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Ant Colony Optimization

Ant Colony Optimization (ACO)

• Developed by Marco Dorigo in the early 1990s.
• A probabilistic optimization technique inspired by

the interaction of ants in nature.

• Individual ants are blind and dumb, but ant

colonies show complex and smart behavior by as a result of low-level based communications.

• Useful for computational problems which can be

reduced to finding good paths in graphs.

Ants search for food

• Ants wander randomly in their search for food.
• If an ant finds food it returns home laying down a

pheromone trail on its way back.

• Other ants stumble upon the trail and start

following this pheromone trail.

• If the other ants successfully followed the trail,

they will also return home and also deposit pheromones on their way back ( reinforcing the trail).

Ants and pheromones

• Ants follow pheromone trails.
• The higher the amount of pheromone on a trail,

the higher the probability of ants following it.

• Pheromones defuse over time, so when a food

source is exhausted, the trail will no longer be reinforced and slowly dissipates.

• When an established path to a food source is

blocked, the ants leave the path to explore new routes.

(applet: the principle in action)

The shortest path

Three reasons why ants find the shortest path:

• Earlier pheromone (the trail is completed earlier)
• More pheromone (higher ant density)
• Younger pheromone (less diffusion)

Soon, the ants will find the shortest path between

their home and the food.

Idea: use this principle to find the shortest paths

of graphs!!!

The ACO algorithm

initialize pheromones tij

for each iteration do

for k = 1 to number of ants do set out ant k at start node while ant k has not build a solution do choose the next node of the path enddo enddo update pheromones

enddo

return best solution found

Ants choosing their path

• When ant k is located at a node vi the probability pjk^ of

choosing vj as the next node is:

With:

• Ni : the set of nodes that ant k can reach from vi (tabu list)
• ij : the heuristical desirability for choosing edge (i,j)
• tij : the amount of pheromone on edge (i,j)
•  and  : relative influence of heuristics vs. pheromone

 (^)  

i

i m N im im

ij ij k

if j N

if j N p (^) k j i

0

 

 

t 

t 

Pheromone update (1)

The pheromone on each edge is updated as:

t ij  ( 1  )t ij  t ij

With:

•  : the evaporation rate of the ‘old’ pheromone
• ∆tij : the ‘new’ pheromone that is deposited by all ants on edge (i,j) calculated as:



m

k

k ij (^) ij 0

t t

Using heuristic information

• The attractiveness ij of edge (i,j) is computed by

some heuristic indicating the a priori desirability of that move.

• The pheromone trail level tij of edge (i,j) indicates

how proficient it has been in the past.

•  = 0 represents a greedy approach and  = 0

represents rapid selection of tours that may not be optimal.

• Thus, there is a tradeoff between speed and

quality.

Example: TSP

• Desirability ij = 1/dij
• Tabu list contains all cities an ant has visited already.
• N = e
• Adding “elitist ant”: best t (^) ij  ( 1  )t ij  t ij  b  t ij





    otherwise

best Q^ Lbest if i j best ij 0

/ ( , ) t

•  = 1
•  = 5
•  = 0.
• Q = 100
• t 0 = 10-^6
• b = 5

Solving a problem by ACO

• Represent the problem in the form a weighted

graph, on which ants can build solutions.

• Define the meaning of the pheromone trails.
• Define the heuristic preference for the ant while

constructing a solution.

• Choose a specific ACO algorithm and apply to

problem being solved.

• Tune the parameters of the ACO algorithm.

Applications of ACO

• Scheduling
• Routing problems
  • Traveling Salesman Problem (TSP)
  • Vehicle routing
  • Network routing
• Set-problems
  • Multi-Knapsack
  • Max Independent Set
  • Set Covering
• Other
  • Shortest Common Sequence
  • Constraint Satisfaction
  • 2D-HP protein folding

Further reading

General Ant Colony Optimization:

• http://en.wikipedia.org/wiki/Ant_colony_optimization
• http://www.aco-metaheuristic.org/

Simulation applications:

• http://www.nightlab.ch/antsim/