Combinatorial Landscapes & Evolutionary Algorithms

Prof. Giuseppe Nicosia University of Catania Department of Mathematics and Computer Science [email protected]

www.dmi.unict.it/~nicosia 30/04/2020 1 DMI Università di Catania

Talk Outline

1. Combinatorial Landscapes 2. Evolutionary Computing DMI Università di Catania 30/04/2020 2

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 points) and (or nodes or –

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 T T T F F T F F f sat (F,s) 1 2 1 2 30/04/2020 6 DMI Università di Catania

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

search process is involved.

one since no • 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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DMI Università di Catania

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. 30/04/2020 DMI Università di Catania 12

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 a is defined by a quadruplet

search process

.

(S, n, f,



)

where • 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 DMI Università di Catania 30/04/2020 15

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. 30/04/2020 18 DMI Università di Catania

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); } DMI Università di Catania 30/04/2020 19