Matching Points with Rectangles and Squares Sergey Bereg, Nikolaus - - PowerPoint PPT Presentation

university-logo Introduction Rectangles Squares

Matching Points with Rectangles and Squares

Sergey Bereg, Nikolaus Mutsanas & Alexander Wolff SOFSEM’06

Bereg, Mutsanas & Wolff 1 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares

Outline

Introduction Matching in graphs and in the plane Already known... Open Problems Rectangles General position 1/2-Approximation Squares Is there a strong realization? Application to map-labeling NP-Completeness

Bereg, Mutsanas & Wolff 2 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Matching in graphs and in the plane Already known... Open Problems

Matching in graphs

Maximum Matching [Micali & Vazirani] O(√nm) Euclidean Minimum-Weight Perfect Matching [Vaidya] O(n2.5 log4 n) [Varadarajan & Agarwal] O((n/ε3) log6 n) Matching with segments, rectangles, squares, disks...

Bereg, Mutsanas & Wolff 3 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Matching in graphs and in the plane Already known... Open Problems

Matching in the plane

Definition Matching is perfect: covers all points. Matching is strong: no overlap.

Bereg, Mutsanas & Wolff 4 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Matching in graphs and in the plane Already known... Open Problems

Matching in the plane

not perfect

Definition Matching is perfect: covers all points. Matching is strong: no overlap.

Bereg, Mutsanas & Wolff 4 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Matching in graphs and in the plane Already known... Open Problems

Matching in the plane

strong and perfect

Definition Matching is perfect: covers all points. Matching is strong: no overlap.

Bereg, Mutsanas & Wolff 4 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Matching in graphs and in the plane Already known... Open Problems

Already known...

Let P be a set of 2n points in the plane. Theorem (Rendl & Woeginger) It is NP-hard to decide whether P admits a strong rectilinear segment matching. Theorem (Ábrego et al.) If P is in general position (no two points on a horiz./vert. line), then P admits a perfect disk matching and a perfect square matching. a strong disk matching covering at least 25% of P. a strong square matching covering at least 40% of P.

Bereg, Mutsanas & Wolff 5 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Matching in graphs and in the plane Already known... Open Problems

Already known...

Let P be a set of 2n points in the plane. Theorem (Rendl & Woeginger) It is NP-hard to decide whether P admits a strong rectilinear segment matching. Theorem (Ábrego et al.) If P is in general position (no two points on a horiz./vert. line), then P admits a perfect disk matching and a perfect square matching. a strong disk matching covering at least 25% of P. a strong square matching covering at least 40% of P.

Bereg, Mutsanas & Wolff 5 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Matching in graphs and in the plane Already known... Open Problems

Already known...

Let P be a set of 2n points in the plane. Theorem (Rendl & Woeginger) It is NP-hard to decide whether P admits a strong rectilinear segment matching. Theorem (Ábrego et al.) If P is in general position (no two points on a horiz./vert. line), then P admits a perfect disk matching and a perfect square matching. a strong disk matching covering at least 25% of P. a strong square matching covering at least 40% of P.

Bereg, Mutsanas & Wolff 5 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Matching in graphs and in the plane Already known... Open Problems

Already known...

Let P be a set of 2n points in the plane. Theorem (Rendl & Woeginger) It is NP-hard to decide whether P admits a strong rectilinear segment matching. Theorem (Ábrego et al.) If P is in general position (no two points on a horiz./vert. line), then P admits a perfect disk matching and a perfect square matching. a strong disk matching covering at least 25% of P. a strong square matching covering at least 40% of P.

Bereg, Mutsanas & Wolff 5 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Matching in graphs and in the plane Already known... Open Problems

Open Problems

Questions – How many points can be matched strongly? – Does a given matching have a strong realization? matching size

• ex. strong realization?

segments 100% O(n log n) rectangles ? O(n log n) squares n2?n2 ? disks ? ?

Bereg, Mutsanas & Wolff 6 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Matching in graphs and in the plane Already known... Open Problems

Open Problems

Questions – How many points can be matched strongly? – Does a given matching have a strong realization? matching size

• ex. strong realization?

segments 100% O(n log n) rectangles 50% O(n log n) squares n2?n2 O(n2 log n) disks ? ?

