Title: TSPortraits of Knots and Links
1TSPortraits of Knots and Links
- Robert Bosch
- May 13, 2009
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5Hands
INFORMS 06 Pittsburgh, PA
6Whats Inside?
INFORMS 06 Pittsburgh PA
JMM 2007 San Diego CA
Bridges 2007 Donostia, Spain
7This is!
INFORMS 06 Pittsburgh PA
JMM 2007 San Diego CA
Bridges 2007 Donostia, Spain
8One loop variation 1
INFORMS 06 Pittsburgh PA
9One loop variation 2
INFORMS 06 Pittsburgh PA
10One loop variation 3
INFORMS 06 Pittsburgh PA
11Knot?
CPAIOR06 Cork, Ireland
JMM 2007 New Orleans, LA
Bridges 2007 Donostia, Spain
12One fish, two fish, red fish, black fish
JMM 2008 San Diego, CA
Bridges 2008 Leeuwarden, NL
13Outside
Ring
JMM 2008 San Diego, CA
Bridges 2008 Leeuwarden, NL
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21Solving TSPs with IP
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23Variables xe 1 if edge e is used in the tour
24Variables xe 1 if edge e is used in the tour
x3,8 1
(e 3,8)
25Variables xe 1 if edge e is used in the tour
x9,11 1
(e 9,11)
x3,8 1
(e 3,8)
26Variables 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)
27Objective minimize length of tour
total cost S ( ce xe all edges e )
28Constraints each city v must be touched
twice
- ( xe all edges e that touch v ) 2
29Constraints no subtours!
S ( xe all e with both endpoints in S ) lt
S
30Solving 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
31An example
32Stage 1 6 subtours.
33Stage 1 6 subtours.
x13,19
x13,25
x19,25 lt 3
34Stage 2 4 more subtours.
35Stage 3 4 more subtours.
36Stage 4 3 more subtours.
37Stage 5 success!
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39Definition
c(lA,B) all edges e that cross line lA,B
40Definition
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
41Definition
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
42The 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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61References
- 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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