1. Ant Colony Optimization: an introduction
Daniel Chivilikhin

2. Outline Biological inspiration of ACO
Solving NP-hard combinatorial problems
The ACO metaheuristic
ACO for the Traveling Salesman Problem

3. Outline Biological inspiration of ACO
Solving NP-hard combinatorial problems
The ACO metaheuristic
ACO for the Traveling Salesman Problem

4. Biological inspiration: from real to artificial ants

5. Ant colonies Distributed systems of social insects
Consist of simple individuals
Colony intelligence >> Individual intelligence

6. Ant Cooperation
Stigmergy – indirect communication between individuals (ants)
Driven by environment modifications

7. Denebourg’s double bridge experiments
Studied Argentine ants I. humilis
Double bridge from ants to food source

8. Double bridge experiments: equal lengths (1)

9. Double bridge experiments: equal lengths (2)
Run for a number of trials
Ants choose each branch ~ same number of trials

10. Double bridge experiments: different lengths (2)

11. Double bridge experiments: different lengths (2)
The majority of ants follow the short path

12. Outline Biological inspiration of ACO
Solving NP-hard combinatorial problems
The ACO metaheuristic
ACO for the Traveling Salesman Problem

13. Solving NP-hard combinatorial
problems

14. Combinatorial optimization
Find values of discrete variables
Optimizing a given objective function

15. Combinatorial optimization
Π = (S, f, Ω) – problem instance
S – set of candidate solutions
f – objective function
Ω – set of constraints
– set of feasible solutions (with respect to Ω)
Find globally optimal feasible solution s*

16. NP-hard combinatorial problems
Cannot be exactly solved in polynomial time
Approximate methods – generate near-optimal solutions in reasonable time
No formal theoretical guarantees
Approximate methods = heuristics

17. Approximate methods
Constructive algorithms
Local search

18. Constructive algorithms
Add components to solution incrementally
Example – greedy heuristics:
add solution component with best heuristic estimate

19. Local search Explore neighborhoods of complete solutions
Improve current solution by local changes
first improvement
best improvement

20. What is a metaheuristic?
A set of algorithmic concepts
Can be used to define heuristic methods
Applicable to a wide set of problems

21. Examples of metaheuristics
Simulated annealing
Tabu search
Iterated local search
Evolutionary computation
Ant colony optimization
Particle swarm optimization
etc.

22. Outline Biological inspiration of ACO
Solving NP-hard combinatorial problems
The ACO metaheuristic
ACO for the Traveling Salesman Problem

23. The ACO metaheuristic

24. ACO metaheuristic
A colony of artificial ants cooperate in finding good solutions
Each ant – simple agent
Ants communicate indirectly using stigmergy

25. Combinatorial optimization problem mapping (1)
Combinatorial problem (S, f, Ω(t))
Ω(t) – time-dependent constraints
Example – dynamic problems
Goal – find globally optimal feasible solution s*
Minimization problem
Mapped on another problem…

26. Combinatorial optimization problem mapping (2)
C = {c1, c2, …, cNc} – finite set of components
States of the problem:
X = {x = <ci, cj, …, ch, …>, |x| < n < +∞}
Set of candidate solutions:

27. Combinatorial optimization problem mapping (3)
Set of feasible states:
We can complete into a solution satisfying Ω(t)
Non-empty set of optimal solutions:

28. Combinatorial optimization problem mapping (4)
X – states
S – candidate solutions
– feasible states
S* – optimal solutions

29. Combinatorial optimization problem mapping (5)
Cost g(s, t) for each
In most cases – g(s, t) ≡ f(s, t)
GC = (C, L) – completely connected graph
C – set of components
L – edges fully connecting the components (connections)
GC – construction graph

30. Combinatorial optimization problem mapping (last )
Artificial ants build solutions by performing randomized walks on GC(C, L)

31. Construction graph
Each component ci or connection lij have associated:
heuristic information
pheromone trail

32. Heuristic information
A priori information about the problem
Does not depend on the ants
On components ci – ηi
On connections lij – ηij
Meaning:
cost of adding a component to the current solution

33. Pheromone trail Long-term memory about the entire search process
On components ci – τi
On connections lij – τij
Updated by the ants

34. Artificial ant (1) Stochastic constructive procedure
Builds solutions by moving on GC
Has finite memory for:
Implementing constraints Ω(t)
Evaluating solutions
Memorizing its path

