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By:Omkar Thakoor
Prakhar Jain
Utkarsh Diwaker
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 Swarm Intelligence : Definition
 SI as a discipline of AI
 Ant Colony Optimization(ACO) :
 Introduction
 The ACO Algorithm
 ACO for subset-problem
 Maximum Independent Set Problem(MISP)
 Solving MISP using ACO
 Summary
 References
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According to Bonabeau et al, it is
“The emergent collective intelligence of
groups of simple agents”
•
Refers to the collective behaviors that result from the
local interactions of the individuals with each other
and with their environment.
 Swarm intelligence as a discipline of Artificial
Intelligence, deals with natural and artificial
systems composed of many individuals that
coordinate using decentralized control and selforganization.
 Basic Philosophy of Swarm Intelligence : Observe
the behaviour of social animals and try to mimic
those animals on computer systems
Ants taking prey
Heards of animals
schools of fish
flocks of birds
•
•
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Composed of many individuals
the individuals are relatively homogeneous (i.e., their
computing behaviour is governed by same set of rules.)
the interactions among the individuals are based on
simple behavioral rules that exploit only local
information that the individuals exchange directly or via
the environment
the overall behaviour of the system results from the
interactions of individuals with each other and
with their environment, that is, the group
behavior self-organizes
•
The characterizing property of a swarm intelligence
system is its ability to act in a coordinated way without
the presence of a coordinator or of an external
controller.
•
Many examples can be observed in nature of swarms
that perform some collective behavior without any
individual controlling the group, or being aware of the
overall group behavior.
•
Notwithstanding the lack of individuals in charge of the
group, the swarm as a whole can show an intelligent
behavior.
•
This is the result of the interaction of spatially
neighboring individuals that act on the basis of simple
rules.
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Most often, the behavior of each individual of the swarm
is described in probabilistic terms: Each individual has a
stochastic behavior that depends on his local perception
of the neighborhood.
 Inspired by foraging behavior of ants.
 Ants find shortest path to food source from nest.
 Ants deposit pheromone along traveled path which is
used by other ants to follow the trail.
 This kind of indirect communication via the local
environment is called stigmergy.
 Has adaptability, robustness and redundancy.
ANTS
Why are ants interesting?
 ants solve complex tasks by simple local means
 ant productivity is better than the sum of their single
activities
 ants are ‘grand masters’ in search and exploitation
Which mechanisms are important?
 cooperation and division of labour
 pheromones
 2 ants start with equal probability of going on either
path.
 The ant on shorter path has a shorter to-and-fro time
from it’s nest to the food.
 The density of pheromone on the shorter path is
higher because of 2 passes by the ant (as compared to 1
by the other).
 The next ant takes the shorter route.
 Over many iterations, more ants begin using the path
with higher pheromone, thereby further reinforcing it.
 After some time, the shorter path is almost exclusively
used.
 Probabilistic technique for solving computational
problems which can be reduced to finding good paths
through graphs.
 An ant corresponds to a simple computational agent in
the ACO algorithm.
 It iteratively constructs a solution for the problem at
hand.
 The intermediate solutions are referred to as solution
states.
 At each iteration of the algorithm, each ant moves
from a state x to state y, corresponding to a more
complete intermediate solution
 Thus, each ant computes a set of feasible expansions
to its current state in each iteration, and moves to one
of these in probability
 For ant k ,
pkxy = probability of moving from state x to state y
which depends on the combination of two values, viz.
 The attractiveness Ƞxy of the move, computed by some
heuristic indicating the a priori desirability of that move
and
 The trail level τ xy of the move, indicating how proficient
it has been in the past to make that particular move.
 Trails are updated usually when all ants have completed
their solution, increasing or decreasing the level of trails
corresponding to moves that were part of "good" or "bad"
solutions, respectively.
 The kth ant moves from state to state with probability pkxy
 τ xy is the amount of pheromone deposited for transition from
state x to state y
 Ƞxy is the desirability of state transition xy (a priori
knowledge)
 0 ≤ α is a parameter to control the influence of τ xy
 β ≥ 1 is a parameter to control the influence of Ƞxy
 In each iteration, the pheromone values are updated by all
the ants that have built a solution in the iteration. The
pheromone τ ij on the edge joining node i and node j is
updated as follows
τ ij = (1 – σ) τ ij + ∑k ∆τijk
 σ is the pheromone evaporation coefficient,
 The summation is over the no of ants
 ∆τijk is the pheromone quantity laid by ant k on the edge
joining node i and node j and is given by
∆τijk = Q(Lk)
(Lk is the cost of the kth ant's tour (typically length) and Q is a constant.)
procedure ACO_MetaHeuristic
while(not_termination)
generateSolutions()
daemonActions()
pheromoneUpdate()
end while
end procedure
 In the Ant System seen so far, the pheromone is laid on
paths while for subset problems no path exists
connecting the items.
 A subset-based Ant System adapts the central idea in
the following way: “the more pheromone on a
particular item, the more profitable that item is. ”
 In other words, we move the pheromone from paths
to items.
