Genetic Operators for TSP
Chapter 8 in Michalewicz and Fogel,
How to Solve It: Modern Heuristics,
Springer, 2000
Size of TSP applications
Some examples
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Circuit board drilling: 17,000 cities
X-ray crystallography: 14,000 cities
VLSI fabrication (assembly): 1.3 million cities
Genome sequencing: several million cities
Instances with several hundred cities can be
solved optimally using general purpose
optimization software
Size of TSP applications (cont.)
Applegate et al. (2003) solved a one million city
instance using the cutting plane method due to
Dantzig et al. in 2 days (with 0.3% gap) and 308
days (with 0.04% gap)
However, the algorithms they used are very
difficult to understand and implement, and their
computing facilities are highly sophisticated
A simple minded GA can solve instances with
several thousand cities in under an hour (different
operators and solution quality)
Evolutionary algorithms
Schema theorem and building block hypothesis
are valid when a GA uses
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Binary coding as the representation scheme
Crossover and mutation operators that are suitable for
binary coding
Those algorithms that are based on GA notions,
but use other forms of coding and operators are
referred to as “evolutionary algorithms”
Most applications for TSP (and other ordering
problems) are EAs
Representation schemes and
genetic operators for TSP
The most natural representation of a TSP tour is a
permutation of cities
Binary coding of a TSP tour is possible but not
suitable because of ordering dependencies
Traditional crossover and mutation will result in
illegal tours (duplicates and omissions) that
require extensive repair
So, we should develop coding schemes and
genetic operators that are appropriate for the
problem at hand
Adjacency representation (AR)
AR encodes the tour as a list of n cities
City j is in position i if and only if the tour leads
from city i to city j
For example, the tour
1-5-2-6-3-4-1
is represented by the list
(5 6 4 1 2 3)
Each tour has one AR, but some lists can represent
illegal tours, e.g.the list
(5 4 6 1 2 3)
yields the cycle
1-5-2-4-1
Advantage: Each gene represents an edge
Crossover operators for AR:
Alternating-edges crossover (AEC)
AEC starts with a random edge from one parent
and proceeds by selecting edges from the two
parents in an alternating manner, e.g.
list
tour
Parent 1:
(5 6 4 1 2 3)
1-5-2-6-3-4-1
Parent 2:
(4 5 6 2 3 1)
3-6-1-4-2-5-3
Offspring 1:
Offspring 2:
(5 6 4 2 3 1)
(4
1? )
1-5-3-4-2-6-1
If a new edge from a parent causes a cycle, one of
the remaining edges is selected at random
Crossover operators for AR:
Subtour-chunks crossover (SCC)
SCC starts by choosing a random-length subtour
(or subpath) from one parent and proceeds by
choosing subtours from the two parents in an
alternating manner
It extends the tour by choosing edges from
alternating parents (such an edge defines the
starting point of the next subtour)
Again, if an edge from a parent causes a cycle, one
of the remaining edges is selected at random
Note that SCC is a generalization of AEC
Crossover operators for AR:
Heuristic crossover (HC)
HC first chooses a random starting city
It then compares the two edges emanating from
this city in two parents and selects the shorter edge
The city on the other end of the selected edge
becomes the starting city and the process is
repeated
Again, a random edge is introduced to avoid a
cycle
Crossover operators for AR:
Heuristic crossover (cont.)
A modification of HC uses two rules
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If shorter of the two edges causes a cycle, then try to
take the other (longer) edge
If the longer edge also causes a cycle, then choose the
shortest of the q randomly selected edges
This heuristic tries to combine short subpaths from
the parents
It may leave undesirable crossings of edges
Mutation operator for AR:
A heuristic based on 2-opt (HM)
HM can be used for local improvement (as a form
of mutation)
It randomly selects two edges, {i, j} and {k, m}
If dist(i, j)+dist(k, m)>dist(i, m)+dist(k, j), then the
edges {i, j} and {k, m} are replaced by the edges
{i, m} and {k, j}
Performance of AR
Genes in AR represent edges that count for fitness
It allows us to look for patterns in good solutions,
e.g. (# # 4 # 2 #) denotes all tours that have the
edges {3, 4} and {5, 2}
However, results obtained with AR and its
operators are not very good
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AEC disrupts good patterns
SCC does better than AEC possibly because its rate of
disruption is lower
Even with HC, percent deviation from optimum is 1627% for problems with 50-200 cities
Ordinal representation (OR)
OR encodes the tour as a list of n cities
Given an ordered list C of cities as a reference
point, the i-th element of a chromosome is in the
range from 1 to n-i+1
For example, given
C=(1 2 3 4 5 6)
the tour
1-2-5-3-6-4-1
is represented by the list
(1 1 3 1 2 1)
Advantage: Traditional 1- and 2-point crossover
operators generate legal tours
Ordinal representation (cont.)