Bereg, Mutsanas & Wolff 6 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

General position

Bereg, Mutsanas & Wolff 7 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

General position

Bereg, Mutsanas & Wolff 7 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

General position

Bereg, Mutsanas & Wolff 7 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

General position

Bereg, Mutsanas & Wolff 7 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

No general position

Bereg, Mutsanas & Wolff 8 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

No general position

Bereg, Mutsanas & Wolff 8 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

No general position

Bereg, Mutsanas & Wolff 8 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

No general position

Bereg, Mutsanas & Wolff 8 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

No general position

π 2

Bereg, Mutsanas & Wolff 8 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

1/2-Approximation

Divide into subsets → match subsets → put together

Bereg, Mutsanas & Wolff 9 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

1/2-Approximation

Divide into subsets → match subsets → put together

H1 V1 H2 V2

Bereg, Mutsanas & Wolff 9 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

1/2-Approximation

Divide into subsets → match subsets → put together

H1 V1 H2 V2

Bereg, Mutsanas & Wolff 9 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

1/2-Approximation

Divide into subsets → match subsets → put together

H1 V1 H2 V2

Bereg, Mutsanas & Wolff 9 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

1/2-Approximation

Divide into subsets → match subsets → put together

Bereg, Mutsanas & Wolff 9 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

1/2-Approximation

Divide into subsets → match subsets → put together

Bereg, Mutsanas & Wolff 9 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

1/2-Approximation - worst case

(almost) Worst Case

Bereg, Mutsanas & Wolff 10 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

1/2-Approximation - worst case

(almost) Worst Case

H1 V1 H2 V2 H3 V3 H4 V4

Bereg, Mutsanas & Wolff 10 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

1/2-Approximation - worst case

(almost) Worst Case

H1 V1 H2 V2 H3 V3 H4 V4

Bereg, Mutsanas & Wolff 10 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

1/2-Approximation - worst case

(almost) Worst Case

H1

Bereg, Mutsanas & Wolff 10 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

1/2-Approximation - worst case

(almost) Worst Case

H1

Bereg, Mutsanas & Wolff 10 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

1/2-Approximation - worst case

Worst Case Matching with n/2 points.

Bereg, Mutsanas & Wolff 11 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

1/2-Approximation - worst case

Worst Case Matching with n/2 points.

Bereg, Mutsanas & Wolff 11 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares General position 1/2-Approximation

1/2-Approximation - worst case

Worst Case Optimal matching with n − 2 points.

Bereg, Mutsanas & Wolff 11 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Minimal squares

Minimal squares: points lie on the boundary.

p q α

β Bereg, Mutsanas & Wolff 12 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Minimal squares

Minimal squares: points lie on the boundary.

p q α

β Bereg, Mutsanas & Wolff 12 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Sliding squares

Sliding area p q p q α β α Q Kernel

Bereg, Mutsanas & Wolff 13 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Is there a strong realization?

Bereg, Mutsanas & Wolff 14 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Is there a strong realization?

Bereg, Mutsanas & Wolff 14 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Is there a strong realization?

Bereg, Mutsanas & Wolff 14 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Help from map-labeling

Labeling rectilinear segments Given: Set of rectilinear segments, B ∈ R. Question: Is there a labeling of height B?

Bereg, Mutsanas & Wolff 15 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Help from map-labeling

Labeling rectilinear segments Given: Set of rectilinear segments, B ∈ R. Question: Is there a labeling of height B?

Bereg, Mutsanas & Wolff 15 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Help from map-labeling

Labeling rectilinear segments Given: Set of rectilinear segments, B ∈ R. Question: Is there a labeling of height B?

B B

Bereg, Mutsanas & Wolff 15 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Help from map-labeling

Labeling rectilinear segments Given: Set of rectilinear segments, B ∈ R. Question: Is there a labeling of height B? Theorem (Kim, Shin & Yang) Rectilinear segment labeling is solvable in O(n2 log n)

Bereg, Mutsanas & Wolff 15 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Squares - canonical form

Let squares slide for vertical kernels leftwards as far as possible. for horizontal kernels downwards as far as possible. When does a square stop sliding?

Bereg, Mutsanas & Wolff 16 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Squares - canonical form

Let squares slide for vertical kernels leftwards as far as possible. for horizontal kernels downwards as far as possible. When does a square stop sliding?

