 ∨

x

3

                

v a r i a b l e g a d g e t s

x4 x1 ¬x2 x3

11

Example

MONOTONE-3-SAT formula: (x1 ∨ x2 ∨ x3)∧ (x2 ∨ x3 ∨ x4)∧ (¬x1 ∨ ¬x2 ∨ ¬x4)

¬x1 ∨ ¬x2 ∨ ¬x4 x

2 ∨

x

3 ∨

x

4

                                                                    

c l a u s e g a d g e t s

x

1 ∨

x

2 ∨

x

3

                

v a r i a b l e g a d g e t s

x4 x1 ¬x2 x3

11

Example

MONOTONE-3-SAT formula: (x1 ∨ x2 ∨ x3)∧ (x2 ∨ x3 ∨ x4)∧ (¬x1 ∨ ¬x2 ∨ ¬x4)

¬x1 ∨ ¬x2 ∨ ¬x4 x

2 ∨

x

3 ∨

x

4

                                                                    

c l a u s e g a d g e t s

x

1 ∨

x

2 ∨

x

3

                

v a r i a b l e g a d g e t s

x4 x1 ¬x2 ¬x3

11

Example

MONOTONE-3-SAT formula: (x1 ∨ x2 ∨ x3)∧ (x2 ∨ x3 ∨ x4)∧ (¬x1 ∨ ¬x2 ∨ ¬x4)

¬x1 ∨ ¬x2 ∨ ¬x4 x

2 ∨

x

3 ∨

x

4

                                                                    

c l a u s e g a d g e t s

x

1 ∨

x

2 ∨

x

3

                

v a r i a b l e g a d g e t s

x1 ¬x2 ¬x3 ¬x4

11

STICKfix

AB without isolated vertices

O(|A||B|) O(|E|) NP-complete NP-complete NP-complete

O((|A| + |B|)2)

in general: w/o isolated vtc.:

STICKfix

⋆

STICK⋆ ⋆ A AB ?

12

Uniqueness Lemma

Lemma:

13

Uniqueness Lemma

Lemma: In all stick representations of an instance of STICKAB, the order of vertices on the ground line is the same after removing all isolated vertices. This order can be found in time O(|E|).

13

Uniqueness Lemma

Lemma: In all stick representations of an instance of STICKAB, the order of vertices on the ground line is the same after removing all isolated vertices. This order can be found in time O(|E|). Proof (Sketch):

13

Uniqueness Lemma

Lemma: In all stick representations of an instance of STICKAB, the order of vertices on the ground line is the same after removing all isolated vertices. This order can be found in time O(|E|). Proof (Sketch):

• Assume: stick representations Γ1, Γ2 have different order

13

Uniqueness Lemma

Lemma: In all stick representations of an instance of STICKAB, the order of vertices on the ground line is the same after removing all isolated vertices. This order can be found in time O(|E|). Proof (Sketch):

• Assume: stick representations Γ1, Γ2 have different order

Γ1 Γ2

13

Uniqueness Lemma

Lemma: In all stick representations of an instance of STICKAB, the order of vertices on the ground line is the same after removing all isolated vertices. This order can be found in time O(|E|). Proof (Sketch):

• Assume: stick representations Γ1, Γ2 have different order
• W.l.o.g. a < b in Γ1 and b < a in Γ2

⇒ ab / ∈ E Γ1 Γ2

13

Uniqueness Lemma

Lemma: In all stick representations of an instance of STICKAB, the order of vertices on the ground line is the same after removing all isolated vertices. This order can be found in time O(|E|). Proof (Sketch):

• Assume: stick representations Γ1, Γ2 have different order
• W.l.o.g. a < b in Γ1 and b < a in Γ2

⇒ ab / ∈ E Γ1 Γ2

13

Uniqueness Lemma

Lemma: In all stick representations of an instance of STICKAB, the order of vertices on the ground line is the same after removing all isolated vertices. This order can be found in time O(|E|). Proof (Sketch):

• Assume: stick representations Γ1, Γ2 have different order
• W.l.o.g. a < b in Γ1 and b < a in Γ2

⇒ ab / ∈ E

a b b a

Γ1 Γ2

13

Uniqueness Lemma

Lemma: In all stick representations of an instance of STICKAB, the order of vertices on the ground line is the same after removing all isolated vertices. This order can be found in time O(|E|). Proof (Sketch):

• Assume: stick representations Γ1, Γ2 have different order
• W.l.o.g. a < b in Γ1 and b < a in Γ2

⇒ ab / ∈ E

a b b a

Γ1 Γ2

• b′ is adjacent to a ⇒ b′ < b

13

Uniqueness Lemma

Lemma: In all stick representations of an instance of STICKAB, the order of vertices on the ground line is the same after removing all isolated vertices. This order can be found in time O(|E|). Proof (Sketch):

• Assume: stick representations Γ1, Γ2 have different order
• W.l.o.g. a < b in Γ1 and b < a in Γ2

⇒ ab / ∈ E

a b b a

Γ1 Γ2

• b′ is adjacent to a ⇒ b′ < b

b′ 13

Uniqueness Lemma

Lemma: In all stick representations of an instance of STICKAB, the order of vertices on the ground line is the same after removing all isolated vertices. This order can be found in time O(|E|). Proof (Sketch):

• Assume: stick representations Γ1, Γ2 have different order
• W.l.o.g. a < b in Γ1 and b < a in Γ2

⇒ ab / ∈ E

a b b a

Γ1 Γ2

• b′ is adjacent to a ⇒ b′ < b

b′ b′ 13

Uniqueness Lemma

Lemma: In all stick representations of an instance of STICKAB, the order of vertices on the ground line is the same after removing all isolated vertices. This order can be found in time O(|E|). Proof (Sketch):