For example, given C=(1 2 3 4 5 6)
list
tour
Parent 1:
(1 4 1 | 3 1 1)
1-5-2-6-3-4-1
Parent 2:
(3 5 1 | 2 1 1)
3-6-1-4-2-5-3
Offspring 1:
Offspring 2:
(1 4 1 | 2 1 1)
(3 5 1 | 3 1 1)
1-5-2-4-3-6-1
3-6-1-5-2-4-3
Note that the “head” subpaths are preserved, but
the “tail” subpaths are disrupted essentially at
random, resulting in poor performance
Path representation (PR)
PR is the most natural representation of a tour
For example, the tour
1-5-2-6-3-4-1
is represented by the list
(1 5 2 6 3 4)
Traditional crossover operators generate illegal
tours
Edge information is carried not by a single gene
but by a pair of genes, hence epistasis is strong
Crossover operators for PR:
Partially mapped crossover (PMX)
PMX takes a subpath between two random cut
points from one parent, completes the tour by
preserving the order and position of as many cities
as possible from the other parent
Illegal tours are avoided by using a mapping, e.g.
Parent 1:
Parent 2:
(1 | 5 2 6 | 3 4)
(3 | 6 1 4 | 2 5)
mapping
56
Offspring 1:
Offspring 2:
(3 | 5 2 6 | 1 4)
(2 | 6 1 4 | 3 5)
2 1
64
Crossover operators for PR:
Partially mapped crossover (cont.)
PMX preserves the “mid portion” (one subpath)
from the first parent
With the help of the mapping, it also tries to
preserve the order and position of cities in the
“side portions” (the other subpath) from the
second parent
Is it meaningful to preserve the absolute position
of a city in the list, when in fact the list represents
a tour? (Is the tour 1-5-2-6-3-4-1 not the same as
the tour 6-3-4-1-5-2-6?)
Crossover operators for PR:
Order crossover (OX)
OX also takes a subpath between two random cut
points from one parent, completes the tour by
preserving the relative order of as many cities as
possible from the other parent
Illegal tours are avoided by ommission, e.g.
Parent 1:
Parent 2:
(1 | 5 2 6 | 3 4)
(3 | 6 1 4 | 2 5)
Offspring 1:
Offspring 2:
(4 | 5 2 6 | 3 1)
(2 | 6 1 4 | 3 5)
using 3-1-4
using 3-5-2
Crossover operators for PR:
Variants of OX
Order-based crossover: OBC selects several
random positions, and the order of cities in
selected positions in one parent is imposed on the
corresponding cities in the other parent
Position-based crossover: Instead of selecting one
subpath to be copied, PBC randomly selects
several cities for this purpose
OBC and PBC are essentially equivalent
Crossover operators for PR:
Cycle crossover (CX)
Starting with city i from the first parent, CX takes
the next city j from the second parent as the one in
the same position as city i, and places j in the same
position it has in the first parent
CX also preserves absolute positions of cities
When a cycle occurs, remaining cities are filled in
from the second parent, e.g.
Parent 1: (1 5 2 6 3 4)
Parent 2: (3 6 1 4 2 5)
Offspring 1: (1 6 2 4 3 5)
Offspring 2: (3 5 1 6 2 4)
Mutation operators for PR:
Inversion
Inversion reverses the subpath between two
randomly selected cut points
For example, the list (1 | 5 2 6 | 3 4) becomes
(1 | 6 2 5 | 3 4) after inversion
Inversion guarantees a legal tour
Experiments show that it outperforms a “cross and
correct” operator
Increasing the number of cut points decreases the
performance
Mutation operators for PR:
Insertion, displacement and reciprocal exchange
Insertion selects a city and inserts it at a random
position
Displacement selects a subpath and inserts it at a
random position
Reciprocal exchange swaps two cities
Edge preservation in TSP
Crossover operators discussed so far try to
preserve order and/or position of cities
Basic building blocks of TSP that determine the
fitness value are not the cities but the edges
A good operator should extract edge information
from parents as much as possible and pass it to
offspring
Experimental results show that OX, which
preserves relative order of cities (if not the edges),
does 11% and 15% better than PMX and CX
Edge preserving heuristic
crossovers
These operators transfer about 60% of edges from parents
Edge recombination (ER)
operator
ER builds an offspring from the edges present in
both parents
It uses an edge list for this purpose, e.g.
for parents
(1 2 3 4 5 6 7 8 9)
(4 1 2 8 7 6 9 3 5)
the edge list is
ER operator (cont.)
