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Ant Colony Optimization:
an introduction
Daniel Chivilikhin
03.04.2013
2
Outline
1. Biological inspiration of ACO
2. Solving NP-hard combinatorial problems
3. The ACO metaheuristic
4. ACO for the Traveling Salesman Problem
3
Outline
1. Biological inspiration of ACO
2. Solving NP-hard combinatorial problems
3. The ACO metaheuristic
4. 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
1. Biological inspiration of ACO
2. Solving NP-hard combinatorial problems
3. The ACO metaheuristic
4. 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*
SS ⊆
~
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
1. Biological inspiration of ACO
2. Solving NP-hard combinatorial problems
3. The ACO metaheuristic
4. 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:
XS ⊆
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:
XX ⊆
~
Xx
~
∈
SXS 
~*
⊆
28
Combinatorial optimization
problem mapping (4)
• X – states
• S – candidate solutions
• – feasible states
• S* – optimal solutions
X
~
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
Ss∈
Ss
~
∈∀
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
1. Biological inspiration of ACO
2. Solving NP-hard combinatorial problems
3. The ACO metaheuristic
4. 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
• 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 :
∑
∈
⋅
⋅
=
k
iNl
ilil
ijijk
ijp βα
βα
ητ
ητ
k
iN
k
iNj ∈
56
Tour construction:
comprehension
• α = 0 – greedy algorithm
• β = 0 – only pheromone is at work
 quickly leads to stagnation
∑
∈
⋅
⋅
=
k
iNl
ilil
ijijk
ijp βα
βα
ητ
ητ
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:
( ) Lji
m
k
k
ijijij ∈∀∆+← ∑=
,,
1
τττ
( )


 ∈
=∆
otherwise
TjiC kk
k
ij
,0
,,/1
τ
59
Elitist Ant System
• Best-so-far ant deposits pheromone on each
iteration:
best
ij
m
k
k
ijijij e ττττ ∆+∆+← ∑=1
( )


 ∈
=∆
otherwise
TjiC bestbest
best
ij
,0
,,/1
τ
60
Rank-based Ant System
• Rank all ants
• Each ant deposits amounts of pheromone
proportional to its rank
61
MAX-MIN Ant System
1. Only iteration-best or best-so-far ant
deposits pheromone
2. 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
{ }



 ≤
= ∈
otherwiseSystemAntlike
qqif
j
ililNl k
i
,
,maxarg 0
β
ητ
1.Tour Construction
2. Local pheromone update:
τij ← (1 – ξ)τij + ξτ0,
64
Comparing Ant System variants
65
State of the art in TSP
• CONCORDE http://www.tsp.gatech.edu/
concorde.html
• Solved an instance of 85900 cities
• Computation took 286-2719 CPU days!
66
Current ACO research activity
• New applications
• Theoretical proofs
67
Further reading
• M. Dorigo, T. Stützle. Ant Colony
Optimization. MIT Press, 2004.
• http://iridia.ulb.ac.be/~mdorigo/ACO/
68
Next time…
• Some proofs of ACO convergence
69
Thank you!
Any questions?
This presentation is available at:
http://rain.ifmo.ru/~chivdan/presentations/
Daniel Chivilikhin [mailto: chivdan@gmail.com]
70
Used resources
• http://teamaltman.com/wp-content/uploads
/2011/06/Uncertainty-Ant-Apple-1024x1024.jpg
• http://myrealestatecoach.files.wordpress.com/2012/04/ant.jpg
• http://ars.els-cdn.com/content/image/1-s2.0-S1568494613000264-gr3.jpg
• http://www.theorie.physik.uni-goettingen.de/forschung/ha/talks/stuetzle.pdf
• http://www.buyingandsellingwebsites.com/wp-content/uploads/2011/12/Ant-150.jpg
• http://moodle2.gilbertschools.net/moodle/file.php/1040/Event-100_Days/Ant_Hormiga.gif
• http://4.bp.blogspot.com/_SPAe2p8Y-kg/TK3hQ8BUMtI/AAAAAAAAAHA/p75K-GcT_oo/s160
• http://1.bp.blogspot.com/-tXKJQ4nSqOY/UQZ3vabiy_I/AAAAAAAAAjA
/IK8jtlhElqk/s1600/16590492-illustration-of-an-ant-on-a-white-background.jpg

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Aco 03-04-2013