A Hybrid Heuristic for the Traveling Salesman Problem

R. Baraglia, J. I. Hildalgo, R. Perego

CMPSC 580, Spring 2006

I didn’t even think “difficultly” was a word.

Genetic Algorithms(from Eiben and Schoenauer)

Simple Example

Find the 8-bit vector which maximizes the number of 1’s in the vector – the “MAXONES” problem.

Answer is obvious: 11111111

Can a genetic algorithm find it?

Example (continued)

Initialize population (random):

10010101 11001011

01001010 01011001

01010100 10001110

Example (continued)

Evaluate fitness:

f(10010101) = 4 f(11001011) = 5

f(01001010) = 3 f(01011001) = 4

f(01010100) = 3 f(10001110) = 3

Example (continued)

Select genitors (parents):

Random, but biased toward fitter individuals:

f(10010101) = 4 f(11001011) = 5

f(01001010) = 3 f(01011001) = 4

f(01010100) = 3 f(10001110) = 3

Example (continued)

10010101 11001011

10010101 10010011

Variation operators: crossover and mutation

10110101 10010001

Example (concluded)

Evaluate and replace (next generation):

10010101(4) 11001011(5)

01001010(3) 10110101(5)

11010101(5) 10010010(3)

Survivors

Offspring

…Now repeat: select, mate, replace…

Traveling Salesperson

Here’s a nice applet from a U. Texas/Arlington course Web site:

http://www-cse.uta.edu/~cook/ai1/lectures/applets/gatsp/TSP.html

Compact GA (CGA)

Instead of maintaining a population of individuals, just keep a description, or model, of the population. Example:

A Possible Population:1 2 3 4 50 1 0 0 11 1 0 0 01 0 0 0 10 1 0 0 01 1 1 0 1

Model:(.6, .8, .1, 0, .6)

For each bit position, maintain theprobability that some element ofthe population has a “1” in that position.Example (with 5-bit chromosomes):

Compact GA (continued)

Start with uniform random model:

(.5, .5, .5, .5, .5)

Randomly generate two (or more) individuals according to this probability distribution; find fitness:

1 1 0 1 0; fitness = 3.21

0 1 1 0 0; fitness = 1.98

Compact GA (continued)

Old model: (.5, .5, .5, .5, .5)

1 1 0 1 0; fitness = 3.21 (winner)

0 1 1 0 0; fitness = 1.98 (loser)

Reward 1’s in winner; penalize 1’s in loser:

New model: (.6, .5, .4, .6, .5)

Repeat the process using the new probabilities

Compact GA (continued)

Ideally, process will converge– this is the “solution” found by the CGA:

(.5, .5, .5, .5, .5)

(.6, .5, .4, .6, .5)

(.7, .4, .3, .6, .5)

…

(1, 0, 0, 1, 1) [“solution”]

Compact GA for TSPSave a description of population in the form

of a probability matrix. Example:

1

2

34

5

Population: 1,4,5,2,3 1,2,5,3,4 1,5,3,4,2 1,3,5,2,4

0

.5 0

.5 .25 0

.75 .5 .5 0

.25 .75 .75 .25 0

1

2

3

4

5

1 2 3 4 5

Probability of edge (i,j):

CGA for TSP (continued)

Constructing a new tour:

Pick random start city:

(4, _

Pick another city:

1

Decide whether to include

1 with a 75% probability:

YES

0

.5 0

.5 .25 0

.75 .5 .5 0

.25 .75 .75 .25 0

1

2

3

4

5

1 2 3 4 5

Probability of edge (i,j):

CGA for TSP (continued)

(4, 1, _

Pick another city:

5

Decide whether to include

1 with a 25% probability:

NO

So try another city, … etc.

0

.5 0

.5 .25 0

.75 .5 .5 0

.25 .75 .75 .25 0

1

2

3

4

5

1 2 3 4 5

Probability of edge (i,j):

CGA for TSP (continued)

0

.5 0

.5 .25 0

.75 .5 .5 0

.25 .75 .75 .25 0

1

2

3

4

5

1 2 3 4 5

Probability of edge (i,j):

Constructing a new tour:

Process might get “stuck”(repeated “NO” results). In thiscase, pick the city with highestprobability of being connectedto most recently added city:

(4,1,3,5,…

Hybrid Approach

Combine Genetic Algorithm (problem-independent method) with heuristic specifically designed for TSP.

Make one “genetic step”, then improve the solution using the heuristic.

A Simple Heuristic for Improving TSP Solutions

2-opt: look at all possible pairs of edges in the tour, see if tour length can be improved by reordering cities as follows. Repeat until no more improvement is possible:

(Can have 3-opt, 4-opt, etc.; more expensive)

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