23/10/2013
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Università di Catania
1
Combinatorial Landscapes &
Evolutionary Algorithms
Prof. Giuseppe Nicosia
University of Catania
Department of Mathematics and Computer Science
nicosia@dmi.unict.it
www.dmi.unict.it/~nicosia
23/10/2013
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Talk Outline
1.
Combinatorial Landscapes
2.
Evolutionary Computing
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1. Combinatorial Landscapes
The notion of landscape is among the rare
existing concepts which help to understand
the behaviour of search algorithms
and
heuristics and
to characterize the difficulty
of a combinatorial problem.
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Search Space
Given a
combinatorial problem
P
, a
search
space
associated to a mathematical formulation
of
P
is defined by a couple
(S,f)
–
where
S
is a finite
set of configurations
(or
nodes
or
points) and
–
f
a
cost function
which associates a real number to
each configurations of
S
.
For this structure two most common measures
are
the minimum and the maximum costs
.In
this case we have the
combinatorial
optimization problems
.
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Example: K
-
SAT
An instance of the K
-
SAT problem consists of a
set V of variables, a collection C of clauses over
V such that each clause c
C has |c|= K.
The problem is to find a satisfying truth
assignment for C.
The search space for the 2
-
SAT with |V|=2 is
(S,f) where
–
S
={ (T,T), (T,F), (F,T), (F,F) } and
–
the cost function
for 2
-
SAT computes only the
number of satisfied clauses
f
sat
(s)= #SatisfiedClauses(F,s), s
S
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An example of Search Space
Let we consider F = (A
B)
(
A
B)
A B
f
sat
(F,s)
T T
1
T F
2
F T
1
F F
2
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Search Landscape
•
Given a search space
(S,f)
, a
search landscape
is defined by a triplet
(S,n,f)
where
n
is a
neighborhood function
which verifies
n : S
2
S
-
{ 0}
•
This landscape, also called
energy landscape
,
can be considered as a
neutral
one since no
search process is involved.
•
It can be conveniently viewed as
weighted
graph
G=(S, n , F)
where the weights are
defined on the nodes, not on the edges.
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Example and relevance of
Landscape
The search Landscape for the K
-
SAT problem
is a
N dimensional hypercube
with
N = number of variables = |V| .
•
Combinatorial optimization problems are often
hard to solve
since such problems may have
huge and complex search landscape
.
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Hypercubes
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Solvable &
Impossible
•
The New York Times
, July
13, 1999
“
Separating
Insolvable and Difficult
”.
•
B. Selman, R. Zecchina,
et al.
“
Determing
computational complexity
from characteristic
‘phase transitions’
”,
Nature
, Vol. 400, 8 July
1999,
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Phase Transition,
=4.256
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Characterization of the Landscape in terms of
Connected Components
Number of solutions, number of connected components and CCs'
cardinality versus
for
#3
-
SAT
problem with
n=10
variables.
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CC's cardinality at phase transition
(3)=4.256
Number of Solutions, number of connected components and CC's
cardinality at phase transition
(3)=4.256
versus number of variables
n
for
#3
-
SAT problem
.
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Process Landscape
Given a search landscape (S, n, f), a
process
landscape
is defined by a quadruplet
(S, n, f,
)
where
is a
search process
.
•
The process landscape represents
a particular view of
the neutral landscape (S, n, f) seen by a search
algorithm
.
•
Examples of search algorithms:
–
Local Search Algorithms.
–
Complete Algorithms (e. g. Davis
-
Putnam algorithm).
–
Evolutionary Algorithms
: Genetic Algorithms, Genetic
Programming, Evolution Strategies, Evolution Programming,
Immune Algorithms.
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2. Evolutionary Algorithms
EAs are optimization methods based on
an evolutionary metaphor that showed
effective in solving difficult problems.
“Evolution is the natural way to program”
Thomas Ray
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Evolutionary Algorithms
1. Set of candidate solutions (
individuals
):
Population
.
2. Generating candidates by:
–
Reproduction
: Copying an individual.
–
Crossover
:
2 parents
2 children.
–
Mutation
: 1 parent
1 child.
3. Quality measure of individuals:
Fitness function
.
4.
Survival
-
of
-
the
-
fittest
principle.
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Main components of EAs
1. Representation of individuals:
Coding
.
2. Evaluation method for individuals:
Fitness
.
3. Initialization procedure for the
1st generation
.
4. Definition of variation operators (
mutation
and
crossover
).
5. Parent (
mating
) selection mechanism.
6. Survivor (
environmental
) selection mechanism.
7.
Technical parameters
(e.g. mutation rates, population size).
Experimental tests, Adaptation based on measured quality,
Self
-
adaptation based on evolution.
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Mutation and Crossover
EAs manipulate partial
solutions in their search for
the overall optimal solution
.
These partial solutions or
`
building blocks
' correspond to
sub
-
strings of a trial solution
-
in
our case local sub
-
structures
within the overall conformation.
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Algorithm Outline
procedure EA; {
t = 0;
initialize population (P(t), d);
evaluate P(t);
until (done) {
t = t + 1;
parent_selection P(t);
recombine (P(t), p
cross
);
mutate ( P(t), p
mut
);
evaluate P(t);
survive P(t);
}
}
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