ER selects an initial city from one of the parents,
preferably the one having the lowest degree in the
edge list, which is city 7 in our example
According to the edge list, city 7 can be connected
to 6 or 8, both having a degree of three
Breaking the tie randomly, suppose ER chooses 6
City 6 can be connected to 5 with degree three or
9 with degree four, so ER chooses 5
Offspring obtained at the end is (7 6 5 4 1 2 8 9 3)
with only one non-parental edge {3, 7}
ER operator (cont.)
The reason behind giving priority to a low degree
city is to increase the chance of completing the
tour by using the edges present in the parents
ER resorts to random choice of the next city only
when an “edge failure occurs,” i.e. when the
current city does not have a continuing edge since
the cities it can be connected to are already visited
This way, ER can transfer more than 95% of edges
from parents to a single offspring
Enhanced ER
In choosing the next city from the edge list, give
priority to edges common to both parents, e.g.
City 4: edges to other cities: 3 -5 1
-5 means that edge {4, 5} is in both parents
This is valid for cities having a degree of three,
because degree four means no edge is common,
and degree two means both edges are common
In addition, random choice of next city in case of
edge failure can be replaced by a better choice
Matrix representation (MR)
Binary n x n matrix M represents ordering
information, where mij=1 if and only if city i
precedes city j,
e.g. the tour
1-5-2-6-3-4-1
is represented as
1
1 0
2 0
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3 0
4 0
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5 0
6 0
2 3 4 5 6
1 1 1 1 1
0 1 1 0 1
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0 0 1 0 0
0 0 0 0 0

1 1 1 0 1
0 1 1 0 0
Not very practical due to memory requirements
Other possibilities
EAs based on divide and conquer strategies and
bisection methods
Operators based on conventional heuristics
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Edge exchange crossover (EEX)
Subtour exchange crossover (SXX)
Edge assembly crossover (EAX): to be discussed
in detail
Inver-over operator
Incorporating local search (local imrovement)
Crossover with conventional
heuristics
Nearest neighbor crossover (NNX) and Greedy
crossover (GX)
Applied to union graph of parental edges to
generate one offspring
Deterministic choice of edges from the union
graph
2-opt based improvement of offspring
Combined use of NNX (95%) and GX (5%)
0.3% deviation from optimal for n<250
Inver-over operator
Combination of inversion (mutation) and crossover
Each individual competes only with its offspring
Parallel hillclimbing where each hillclimber
performs a variable number of edge swaps in the
spirit of Lin-Kernighan algorithm
High selection pressure
Adaptive: (1) the number of inversions applied to a
single individual and (2) the segment to be inverted
are determined by another randomly selected
individual (second parent)
Inver-over algorithm
Inver-over example
Given S’=(2 3 9 4 1 5 8 6 7) and c=3
If rand()<p, then c’=8 is selected from S’ and
S’=(2 3 8 5 1 4 9 6 7) after random inversion
If rand()>p, then a second individual, say
(1 6 4 3 5 7 9 2 8) is selected, c’=5 next to c=3,
S’=(2 3 5 1 4 9 8 6 7) after guided inversion
where the edge {3, 5} is introduced from the
second parent
This is repeated until selected c’ is next to c in the
first individual, then S’ is evaluated
Inver-over performance
Percent deviation from Help-Karp lower bound is
< 0.63% for 442-city instance and < 3% for
10,000-city instance (better than most EAs)
Solves 105, 442 and 2392-city instances in 3-4
seconds, < 3 minutes and < 90 minutes (faster than
most EAs)
Has only three parameters to set: population size,
probability p of generating random inversion (as
opposed to guided inversion), number of iterations
until termination
Incorporating local search
It is possible to improve individuals produced in
initial population and by recombination
Conventional deterministic heuristics such as 2opt, 3-opt, and Lin-Kernighan can be used for this
purpose
Applying these to offspring produced by crossover
constitutes a form of mutation (Lamarckian
evolution)
Some EAs with local search are found to perform
better than pure local search with multi starts