# Combinatorial Landscapes &

Τεχνίτη Νοημοσύνη και Ρομποτική

23 Οκτ 2013 (πριν από 4 χρόνια και 8 μήνες)

114 εμφανίσεις

23/10/2013

DMI
-

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

DMI
-

Università di Catania

2

Talk Outline

1.
Combinatorial Landscapes

2.
Evolutionary Computing

23/10/2013

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Università di Catania

3

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.

23/10/2013

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Università di Catania

4

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
.

23/10/2013

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Università di Catania

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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

23/10/2013

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Università di Catania

7

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.

23/10/2013

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Università di Catania

8

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
.

23/10/2013

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9

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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11

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.

23/10/2013

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Università di Catania

13

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
.

23/10/2013

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Università di Catania

14

Process Landscape

Given a search landscape (S, n, f), a
process
landscape

(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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15

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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Università di Catania

17

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
-

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18

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.

23/10/2013

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Università di Catania

19

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);

}

}