Solving NP Complete Problems using GA

Mubasher Baig

Asif Jamshed

Umar Faiz

Solving NP Complete Problems using GA

• Two basic steps to solving the traveling salesman problem using a GA.

Solving NP Complete Problems using GA

• First step– Create a group of many random tours in what

is called a population. • These tours are stored as a sequence of numbers.

Solving NP Complete Problems using GA and FT

• Second step– Randomly select tours for crossover– Crossover creates 2 new solutions children

in the hope that they create an even better solution. (single point crossover?)

Solving NP Complete Problems using GA and FT

• Size of the population may be changed by changing CONST variable (POP_SIZE) in the code

• Maximum distance between two cities may be controlled by changing the value of associated CONST (MAX_DIST).

Implementation

• Take input on the number of cities from the user

• Generate a (symmetric) square distance matrix representing the distance between each city.

Implementation

• Generate the initial population – generating random permutations is a difficult

since we have to be careful that an older number is not repeated.

• Both the distance matrix and the initial population is displayed to the user.

• Run the genetic algorithm

Implementation

/*The general form of a genetic algorithm:t := 0;initialize(P(0));evaluate(P(0));while not done do P'(t) := propotional_selection(P(t)); P'(t) := crossover(P'(t)); P'(t) := mutation(P'(t)); evaluate(P'(t)); P(t + 1) := P'(t); t := t + 1;od*/

Implementation

#include<stdlib.h>

#include<iostream.h>

#include<time.h>

const int MAX_DISTANCE = 30;

const int POP_SIZE = 100;

Implementation

– //initiate a symmetric matrix with random distances– short int** generateDistanceMatrix(short int);

– //display the distance matrix– void display(short int** matrix, short int size);

– //free the allocated memory– void deleteMatrix(short int** matrix, short int size);

– //returns a permutation of [1,n]– short int* generateRandomPermutation(short int n,

bool*);

Implementation

– //initialize the population with random tours– short int** initialize(short int cities);

– void displayPopulation(short int**, short int);

– //evaluation function returns the shortest permutation– short int* evaluate(short int** pop, short int**

distances, int n);

– //calculates the length of the supplied tour– int tourLength(short int* tour, short int** distances,

short int n);

Implementation

void displayTour(short int* tour, short int n);

int main(){

//give a seed for random number generationsrand(time(NULL));short int cities;cout << "Enter the number of cities: ";cin >> cities;

//randomly assign distances between citiesshort int** distanceMatrix = generateDistanceMatrix(cities);

cout << "\nThe Distance Matrix:" << endl;display(distanceMatrix, cities);}

Implementation

short int** generateDistanceMatrix(short int cities){

int i, j;

short int** matrix = new short int* [cities];for(i = 0; i < cities; i++)

matrix[i] = new short int[cities];

//assign random weights to upper half and copy to lower halffor(i = 0; i < cities; i++){

for(j = i; j < cities; j++){

matrix[i][j] = rand() % MAX_DISTANCE + 1;matrix[j][i] = matrix[i][j];

}}//distance to itself is 0for(i = 0; i < cities; i++)

matrix[i][i] = 0return matrix;

}

Implementation

void display(short int** matrix, short int size){

int i, j;

for(i = 0; i < size; i++){

for(j = 0; j < size; j++){

//for formattingif(matrix[i][j] < 10)

cout << " ";cout << matrix[i][j] << " ";

}

cout << endl;}return;

}

Implementation

void deleteMatrix(short int** matrix, short int size){

for(int i = 0; i < size; i++)delete matrix[i];

delete matrix;

return;}short int** initialize(short int cities){

short int** tour = new short int* [POP_SIZE];bool* node = new bool[cities];

//assign random permutationsfor(int i = 0; i < POP_SIZE; i++)

tour[i] = generateRandomPermutation(cities, node);

delete node;return tour;

}

Implementation

short int* generateRandomPermutation(short int n, bool* node){

short int* permutation = new short int[n];int i;

//flags for the numbers already usedfor(i = 0; i < n; i++)

node[i] = true;

int pointer = 0;int jump = 0;for(i = 0; i < n; i++){

//jump size determined randomlyjump = (rand() % (n - 1));

//find the number not yet used which is at//a distance 'jump' with wrap aroundwhile(jump >= 0){

pointer = (pointer + 1) % n;if(node[pointer])

jump--;

}//assign that number to the position//and mark it 'used' in then odes arraypermutation[i] = pointer + 1;node[pointer] = false;

}return permutation;

}

Implementation

short int* generateRandomPermutation(short int n, bool* node){

short int* permutation = new short int[n];int i;

//flags for the numbers already usedfor(i = 0; i < n; i++)

node[i] = true;

int pointer = 0;int jump = 0;for(i = 0; i < n; i++){

//jump size determined randomlyjump = (rand() % (n - 1));

//find the number not yet used which is at//a distance 'jump' with wrap aroundwhile(jump >= 0){

pointer = (pointer + 1) % n;if(node[pointer])

jump--;

}//assign that number to the position//and mark it 'used' in then odes arraypermutation[i] = pointer + 1;node[pointer] = false;

}return permutation;

}

Implementation

void displayPopulation(short int** tours, short int n){

for(int i = 0; i < POP_SIZE; i++){

for(int j = 0; j < n; j++){

//for formattingif(tours[i][j] < 10)

cout << " ";cout << tours[i][j] << " ";

}cout << endl;

}return;

}

Implementation

short int* evaluate(short int** pop, short int** distances, int n){

//set the best permutation to the first oneshort int* best = pop[0];

//then iterate through all of the population//to fnd the shotest tourfor(int i = 0; i < POP_SIZE; i++){

if(tourLength(pop[i], distances, n) < tourLength(best, distances, n))

best = pop[i];}

return best;}

Implementation

//calculate the tour length

int tourLength(short int* best, short int** distances, short int n)

{

int length = 0;

//find the length of the shortest cycle

for(int i = 0; i < n; i++)

length += distances[i][(i+1) % n];

return length;

}

Implementation

void displayTour(short int* tour, short int n){

for(int i = 0; i < n; i++){

//for formattingif(tour[i] < 10)

cout << " ";cout << tour[i] << " ";

}cout << "\b";return;

}

Implementation