 For the subset problem, the Ant system considers a special
type of local heuristic which takes into account both,
problem knowledge and the partial solution being built by
a particular ant k
 The intensity of pheromone trail on item i at time t+1 is
given by :τ i(t+1) = (1 – σ) τ i(t) + ∑k ∆τik(t)
 ∆τik(t) is the quantity of pheromone trail laid on item i
by the k-th ant at time t
This quantity is given by :∆τik(t) =
G(Lk) , if k-th ant incorporates item i
0
, otherwise
{
 The function Q depends upon the problem and gives the
amount of pheromone added to item i
 Usually Q(Lk) = M/Lk or M*Lk for minimization and
maximization problems respectively
 M is a constant.
 Lk depends on the objective.
 The heuristic value for the item i ∊ S - Ŝk(t) , is
defined as a function of the partial solution Ŝk(t) built
by ant k at time t.
 Now, for a partial solution Ŝk (t) = {i1, …, ij} , the probability
of selecting ip as the next item (p ∊ {j+1, j+2, …, n}) is given
by :-
 allowedk(t) ⊆ S – Ŝk (t) is the set of remaining feasible items
 τ i (t) is the amount of pheromone on item i
p
 Ƞi (Ŝk (t)) represents the heuristic value for item i based on
p
the solution being built by the k-th ant
 Thus, the higher the value of τ i (t) and Ƞi (Ŝk (t)), the more
p
p
profitable it is to include ip in the partial solution
 The maximum independent set problem (MISP)
consists of finding the largest subset of vertices of a
graph such that none of them are connected by an
edge (i.e., all vertices are independent of each other).
 If G = <V, E> denotes a graph where V is the set of
nodes and E the set of edges, the problem is to
determine a set V* ⊆ V such that ∀ i,j ∊ V* the edge
<i,j> ∉ E and | V* | is maximum.
 Let Fk(t) be the set of remaining feasible items with respect
to Ŝk(t) : the solution being built by ant k at time t.
 The local heuristic for the MISP can be defined as
Ƞi(Ŝk (t)) = |Fi|
where Fi represents Fk(t+1) in case item i is added to Ŝk(t)
 Then the local heuristic aims at assigning higher scores to
that item (say i) which yields a larger Fi. Thus, larger the
value of Fk(t+1), the larger the set of remaining items for
completing Ŝk after the inclusion of item i.
 The probability for item selection was given
previously where
 allowedk(t) = V - Ŝk (t) – Uk(t)
 Uk(t) = { j | ( (j,i) ∊ E ∨ (i,j) ∊ E) ∧ i ∊ Ŝk (t) } , i.e.,
the set of infeasible items with respect to Ŝk (t).
 Function Q is defined as Q(Lk) = M*Lk,
where M = 1/|v| and Lk, the objective value, is the
cardinality of the set of vertices conforming the
solution obtained by the ant k
 Let us consider the following example concerning the
heuristic defined above
 Figure shows a small MISP instance where |V| = 8.
 Let us suppose that in time the partial solution being
built by the k-th ant is Ŝk (t) = {2}, then
Fk(t) = V – {2} – {1,8} = {3, 7, 4, 5, 6}
 the set {1,8} represents the subset of infeasible items
due to the inclusion of item 2 in the partial solution
 Now the subset {3, 7, 4, 5, 6} is the set of current
feasible items and the corresponding heuristic values
are as follows :
Ƞ3(Ŝk(t)) = |F3| = |{4, 5}| = 2
Ƞ4(Sk(t)) = |F4| = |{3, 6, 7}| = 3
Ƞ5(Sk(t)) = |F5| = |{3}| = 1
Ƞ6(Sk(t)) = |F6| = |{4, 7}| = 2
Ƞ7(Sk(t)) = |F7| = |{4, 6}| = 2
Therefore, the highest score is obtained by item i = 4
possessing the biggest set of feasible items for the
next selection step
Let Ŝk(t) be the solution being built by ant k at time t.
Define Ƞv(Ŝk(t)) =
dG(v) is the degree of vertex v, and NG(v) is the neighbour set
of vertex v
It can be seen that this heuristic is different from the previous
in that it doesn’t depend on the solution being built.
It can be shown that higher the value of Ƞv(Ŝk(t)), better
the chances of ‘v’ being present in the optimal solution.
Many other such heuristics can be and in fact are used in
practice.
 Nature is very Intelligent and we can still learn a lot of
intelligent things from nature
 Individual agents in the group seem to have no
intelligence but group as a whole shows some
intelligence
 The intelligence of the group come from various
simple rules followed by individual agents.
 Has wide variety of applications.
 G. Leguizamon, Z. Michalewicz and Martin Schutz, "An
ant system for the maximum independent set problem,"
Proceedings of the 2001 Argentinian Congress on
Computer Science, vol.2, pp.1027-1040, 2001
 Hwayong Choi, Namsu Ahn, Sungsoo Park, “An Ant Colony
Optimization Approach for the Maximum Independent Set
Problem”, Computational Intelligence and Multimedia
Applications, 2003.ICCIMA 2003. Proceedings. Fifth
International Conference
 http://en.wikipedia.org/wiki/Swarm_intelligence
 http://www.scholarpedia.org/article/Swarm_intelligence