Existence of Simple Tours through Imprecise Points Maarten L - - PowerPoint PPT Presentation

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Existence of Simple Tours through Imprecise Points Maarten L - - PowerPoint PPT Presentation

Existence of Simple Tours through Imprecise Points Maarten L offler Center for Geometry, Imaging and Virtual Environments Utrecht University 1-1 Overview Imprecise points Imprecise simple polygons NP-hardness proof Planar


slide-1
SLIDE 1

1-1

Existence of Simple Tours through Imprecise Points

Maarten L¨

  • ffler

Center for Geometry, Imaging and Virtual Environments

Utrecht University

slide-2
SLIDE 2

2-1 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007
  • Imprecise points

Overview

  • Imprecise simple polygons
  • NP-hardness proof
  • Planar 3-SAT
  • Variables as scissors
  • Other results and consequences
  • Clauses
  • Further details
slide-3
SLIDE 3

3-1 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007
  • Unknown location

Imprecise Points

  • Known region of

possible locations

  • Regions are simple

geometric objects

  • Disc
  • Square
  • Rectangle
  • Convex polygon
  • Line segment
slide-4
SLIDE 4

3-2 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007
  • Unknown location

Imprecise Points

  • Known region of

possible locations

  • Regions are simple

geometric objects

  • Disc
  • Square
  • Rectangle
  • Convex polygon
  • Line segment
slide-5
SLIDE 5

3-3 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007
  • Unknown location

Imprecise Points

  • Known region of

possible locations

  • Regions are simple

geometric objects

  • Disc
  • Square
  • Rectangle
  • Convex polygon
  • Line segment
slide-6
SLIDE 6

3-4 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007
  • Unknown location

Imprecise Points

  • Known region of

possible locations

  • Regions are simple

geometric objects

  • Disc
  • Square
  • Rectangle
  • Convex polygon
  • Line segment
slide-7
SLIDE 7

3-5 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007
  • Unknown location

Imprecise Points

  • Known region of

possible locations

  • Regions are simple

geometric objects

  • Disc
  • Square
  • Rectangle
  • Convex polygon
  • Line segment
slide-8
SLIDE 8

4-1 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007
  • Sequence of imprecise

points

Imprecise Simple Polygons

  • Place one vertex in

each region

  • The result should be

a simple polygon

  • This problem is

NP-hard

1 2 3 4 5 6 7 8 9 10

slide-9
SLIDE 9

4-2 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007
  • Sequence of imprecise

points

Imprecise Simple Polygons

  • Place one vertex in

each region

  • The result should be

a simple polygon

  • This problem is

NP-hard

1 2 3 4 5 6 7 8 9 10

slide-10
SLIDE 10

4-3 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007
  • Sequence of imprecise

points

Imprecise Simple Polygons

  • Place one vertex in

each region

  • The result should be

a simple polygon

  • This problem is

NP-hard

1 2 3 4 5 6 7 8 9 10

slide-11
SLIDE 11

5-1 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007
  • Known to be NP-hard

[Lichtenstein 1982]

Planar 3-SAT

  • At most three

variables per clause

  • Variable-clause graph

must be planar

a ∨ b ∨ ¬c ¬a ∨ c ∨ ¬d b ∨ ¬c ∨ e a b c d e

  • Boolean satisfiability

problem

slide-12
SLIDE 12

5-2 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007
  • Known to be NP-hard

[Lichtenstein 1982]

Planar 3-SAT

  • At most three

variables per clause

  • Variable-clause graph

must be planar

a ∨ b ∨ ¬c ¬a ∨ c ∨ ¬d b ∨ ¬c ∨ e a b c d e

  • Boolean satisfiability

problem

slide-13
SLIDE 13

6-1 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007
  • Configuration of four

imprecise points

Scissors Gadget

  • Two distinct possible

solutions

3 2 1 4

  • Each variable will be

represented by a number of scissors

  • Negative slope: false
  • Positive slope: true
slide-14
SLIDE 14

6-2 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007
  • Configuration of four

imprecise points

Scissors Gadget

  • Two distinct possible

solutions

3 2 1 4

  • Each variable will be

represented by a number of scissors

  • Negative slope: false
  • Positive slope: true
slide-15
SLIDE 15

6-3 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007
  • Configuration of four

