TSPortraits of Knots and Links

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TSPortraits of Knots and Links

• Robert Bosch
• May 13, 2009

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Hands
INFORMS 06 Pittsburgh, PA
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Whats Inside?
INFORMS 06 Pittsburgh PA
JMM 2007 San Diego CA
Bridges 2007 Donostia, Spain
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This is!
INFORMS 06 Pittsburgh PA
JMM 2007 San Diego CA
Bridges 2007 Donostia, Spain
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One loop variation 1
INFORMS 06 Pittsburgh PA
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One loop variation 2
INFORMS 06 Pittsburgh PA
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One loop variation 3
INFORMS 06 Pittsburgh PA
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Knot?
CPAIOR06 Cork, Ireland
JMM 2007 New Orleans, LA
Bridges 2007 Donostia, Spain
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One fish, two fish, red fish, black fish
JMM 2008 San Diego, CA
Bridges 2008 Leeuwarden, NL
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Outside
Ring
JMM 2008 San Diego, CA
Bridges 2008 Leeuwarden, NL
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Solving TSPs with IP
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Variables xe 1 if edge e is used in the tour
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Variables xe 1 if edge e is used in the tour
x3,8 1
(e 3,8)
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Variables xe 1 if edge e is used in the tour
x9,11 1
(e 9,11)
x3,8 1
(e 3,8)
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Variables xe 1 if edge e is used in the tour
x1,2 0
(e 1,2)
x9,11 1
(e 9,11)
x3,8 1
(e 3,8)
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Objective minimize length of tour
total cost S ( ce xe all edges e )
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Constraints each city v must be touched
twice

• ( xe all edges e that touch v ) 2

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Constraints no subtours!
S ( xe all e with both endpoints in S ) lt
S
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Solving TSPs with IP (DFJ formulation)
Variables xe 1 if edge e is used in the tour
Objective minimize length of tour
total cost S ( ce xe all edges e )
Constraints each city v must be touched
twice

• ( xe all edges e that touch v ) 2

A
v
Constraints no subtours are allowed
S ( xe all e w/both endpts in S ) lt S
A
S
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An example
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Stage 1 6 subtours.
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Stage 1 6 subtours.
x13,19
x13,25
x19,25 lt 3
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Stage 2 4 more subtours.
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Stage 3 4 more subtours.
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Stage 4 3 more subtours.
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Stage 5 success!
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Definition
c(lA,B) all edges e that cross line lA,B
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Definition
c(lA,B) all edges e that cross line lA,B
Note
A and B lie on the same side of the tour
S (xe all edges e in c(lA,B) ) is even
S (xe all edges e in c(lA,B) ) 2 yA,B
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Definition
c(lA,B) all edges e that cross line lA,B
Equivalently
A and B lie on opposite sides of the tour
S (xe all edges e in c(lA,B) ) is odd
S (xe all edges e in c(lA,B) ) 2 yA,B 1
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The Big Idea
Use
S (xe all edges e in c(lA,B) ) 2 yA,B
and/or
S (xe all edges e in c(lA,B) ) 2 yA,B 1
in the DFJ IP formulation.
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References

• R. Bosch and A. Herman. 2004. Continuous line
  drawings via the
• traveling salesman problem. Operations
  Research Letters 32 302-303.
• C.S. Kaplan and R. Bosch. 2005. TSP art.
  Bridges 2005.
• R. Bosch. 2008. Connecting the dots the ins
  and outs of TSP Art.
• Bridges 2008.
• R. Bosch. 2009. Jordan as a Jordan Curve. In
  Mathematical
• Wizardry for a Gardner, A.K. Peters, Natick,
  MA.

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