Augmenting Polygons with Matchings Alexander Pilz, Jonathan Rollin, - - PowerPoint PPT Presentation

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Augmenting Polygons with Matchings Alexander Pilz, Jonathan Rollin, - - PowerPoint PPT Presentation

Oct 15, 2022 28 likes •398 views

Augmenting Polygons with Matchings Alexander Pilz, Jonathan Rollin, Lena Schlipf, Andr e Schulz EuroCG 2020 Problem Given a simple polygon P (or a geometric graph) edges drawn with straight-lines, noncrossing Problem Given a simple polygon

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SLIDE 1

Augmenting Polygons with Matchings

EuroCG 2020 Alexander Pilz, Jonathan Rollin, Lena Schlipf, Andr´ e Schulz

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SLIDE 2

Problem

Given a simple polygon P (or a geometric graph) edges drawn with straight-lines, noncrossing

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SLIDE 3

Problem

Given a simple polygon P (or a geometric graph) find a geometric matching on the vertices of P, such that no edges cross in the augmentation. − → this is called a compatible matching

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SLIDE 4

Problem

Given a simple polygon P (or a geometric graph) find a geometric matching on the vertices of P, such that no edges cross in the augmentation. − → this is called a compatible matching

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SLIDE 5

Problem

Given a simple polygon P (or a geometric graph) find a geometric matching on the vertices of P, such that no edges cross in the augmentation. − → this is called a compatible matching

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SLIDE 6

Problem

Given a simple polygon P (or a geometric graph) find a geometric matching on the vertices of P, such that no edges cross in the augmentation. − → this is called a compatible matching

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SLIDE 7

Problem

Given a simple polygon P (or a geometric graph) find a geometric matching on the vertices of P, such that no edges cross in the augmentation. − → this is called a compatible matching

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SLIDE 8

Problem

Given a simple polygon P (or a geometric graph)

  • Is there a compatible perfect matching on the vertices of P?
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SLIDE 9

Problem

Given a simple polygon P (or a geometric graph)

  • Is there a compatible perfect matching on the vertices of P?
  • What is the smallest size of a compatible maximal

matching of the vertices of P?

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SLIDE 10

Problem

Given a simple polygon P (or a geometric graph)

  • Is there a compatible perfect matching on the vertices of P?
  • What is the smallest size of a compatible maximal

matching of the vertices of P?

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SLIDE 11

Known results

  • Every polygon with n vertices has a compatible matching
  • f size ≥ n−3

4

and there are polygons with compatible matchings of size ≤ n

3 .

[Aichholzer, Garc´ ıa, Hurtado, Tejel ’11]

  • Deciding whether a geometric matching admits a

compatible perfect matching such that both matchings together are a cycle is NP-complete.

[Akitaya, Korman, Rudoy, Souvaine, T´

  • th ’19]
  • Each geometric matching of even size admits a compatible

perfect matching.

[Ishaque, Souvaine, T´

  • th ’13]
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SLIDE 12

Perfect matchings in polygons

Theorem 1 Given a simple polygon, it is NP-complete to decide whether it admits a compatible perfect matching.

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Perfect matchings in polygons

Theorem 1 Given a simple polygon, it is NP-complete to decide whether it admits a compatible perfect matching. Reduction from POSITIVE PLANAR 1-IN-3SAT

  • planar variable-clause incidence graph
  • only positive literals
  • formula is satisfied if and only if there is exactly one true

variable per clause

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SLIDE 14

Perfect matchings in polygons

Theorem 1 Given a simple polygon, it is NP-complete to decide whether it admits a compatible perfect matching. Reduction from POSITIVE PLANAR 1-IN-3SAT

  • variable gadgets:
  • corner gadget:
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SLIDE 15

Perfect matchings in polygons

Theorem 1 Given a simple polygon, it is NP-complete to decide whether it admits a compatible perfect matching. Reduction from POSITIVE PLANAR 1-IN-3SAT

  • variable gadgets:
  • corner gadget:
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SLIDE 16

Perfect matchings in polygons

Theorem 1 Given a simple polygon, it is NP-complete to decide whether it admits a compatible perfect matching. Reduction from POSITIVE PLANAR 1-IN-3SAT

  • variable gadgets:
  • corner gadget:
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SLIDE 17

Perfect matchings in polygons

Theorem 1 Given a simple polygon, it is NP-complete to decide whether it admits a compatible perfect matching. Reduction from POSITIVE PLANAR 1-IN-3SAT

  • variable gadgets:
  • corner gadget:
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SLIDE 18

Perfect matchings in polygons

Theorem 1 Given a simple polygon, it is NP-complete to decide whether it admits a compatible perfect matching. Reduction from POSITIVE PLANAR 1-IN-3SAT

  • variable gadgets:
  • corner gadget:
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SLIDE 19

Perfect matchings in polygons

Theorem 1 Given a simple polygon, it is NP-complete to decide whether it admits a compatible perfect matching. Reduction from POSITIVE PLANAR 1-IN-3SAT

