By:- Omkar Thakoor Prakhar Jain Utkarsh Diwaker

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

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 self- organization. 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

flocks of birds schools of fish

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. 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, p k xy = probability of moving from state x to state y which depends on the combination of two values, viz. The attractiveness xy 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 k th ant moves from state to state with probability p k xy τ xy is the amount of pheromone deposited for transition from state x to state y xy 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 xy

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 ∆τ ij k σ is the pheromone evaporation coefficient, The summation is over the no of ants ∆τ ij k is the pheromone quantity laid by ant k on the edge joining node i and node j and is given by ∆τ ij k = Q(L k ) ( L k is the cost of the k th 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 ∆τ i k (t) ∆τ i k (t) is the quantity of pheromone trail laid on item i by the k-th ant at time t This quantity is given by :- ∆τ i k (t) = G(L k ), 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(L k ) = M/L k or M*L k for minimization and maximization problems respectively M is a constant. L k depends on the objective. The heuristic value for the item i ∊ S - Ŝ k (t), is defined as a function of the partial solution Ŝ k (t) built by ant k at time t.

Now, for a partial solution Ŝ k (t) = {i 1, …, i j }, the probability of selecting i p as the next item (p ∊ {j+1, j+2, …, n}) is given by :- allowed k (t) ⊆ S – Ŝ k (t) is the set of remaining feasible items τ i p (t) is the amount of pheromone on item i i p ( Ŝ k (t) ) represents the heuristic value for item i based on the solution being built by the k-th ant Thus, the higher the value of τ i p (t) and i p ( Ŝ k (t)), the more profitable it is to include i p 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 = 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 ∉ E and | V * | is maximum.

Let F k (t) be the set of remaining feasible items with respect to Ŝ k (t) : the solution being built by ant k at time t. The local heuristic for the MISP can be defined as i (Ŝ k (t)) = |F i | where F i represents F k (t+1) in case item i is added to Ŝ k (t) Then the local heuristic aims at assigning higher scores to that item (say i) which yields a larger F i. Thus, larger the value of F k (t+1), the larger the set of remaining items for completing Ŝ k after the inclusion of item i.

The probability for item selection was given previously where allowed k (t) = V - Ŝ k (t) – U k (t) U k (t) = { j | ( (j,i) ∊ E ∨ (i,j) ∊ E) ∧ i ∊ Ŝ k (t) }, i.e., the set of infeasible items with respect to Ŝ k (t). Function Q is defined as Q(L k ) = M*L k, where M = 1/|v| and L k, 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 Ŝ k (t) = {2}, then F k (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 :

3 (Ŝ k (t)) = |F 3 | = |{4, 5}| = 2 4 (S k (t)) = |F 4 | = |{3, 6, 7}| = 3 5 (S k (t)) = |F 5 | = |{3}| = 1 6 (S k (t)) = |F 6 | = |{4, 7}| = 2 7 (S k (t)) = |F 7 | = |{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 Ŝ k (t) be the solution being built by ant k at time t. Define v ( Ŝ k (t)) = d G (v) is the degree of vertex v, and N G (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 v ( Ŝ k (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 , 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 Proceedings. Fifth International Conference