Bereg, Mutsanas & Wolff 16 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Squares - canonical form

Let squares slide for vertical kernels leftwards as far as possible. for horizontal kernels downwards as far as possible. When does a square stop sliding?

Bereg, Mutsanas & Wolff 16 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Squares - canonical form

Let squares slide for vertical kernels leftwards as far as possible. for horizontal kernels downwards as far as possible. When does a square stop sliding?

Bereg, Mutsanas & Wolff 16 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Squares - canonical form

Let squares slide for vertical kernels leftwards as far as possible. for horizontal kernels downwards as far as possible. When does a square stop sliding? Observations – The resulting positions can be computed in advance. – Every square has O(n) relevant positions.

Bereg, Mutsanas & Wolff 16 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Squares - Decision Algorithm

Problem Given: P ⊆ R2, matching M ⊆ P

2

• Question: Is there a strong square realization of M?

Do kernels overlap? Calculate relevant positions. Solve decision problem with 2-SAT.max

Bereg, Mutsanas & Wolff 17 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Squares - Decision Algorithm

Problem Given: P ⊆ R2, matching M ⊆ P

2

• Question: Is there a strong square realization of M?

Do kernels overlap? O(n log n) Calculate relevant positions. Solve decision problem with 2-SAT.max

Bereg, Mutsanas & Wolff 17 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Squares - Decision Algorithm

Problem Given: P ⊆ R2, matching M ⊆ P

2

• Question: Is there a strong square realization of M?

Do kernels overlap? O(n log n) Calculate relevant positions. O(n2) Solve decision problem with 2-SAT.max

Bereg, Mutsanas & Wolff 17 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Squares - Decision Algorithm

Problem Given: P ⊆ R2, matching M ⊆ P

2

• Question: Is there a strong square realization of M?

Do kernels overlap? O(n log n) Calculate relevant positions. O(n2) Solve decision problem with 2-SAT. O(kmax·n log n)

Bereg, Mutsanas & Wolff 17 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Squares - Decision Algorithm

Problem Given: P ⊆ R2, matching M ⊆ P

2

• Question: Is there a strong square realization of M?

Do kernels overlap? O(n log n) Calculate relevant positions. O(n2) Solve decision problem with 2-SAT. O(n·n log n)

Bereg, Mutsanas & Wolff 17 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Squares - Decision Algorithm

Problem Given: P ⊆ R2, matching M ⊆ P

2

• Question: Is there a strong square realization of M?

Do kernels overlap? O(n log n) Calculate relevant positions. O(n2) Solve decision problem with 2-SAT. O(n2 log n)

Bereg, Mutsanas & Wolff 17 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Squares - Decision Algorithm

Problem Given: P ⊆ R2, matching M ⊆ P

2

• Question: Is there a strong square realization of M?

Do kernels overlap? O(n log n) Calculate relevant positions. O(n2) Solve decision problem with 2-SAT. O(n2 log n) Conclusion The decision problem can be solved in O(n2 log n).

Bereg, Mutsanas & Wolff 17 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Labeling points with sliding labels

Labeling points with sliding labels:

1SH 1SV

Labels also below / to the left of a point. Variable label sizes. Sliding area can be shortened.

Bereg, Mutsanas & Wolff 18 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Labeling points with sliding labels

Labeling points with sliding labels:

1SH 1SV

Bereg, Mutsanas & Wolff 18 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Labeling points with sliding labels

Labeling points with sliding labels:

1SH 1SV

Bereg, Mutsanas & Wolff 18 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Labeling points with sliding labels

Labeling points with sliding labels:

1SH 1SV

Bereg, Mutsanas & Wolff 18 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Labeling points with sliding labels

Labeling points with sliding labels:

1SH 1SV

Bereg, Mutsanas & Wolff 18 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Labeling points with sliding labels

Labeling points with sliding labels:

1SH 1SV

Bereg, Mutsanas & Wolff 18 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Labeling points with sliding labels

Labeling points with sliding labels:

1SH 1SV

Bereg, Mutsanas & Wolff 18 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

NP-Completeness

ESPSM Given: Point set P ⊆ R2 Question: Does a strong perfect square-matching exist? Theorem (Bereg, Mutsanas & Wolff ’05) ESPSM is NP-hard. Proof. By reduction from PLANAR 3-SAT to ESPSM.