35. Artificial ant (2) Has a start state x Has termination conditions ek
From state xr moves to a node from the neighborhood – Nk(xr)
Stops if some ek are satisfied

36. Artificial ant (3)
Selects a move with a probabilistic rule depending on:
Pheromone trails and heuristic information of neighbor components and connections
Memory
Constraints Ω

37. Artificial ant (4) Can update pheromone on visited components (nodes)
and connections (edges)
Ants act:
Concurrently
Independently

38. The ACO metaheuristic While not doStop(): ConstructAntSolutions()
UpdatePheromones()
DaemonActions()

39. ConstructAntSolutions
A colony of ants build a set of solutions
Solutions are evaluated using the objective function

40. UpdatePheromones Two opposite mechanisms: Pheromone deposit
Pheromone evaporation

41. UpdatePheromones: pheromone deposit
Ants increase pheromone values on visited components and/or connections
Increases probability to select visited components later

42. UpdatePheromones: pheromone evaporation
Decrease pheromone trails on all components/connections by a same value
Forgetting – avoid rapid convergence to suboptimal solutions

43. DaemonActions Optional centralized actions, e.g.: Local optimization
Ant elitism (details later)

44. ACO applications Traveling salesman Quadratic assignment
Graph coloring
Multiple knapsack
Set covering
Maximum clique
Bin packing …

45. Outline Biological inspiration of ACO
Solving NP-hard combinatorial problems
The ACO metaheuristic
ACO for the Traveling Salesman Problem

46. ACO for the Traveling Salesman Problem

47. Traveling salesman problem
N – set of nodes (cities), |N| = n
A – set of arcs, fully connecting N
Weighted graph G = (N, A)
Each arc has a weight dij – distance
Problem:
Find minimum length Hamiltonian circuit

48. TSP: construction graph
Identical to the problem graph
C = N
L = A
states = set of all possible partial tours

49. TSP: constraints All cities have to be visited Each city – only once
Enforcing – allow ants only to go to new nodes

50. TSP: pheromone trails
Desirability of visiting city j directly after i

51. TSP: heuristic information
ηij = 1 / dij
Used in most ACO for TSP

52. TSP: solution construction
Select random start city
Add unvisited cities iteratively
Until a complete tour is built

53. ACO algorithms for TSP Ant System Elitist Ant System
Rank-based Ant System
Ant Colony System
MAX-MIN Ant System

54. Ant System: Pheromone initialization
Τij = m / Cnn,
where:
m – number of ants
Cnn – path length of nearest-neighbor algorithm

55. Ant System: Tour construction
Ant k is located in city i
is the neighborhood of city i
Probability to go to city :

56. Tour construction: comprehension
α = 0 – greedy algorithm
β = 0 – only pheromone is at work
quickly leads to stagnation

57. Ant System: update pheromone trails – evaporation
Evaporation for all connections∀(i, j) ∈ L:
τij ← (1 – ρ) τij,
ρ ∈[0, 1] – evaporation rate
Prevents convergence to suboptimal solutions

58. Ant System: update pheromone trails – deposit
Tk – path of ant k
Ck – length of path Tk
Ants deposit pheromone on visited arcs:

59. Elitist Ant System
Best-so-far ant deposits pheromone on each iteration:

60. Rank-based Ant System Rank all ants
Each ant deposits amounts of pheromone proportional to its rank

61. MAX-MIN Ant System
Only iteration-best or best-so-far ant deposits pheromone
Pheromone trails are limited to the interval [τmin, τmax]

62. Ant Colony System Differs from Ant System in three points:
More aggressive tour construction rule
Only best ant evaporates and deposits pheromone
Local pheromone update

63. Ant Colony System Tour Construction Local pheromone update:
τij ← (1 – ξ)τij + ξτ0,

64. Comparing Ant System variants

65. State of the art in TSP
CONCORDE
Solved an instance of cities
Computation took CPU days!

66. Current ACO research activity
New applications
Theoretical proofs

67. Further reading
M. Dorigo, T. Stützle. Ant Colony Optimization. MIT Press, 2004.

68. Next time…
Some proofs of ACO convergence

69. This presentation is available at:
Thank you!
Any questions?
This presentation is available at:
Daniel Chivilikhin [mailto:

70. Used resources