TravelingSalesman Problem Instance: - PDF document

9/22/09
 Genotypes and Phenotypes Part
II
‐‐
The
Return
of
the
Salesman
 Traveling
Salesman
 Problem
 • Instance:
 • N
cities
with
distances
between
pairs
of
cities
 • Said
another
way:
 • Complete
graph
with
n
vertices
such
that
all
 edges
are
labeled
with
a
cost
value
 1


9/22/09
 Traveling
Salesman
 Problem
 • Solution:
 • Tour
of
the
cities
such
that
each
city
is
visited
once.
 • Said
another
way:
 • A
permutation
of
the
cities.
 • Ordered
list
of
the
cities
 • Is
this
a
large
search
space?
 • n
cities
=
n!
permutation
 Traveling
Salesman
 Problem
 • Output:
 • Distance
traveled
to
complete
the
tour.
 • Recall:
 • Phenotype
==
ordered
tour
of
the
cities
 • TSP
presents
challenges
for
Genetic
Mapping.
 2


9/22/09
 TSP
‐
Path
 Representation
 • Tour
is
represented
as
an
ordered
list
(or
array)
 of
the
cities.
 • Order
in
array
==
order
of
visitation.
 • If
city
i
is
the
jth
element
of
the
array,
city
i
is
the
 jth
city
to
be
visited.
 • Eg.
 3 2 4 1 7 5 8 6 Tour:
3‐2‐4‐1‐7‐5‐8‐6
 TSP
‐
Path
 Representation
 • Most
intuitive
and
common
genotype.
 • But
it
has
it’s
problems:
 3 2 4 1 7 5 8 6 3 2 4 1 7 5 8 6 8 7 6 5 4 3 2 1 3 2 4 1 4 3 2 1 3 2 4 1 3 5 8 6 Not
valid
tours!!!
 3


9/22/09
 TSP
‐
Path
 Representation
 • GeneRepair
[Mitchell,
et.al.]
 • Keep
a
corrective
template
with
a
valid
tour.
 • Identify
duplicate
cities
 • Use
template
to
replace
duplicate
cities.
 TSP
‐
Path
 Representation
 1 2 3 4 5 6 7 8 3 2 4 1 3 7
 5 8 6 4


9/22/09
 TSP
‐
Path
 Representation
 • Genetic
Mapping
responsible
for
doing
the
 repair
on
a
“bad
genome”
 • Most
approaches
that
use
the
path
 representation
use
designer
crossover/ mutation
operators
 • Assure
valid
offspring.
 TSP
‐
binary
 representation
 • Classic
GA
approach
 • Each
city
encoded
by
a
string
of
length
log 2 (n)
‐‐
 chromosomes?
 • Complete
genome
is
concatenation
of
cities
in
 order.
 • Complete
genome
has
length
n
log 2 (n)
 5


9/22/09
 TSP
‐‐
binary
 representation
 • Example
 i City i i City i 1 000 4 011 2 001 5 100 3 010 6 101 • Tour:
1‐2‐3‐4‐5‐6
 • (000
001
010
011
100
101)
 TSP
‐‐
binary
 representation
 • Similar
problem
with
repair.
 • (000
001
010
011


011
101)


1‐2‐3‐4‐5‐6
 • (101


100
011

010
001
000)


6‐5‐4‐3‐2‐1
 • (000
001
010
010
001
000)

1‐2‐3‐3‐2‐1
 Duplicate
cities
 6


9/22/09
 TSP
‐‐
binary
 representation
 • Genetic
Mapping:
 • Decode
binary
‐>
city.
 • Perform
repair
for
“bad
genome”.
 TSP
‐
Adjacency
Representation 
 • Tour
is
represented
as
an
array
of
n
cities.
 • City
j
is
listed
in
position
i,
if
and
only
if,
the
tour
 leads
from
city
i
to
city
j.
 7


9/22/09
 TSP
‐
Adjacency
Representation
 • Example:
 3 5 7 6 4 8 2 1 1
 3
 7
 2
 5
 4
 6
 8
 TSP
‐
Adjacency
Representation
 • Has
same
problem
as
others
PLUS
 • (
3
5
7
6
2
4
1
8)
 ⇒ 
(1
3
7
|
2
5
|
4
6
8)
 3
 1
 2
 4
 8
 6
 7
 5
 8


9/22/09
 TSP
‐
Adjacency
Representation
 • Note:

 • Same
phenotype
as
path
representation
 • Different
Genetic
Mapping.
 • …and
here’s
another
one
 TSP
‐
ordinal
 representation
 • Tour
is
represented
as
an
array
of
n
cities.
 • The
ith
element
is
a
number
in
the
range
from
1
 to
(n
‐
i
+
1)
 • There
exists
an
ordered
list
of
cities
to
use
as
a
 reference
point.
 9


9/22/09
 TSP
‐
ordinal
 representation
 • Example
 • C
=
(

1

2

3

4

5

67

8

9
)
 • Genome:
 1 1 2 1 4 1 3 1 1 1
 2
 4
 3
 8
 5
 9
 6
 7
 TSP
‐
ordinal
 representation
 • Complex?


Perhaps,
but
std.
crossover
works!
 1 1 2 1 4 1 3 1 1 1‐2‐4‐3‐8‐5‐9‐6‐7
 5‐1‐7‐8‐9‐4‐6‐3‐2
 5 1 5 5 5 3 3 2 1 1‐2‐4‐3‐9‐7‐8‐6‐5
 1 1 2 1 5 3 3 2 1 10


9/22/09
 TSP
‐
ordinal
representation
 • Note:

 • Same
phenotype
as
path
representation
 • Even
more
complex
Genetic
Mapping.
 TSP
‐
Matrix
 representation(1)
 • Tour
is
represented
as
a
2D
binary
matrix,
M.
 • M ij 
=
1
if
and
only
if
city
i
is
visited
before
city
j
in
 the
tour.
 • Must
have
the
following
properties:
 • Number
of
1’s
=
n(n‐2)
/
2
 • M ii
 =
0
 • If
M ij 
=
1
and
M jk
 =1
then
M ik 
=
1
 11


9/22/09
 TSP
‐
Matrix
 representation(1)
  0 0 0 1    1 0 1 1    1 0 0 1    0 0 0 0   Tour:
2‐3‐1‐4
 TSP
‐
Matrix
 representation(2)
 • Tour
is
represented
as
a
2D
binary
matrix,
M.
 • M ij
 =
1
if
and
only
if
city
j
follows
city
i
 immediately
on
the
tour.
 • Each
row
and
each
column
must
have
exactly
 one
1
in
it.
 12


9/22/09
 TSP
‐
Matrix
 representation(2)
  0 0 0 1    0 0 1 0    1 0 0 0    0 1 0 0   Tour:
1‐4‐2‐3
 By
the
way
 • Most
promising
approach:
 • Path
representation
with
designer
crossover
/
 mutation
operators.
 • Questions?
 13


9/22/09
 Take
home
messages
 • A
given
phenotype
can
result
from
many
 different
genotypes.
 • Different
genotype‐phenotype
pairs
can
have
 different
Genetic
Mapping.
 • Get
creative
with
your
genetic
mapping!
 • Bad
mojo
is
a
problem
 • Genetic
repair
during
Genetic
Mapping
 • Use
of
designer
crossover
/
mutation
operators.
 Questions
 • Questions?
 14