imprecise points

Scissors Gadget

  • Two distinct possible

solutions

3 2 1 4

  • Each variable will be

represented by a number of scissors

  • Negative slope: false
  • Positive slope: true
slide-16
SLIDE 16

7-1 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Scissors Chain

3 2 1 4 3 2 1 4 1 4 3 2

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SLIDE 17

7-2 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Scissors Chain

3 2 1 4 3 2 1 4 1 4 3 2

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SLIDE 18

7-3 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Scissors Chain

3 2 1 4 3 2 1 4 1 4 3 2

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SLIDE 19

7-4 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Scissors Chain

3 2 1 4 3 2 1 4 1 4 3 2

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SLIDE 20

7-5 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Scissors Chain

3 2 1 4 3 2 1 4 1 4 3 2

slide-21
SLIDE 21

7-6 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Scissors Chain

3 2 1 4 3 2 1 4 1 4 3 2

slide-22
SLIDE 22

7-7 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Scissors Chain

3 2 1 4 3 2 1 4 1 4 3 2

slide-23
SLIDE 23

8-1 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Clause Gadget

  • Two fixed parts of the

tour

1 3 2

  • Configuration of three

imprecise points

  • Three distinct

possible solutions

  • Each solution will be

connected to one of the clause’s variables

slide-24
SLIDE 24

8-2 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Clause Gadget

  • Two fixed parts of the

tour

1 3 2

  • Configuration of three

imprecise points

  • Three distinct

possible solutions

  • Each solution will be

connected to one of the clause’s variables

slide-25
SLIDE 25

8-3 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Clause Gadget

  • Two fixed parts of the

tour

1 3 2

  • Configuration of three

imprecise points

  • Three distinct

possible solutions

  • Each solution will be

connected to one of the clause’s variables

slide-26
SLIDE 26

8-4 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Clause Gadget

  • Two fixed parts of the

tour

1 3 2

  • Configuration of three

imprecise points

  • Three distinct

possible solutions

  • Each solution will be

connected to one of the clause’s variables

slide-27
SLIDE 27

9-1 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Variables and Clauses

slide-28
SLIDE 28

9-2 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Variables and Clauses

slide-29
SLIDE 29

9-3 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Variables and Clauses

slide-30
SLIDE 30

9-4 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Variables and Clauses

slide-31
SLIDE 31

9-5 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Variables and Clauses

slide-32
SLIDE 32

9-6 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Variables and Clauses

slide-33
SLIDE 33

9-7 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Variables and Clauses

slide-34
SLIDE 34

9-8 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Variables and Clauses

slide-35
SLIDE 35

9-9 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Variables and Clauses

slide-36
SLIDE 36

9-10 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Variables and Clauses

slide-37
SLIDE 37

10-1 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007
  • Splitting variables

Further Details

  • Connecting tour parts
  • Variables will occur in

many clauses

  • Chains need to move

vertically

  • Many small pieces of

tour need to become

  • ne big tour
  • Bridges to cross chains
slide-38
SLIDE 38

10-2 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007
  • Splitting variables

Further Details

  • Connecting tour parts
  • Variables will occur in

many clauses

  • Chains need to move

vertically

  • Many small pieces of

tour need to become

  • ne big tour
  • Bridges to cross chains
slide-39
SLIDE 39

10-3 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007
  • Splitting variables

Further Details

  • Connecting tour parts
  • Variables will occur in

many clauses

  • Chains need to move

vertically

  • Many small pieces of

tour need to become

  • ne big tour
  • Bridges to cross chains
slide-40
SLIDE 40

10-4 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007
  • Splitting variables

Further Details

  • Connecting tour parts
  • Variables will occur in

many clauses

  • Chains need to move

vertically

  • Many small pieces of

tour need to become

  • ne big tour
  • Bridges to cross chains
slide-41
SLIDE 41

11-1 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Conclusions

  • Imprecision model: any other connected region
  • Other results
  • Finding the shortest simple tour through n regions
  • Finding a simple polygon with imprecise

vertices as vertical line segments is NP-hard

slide-42
SLIDE 42

12-1 Existence of Simple Tours through Imprecise Points Maarten L¨

  • ffler, March 7, 2007

Questions?