Bereg, Mutsanas & Wolff 19 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

NP-Completeness

ESPSM Given: Point set P ⊆ R2 Question: Does a strong perfect square-matching exist? Theorem (Bereg, Mutsanas & Wolff ’05) ESPSM is NP-hard. Proof. By reduction from PLANAR 3-SAT to ESPSM.

Bereg, Mutsanas & Wolff 19 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

NP-Completeness

ESPSM Given: Point set P ⊆ R2 Question: Does a strong perfect square-matching exist? Theorem (Bereg, Mutsanas & Wolff ’05) ESPSM is NP-hard. Proof. By reduction from PLANAR 3-SAT to ESPSM.

Bereg, Mutsanas & Wolff 19 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Outline of the Reduction

x1 ∨ x2 ∨ x3 x1 ∨ x3 ∨ x4 x1 ∨ x2 ∨ x4 x2 ∨ x3 ∨ x4 x1 x2 x3 x4

Input: planar 3-SAT formula ϕ =

(x1 ∨ x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3) ∧ . . .

Goal: Point set P ⊆ R2 with: P admits s. p. square-matching ⇔ ϕ satisfiable.

Bereg, Mutsanas & Wolff 20 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Outline of the Reduction

x1 ∨ x2 ∨ x3 x1 ∨ x3 ∨ x4 x1 ∨ x2 ∨ x4 x2 ∨ x3 ∨ x4 x1 x2 x3 x4

Input: planar 3-SAT formula ϕ =

(x1 ∨ x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3) ∧ . . .

Goal: Point set P ⊆ R2 with: P admits s. p. square-matching ⇔ ϕ satisfiable.

Bereg, Mutsanas & Wolff 20 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Outline of the Reduction

x1 ∨ x2 ∨ x3 x1 ∨ x3 ∨ x4 x1 ∨ x2 ∨ x4 x2 ∨ x3 ∨ x4 x1 x2 x3 x4

Input: planar 3-SAT formula ϕ =

(x1 ∨ x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3) ∧ . . .

Goal: Point set P ⊆ R2 with: P admits s. p. square-matching ⇔ ϕ satisfiable.

Bereg, Mutsanas & Wolff 20 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Variable Gadget

v

v = true

Bereg, Mutsanas & Wolff 21 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Variable Gadget

v

v = false

Bereg, Mutsanas & Wolff 21 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Outline of the Reduction

x1 ∨ x2 ∨ x3 x1 ∨ x3 ∨ x4 x1 ∨ x2 ∨ x4 x2 ∨ x3 ∨ x4 x1 x2 x3 x4

Input: planar 3-SAT formula ϕ =

(x1 ∨ x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3) ∧ . . .

Goal: Point set P ⊆ R2 with: P admits s. p. square-matching ⇔ ϕ satisfiable.

Bereg, Mutsanas & Wolff 22 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Clause Gadget

a b x c p

Bereg, Mutsanas & Wolff 23 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Clause Gadget

adapter variable clause a b x c

    

p

Bereg, Mutsanas & Wolff 23 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Clause Gadget

a b x c p adapter variable clause

    

Bereg, Mutsanas & Wolff 23 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Clause Gadget

a b x c p stopper adapter variable clause

    

Bereg, Mutsanas & Wolff 23 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Clause Gadget

v a v b x c p u ¬u w w stopper adapter variable clause

    

Bereg, Mutsanas & Wolff 23 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Clause Gadget

u ¬u v a v b x c p w w stopper adapter variable clause

    

Bereg, Mutsanas & Wolff 23 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Clause Gadget

u ¬u a b x c p v v w w stopper adapter variable clause

    

Bereg, Mutsanas & Wolff 23 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Clause Gadget

u ¬u v a v b x c w w p stopper adapter variable clause

    

Bereg, Mutsanas & Wolff 23 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Clause Gadget

u ¬u a b x c w w p v v stopper adapter variable clause

    

Bereg, Mutsanas & Wolff 23 17 Matching Points with Rectangles and Squares

university-logo Introduction Rectangles Squares Is there a strong realization? Application to map-labeling NP-Completeness

Thank you for your attention!

Bereg, Mutsanas & Wolff 24 17 Matching Points with Rectangles and Squares