  • variable gadgets:
  • corner gadget:
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SLIDE 20

Perfect matchings in polygons

Theorem 1 Given a simple polygon, it is NP-complete to decide whether it admits a compatible perfect matching. Reduction from POSITIVE PLANAR 1-IN-3SAT

  • variable gadgets:
  • corner gadget:
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SLIDE 21

Perfect matchings in polygons

Theorem 2 Given a simple polygon, it is NP-complete to decide whether it admits a compatible perfect matching. Reduction from POSITIVE PLANAR 1-IN-3SAT

  • variable gadgets:
  • corner gadget:
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SLIDE 22

Perfect matchings in polygons

Theorem 2 Given a simple polygon, it is NP-complete to decide whether it admits a compatible perfect matching. Reduction from POSITIVE PLANAR 1-IN-3SAT

  • bend in a variable gadget:
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SLIDE 23

Perfect matchings in polygons

Theorem 2 Given a simple polygon, it is NP-complete to decide whether it admits a compatible perfect matching. Reduction from POSITIVE PLANAR 1-IN-3SAT

  • a split gadget:
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SLIDE 24

Perfect matchings in polygons

Theorem 2 Given a simple polygon, it is NP-complete to decide whether it admits a compatible perfect matching. Reduction from POSITIVE PLANAR 1-IN-3SAT

FALSE FALSE TRUE

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SLIDE 25

Perfect matchings in polygons

Theorem 2 Given a simple polygon, it is NP-complete to decide whether it admits a compatible perfect matching. Reduction from POSITIVE PLANAR 1-IN-3SAT

FALSE FALSE TRUE

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SLIDE 26

Perfect matchings in polygons

Theorem 2 Given a simple polygon, it is NP-complete to decide whether it admits a compatible perfect matching. Reduction from POSITIVE PLANAR 1-IN-3SAT

FALSE FALSE TRUE

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SLIDE 27

Maximal matchings in polygons

Theorem 3 Each maximal compatible matching of a polygon with n vertices has size ≥ n−4

8 .

There are polygons with maximal matchings of size ≤ n

6 .

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SLIDE 28

Maximal matchings in polygons

Theorem 3 Each maximal compatible matching of a polygon with n vertices has size ≥ n−4

8 .

There are polygons with maximal matchings of size ≤ n

6 .

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Maximal matchings in polygons

Theorem 3 Each maximal compatible matching of a polygon with n vertices has size ≥ n−4

8 .

There are polygons with maximal matchings of size ≤ n

6 .

+ rectangle + add new edge at reflex angles

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SLIDE 30

Maximal matchings in polygons

Theorem 3 Each maximal compatible matching of a polygon with n vertices has size ≥ n−4

8 .

There are polygons with maximal matchings of size ≤ n

6 .

  • all faces are convex
  • ≤ 2 unmatched vertices per face
  • at most 2 + |E(M)| + n faces
  • unmatched vertices incident to exactly 3 faces

+ rectangle + add new edge at reflex angles

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SLIDE 31

Maximal matchings in polygons

Theorem 3 Each maximal compatible matching of a polygon with n vertices has size ≥ n−4

8 .

There are polygons with maximal matchings of size ≤ n

6 .

⇒ 3 (n − 2 |E(M)|) ≤ 2 (2 + |E(M)| + n)

  • all faces are convex
  • ≤ 2 unmatched vertices per face
  • at most 2 + |E(M)| + n faces
  • unmatched vertices incident to exactly 3 faces
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SLIDE 32

Maximal matchings in polygons

Theorem 3 Each maximal compatible matching of a polygon with n vertices has size ≥ n−4

8 .

There are polygons with maximal matchings of size ≤ n

6 .

⇒ 3 (n − 2 |E(M)|) ≤ 2 (2 + |E(M)| + n)

  • all faces are convex
  • ≤ 2 unmatched vertices per face
  • at most 2 + |E(M)| + n faces
  • unmatched vertices incident to exactly 3 faces

n−4 8

≤ |E(M)|

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SLIDE 33

Maximal matchings in polygons

Theorem 3 Each maximal compatible matching of a polygon with n vertices has size ≥ n−4

8 .

There are polygons with maximal matchings of size ≤ n

6 . n 7

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SLIDE 34

Maximal matchings in polygons

Theorem 3 Each maximal compatible matching of a polygon with n vertices has size ≥ n−4

8 .

There are polygons with maximal matchings of size ≤ n

6 . n 7

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SLIDE 35

Maximal matchings in polygons

Theorem 3 Each maximal compatible matching of a polygon with n vertices has size ≥ n−4

8 .

There are polygons with maximal matchings of size ≤ n

6 . n 7 n 7

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SLIDE 36

One more result

Given a geometric graph G, find a set of compatible edges such that the augmented graph has minimum degree 5.

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SLIDE 37

One more result

Given a geometric graph G, find a set of compatible edges such that the augmented graph has minimum degree 5. Theorem 4 Given a geometric graph G, it is NP-complete to decide whether there is a set of compatible edges E such that G + E has minimum degree 5.