1. Ants and the TSP

2. What’s a TSP? Nodes with edges in between
Visit each node at least once
All nodes reachable
Might make sense on a 2D grid, might not

3. Why TSP for ants? Obvious application Really hard It’s a benchmark
Easy to understand
It works!

4. How do ants solve it? Pheromones placed on edges that are useful
Pheromones evaporate off all edges

5. Local what?
Local minima, maxima, optima

6. Deciding action
Tabu list
Visibility
Amount of pheromone

7. Pheromone intensity Probability(path) = (intensity)a(visibility)b
a = 0 means greedy search
b = 0 means converge quickly
A good balance is most useful

8. Pheromone updates Or 0 if not visited
Updated at the end of an iteration (all ants complete tour)
D intensity = intensity_constant / length_of_tour
Or 0 if not visited
Still need evaporation
New intensity = decay_constant * intensity + D intensity

9. How Many Ants? How about #ants = #cities?
Place randomly or one at each city, doesn’t matter.

10. Another Modification Elitist Ant Will only take the best tour so far
Keeps the current best solution from “getting lost”
More elitists as iterations continue

11. How good is AS? Similar or better than GP for 30-70 cities
On bigger problems
Did not find solutions in 3000 steps
Converged quickly to “good” solutions
Did not do well compared to SP solvers
What is needed to fix it?
Authors decided on local search

12. Before moving on The authors of the Ant System analyzed its behavior
Does not converge closely
Keeps trying new solutions
Avoids maxima traps
Good for converging quickly to “good” solutions
Good for dynamic situations

13. ACS ACS (Ant Colony System) – improved AS Four changes Transitions
Pheromone updates
Local updates
Candidate lists

14. 1) A new transition rule More explicit exploration enforcement
Next city selected from candidate list
If random > threshold, most desireable
Otherwise, weighted random (same as before)

15. 2) A new update rule Intensity formula is the same
D intensity = intensity_constant / length_of_tour
Or 0 if not visited
Only applied to best solution
Other solutions must emerge through local updates

16. 3) Local updates Intensity decays when ant follows edge
Intensity = (1-decay_constant)*intensity + (decay_constant) * leftover_constant
Forces ants to try different paths

17. 4) Candidate list
The q closest unvisited cities are the only ones considered
If no unvisited cities, go to the nearest city

18. How does ACS do? Tested on 2500 iterations with 10 ants for 50 cities
Comparable to or better than standard algorithms.
On larger problems, a local search is needed

19. Unimplemented improvements
Actually use the threshold
Allow the r best ants to udpate
Penalize bad routes
Use better local searches
Test using “dynamic benchmarks”

20. Implemented modifications
Max-Min AS
Only one ant updates
Pheromones bounded to (min, max)
Trails initialized to max.
Trail-smoothing
D intensity a (max – intensity)

21. Another modification AS-rank Uses elitists
Updates proportional to performance

22. The End With local-optimizing help, ants were successful with the TSP.
Questions?