• Assume: stick representations Γ1, Γ2 have different order
• W.l.o.g. a < b in Γ1 and b < a in Γ2

⇒ ab / ∈ E

a b b a

Γ1 Γ2

• b′ is adjacent to a ⇒ b′ < b

b′ b′

• a′ is adjacent to b ⇒ a < a′

13

Uniqueness Lemma

Lemma: In all stick representations of an instance of STICKAB, the order of vertices on the ground line is the same after removing all isolated vertices. This order can be found in time O(|E|). Proof (Sketch):

• Assume: stick representations Γ1, Γ2 have different order
• W.l.o.g. a < b in Γ1 and b < a in Γ2

⇒ ab / ∈ E

a b b a

Γ1 Γ2

• b′ is adjacent to a ⇒ b′ < b

b′ b′

• a′ is adjacent to b ⇒ a < a′

a′ 13

Uniqueness Lemma

Lemma: In all stick representations of an instance of STICKAB, the order of vertices on the ground line is the same after removing all isolated vertices. This order can be found in time O(|E|). Proof (Sketch):

• Assume: stick representations Γ1, Γ2 have different order
• W.l.o.g. a < b in Γ1 and b < a in Γ2

⇒ ab / ∈ E

a b b a

Γ1 Γ2

• b′ is adjacent to a ⇒ b′ < b

b′ b′

• a′ is adjacent to b ⇒ a < a′

a′ a′ 13

Uniqueness Lemma

Lemma: In all stick representations of an instance of STICKAB, the order of vertices on the ground line is the same after removing all isolated vertices. This order can be found in time O(|E|). Proof (Sketch):

• Assume: stick representations Γ1, Γ2 have different order
• W.l.o.g. a < b in Γ1 and b < a in Γ2

⇒ ab / ∈ E

a b b a

Γ1 Γ2

• b′ is adjacent to a ⇒ b′ < b

b′ b′

• a′ is adjacent to b ⇒ a < a′

a′ a′ 13

Linear Program for STICKfix

AB

• No isolated vertices ⇒ Compute ordering v1, . . . , vn

14

Linear Program for STICKfix

AB

• No isolated vertices ⇒ Compute ordering v1, . . . , vn
• Variable xi for the x-coordinate of vi’s footpoint

14

Linear Program for STICKfix

AB

• No isolated vertices ⇒ Compute ordering v1, . . . , vn
• Variable xi for the x-coordinate of vi’s footpoint
• Constraints: x1 < · · · < xn

14

Linear Program for STICKfix

AB

• No isolated vertices ⇒ Compute ordering v1, . . . , vn
• Variable xi for the x-coordinate of vi’s footpoint
• Constraints: x1 < · · · < xn
• For each vertex 2 more constraints incorporating the

predefined stick-lengths: – for intersecting its last neighbor – for not intersecting its first non-neighbor

14

Linear Program for STICKfix

AB

• No isolated vertices ⇒ Compute ordering v1, . . . , vn
• Variable xi for the x-coordinate of vi’s footpoint
• Constraints: x1 < · · · < xn
• For each vertex 2 more constraints incorporating the

predefined stick-lengths: – for intersecting its last neighbor – for not intersecting its first non-neighbor

• Is a system of difference constraints ⇒ can be modeled as

a shortest-path problem in a directed weighted graph

14

Linear Program for STICKfix

AB

• No isolated vertices ⇒ Compute ordering v1, . . . , vn
• Variable xi for the x-coordinate of vi’s footpoint
• Constraints: x1 < · · · < xn
• For each vertex 2 more constraints incorporating the

predefined stick-lengths: – for intersecting its last neighbor – for not intersecting its first non-neighbor ⇒ Solvable in O((|A| + |B|)2) time with Bellman–Ford

• Is a system of difference constraints ⇒ can be modeled as

a shortest-path problem in a directed weighted graph

14

Linear Program for STICKfix

AB

• No isolated vertices ⇒ Compute ordering v1, . . . , vn
• Variable xi for the x-coordinate of vi’s footpoint
• Constraints: x1 < · · · < xn
• For each vertex 2 more constraints incorporating the

predefined stick-lengths: – for intersecting its last neighbor – for not intersecting its first non-neighbor ⇒ Solvable in O((|A| + |B|)2) time with Bellman–Ford ⇒ Isolated vertices make STICKfix

AB NP-hard

• Is a system of difference constraints ⇒ can be modeled as

a shortest-path problem in a directed weighted graph

14

Summary O(|A||B|)

[De Luca et al. GD’18]

O(|A||B|) O(|E|) NP-complete NP-complete NP-complete

O((|A| + |B|)2)

in general: w/o isolated vtc.:

still open STICKfix

⋆

STICK⋆ ⋆ A AB

15

Summary O(|A||B|)

[De Luca et al. GD’18]

O(|A||B|) O(|E|) NP-complete NP-complete NP-complete

O((|A| + |B|)2)

in general: w/o isolated vtc.:

still open

(by reduction from 3-PARTITION)

STICKfix

⋆

STICK⋆ ⋆ A AB

15

Summary O(|A||B|)

[De Luca et al. GD’18]

O(|A||B|) O(|E|) NP-complete NP-complete NP-complete

O((|A| + |B|)2)

in general: w/o isolated vtc.:

still open

(by reduction from 3-PARTITION) (by reduction from MONO-3-SAT)

STICKfix

⋆

STICK⋆ ⋆ A AB

15