Ant Colony Optimization: an introductionDaniel Chivilikhin03.04.2013

OutlineBiological inspiration of ACOSolving NP-hard combinatorial problemsThe ACO metaheuristicACO for the Traveling Salesman Problem

OutlineBiological inspiration of ACOSolving NP-hard combinatorial problemsThe ACO metaheuristicACO for the Traveling Salesman Problem

Biological inspiration: from real to artificial ants

Ant coloniesDistributed systems of social insectsConsist of simple individualsColony intelligence >> Individual intelligence

Ant CooperationStigmergy indirect communication between individuals (ants)Driven by environment modifications

Denebourgs double bridge experimentsStudied Argentine ants I. humilisDouble bridge from ants to food source

Double bridge experiments: equal lengths (1)

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

Double bridge experiments: different lengths (2)

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

OutlineBiological inspiration of ACOSolving NP-hard combinatorial problemsThe ACO metaheuristicACO for the Traveling Salesman Problem

Solving NP-hard combinatorialproblems

Combinatorial optimizationFind values of discrete variablesOptimizing a given objective function

Combinatorial optimization = (S, f, ) problem instanceS set of candidate solutionsf objective function set of constraints set of feasible solutions (with respect to )Find globally optimal feasible solution s*

NP-hard combinatorial problemsCannot be exactly solved in polynomial timeApproximate methods generate near-optimal solutions in reasonable timeNo formal theoretical guaranteesApproximate methods = heuristics

Approximate methodsConstructive algorithmsLocal search

Constructive algorithmsAdd components to solution incrementallyExample greedy heuristics: add solution component with best heuristic estimate

Local searchExplore neighborhoods of complete solutionsImprove current solution by local changesfirst improvementbest improvement

What is a metaheuristic?A set of algorithmic conceptsCan be used to define heuristic methodsApplicable to a wide set of problems

Examples of metaheuristicsSimulated annealingTabu searchIterated local searchEvolutionary computationAnt colony optimizationParticle swarm optimizationetc.

OutlineBiological inspiration of ACOSolving NP-hard combinatorial problemsThe ACO metaheuristicACO for the Traveling Salesman Problem

The ACO metaheuristic

ACO metaheuristicA colony of artificial ants cooperate in finding good solutionsEach ant simple agentAnts communicate indirectly using stigmergy

Combinatorial optimization problem mapping (1)Combinatorial problem (S, f, (t))(t) time-dependent constraintsExample dynamic problemsGoal find globally optimal feasible solution s*Minimization problemMapped on another problem

Combinatorial optimization problem mapping (2)C = {c1, c2, , cNc} finite set of componentsStates of the problem:X = {x = , |x| < n < +}Set of candidate solutions:

Combinatorial optimization problem mapping (3)Set of feasible states:

We can complete into a solution satisfying (t)Non-empty set of optimal solutions:

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

S*

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 graphC set of componentsL edges fully connecting the components (connections)GC construction graph

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

Construction graphEach component ci or connection lij have associated:heuristic informationpheromone trail

Heuristic informationA priori information about the problemDoes not depend on the antsOn components ci iOn connections lij ijMeaning: cost of adding a component to the current solution

Pheromone trailLong-term memory about the entire search processOn components ci iOn connections lij ijUpdated by the ants

Artificial ant (1)Stochastic constructive procedureBuilds solutions by moving on GCHas finite memory for:Implementing constraints (t)Evaluating solutionsMemorizing its path

Artificial ant (2)Has a start state xHas termination conditions ekFrom state xr moves to a node from the neighborhood Nk(xr)Stops if some ek are satisfied

Artificial ant (3)Selects a move with a probabilistic rule depending on:Pheromone trails and heuristic information of neighbor components and connectionsMemoryConstraints

Artificial ant (4)Can update pheromone on visited components (nodes)and connections (edges)Ants act:ConcurrentlyIndependently

The ACO metaheuristicWhile not doStop():ConstructAntSolutions() UpdatePheromones() DaemonActions()

ConstructAntSolutionsA colony of ants build a set of solutionsSolutions are evaluated using the objective function

UpdatePheromonesTwo opposite mechanisms:Pheromone depositPheromone evaporation

UpdatePheromones: pheromone depositAnts increase pheromone values on visited components and/or connectionsIncreases probability to select visited components later

UpdatePheromones: pheromone evaporationDecrease pheromone trails on all components/connections by a same valueForgetting avoid rapid convergence to suboptimal solutions

DaemonActionsOptional centralized actions, e.g.:Local optimizationAnt elitism (details later)

ACO applicationsTraveling salesmanQuadratic assignmentGraph coloringMultiple knapsackSet coveringMaximum cliqueBin packing

OutlineBiological inspiration of ACOSolving NP-hard combinatorial problemsThe ACO metaheuristicACO for the Traveling Salesman Problem

ACO for the Traveling Salesman Problem

Traveling salesman problemN set of nodes (cities), |N| = nA set of arcs, fully connecting NWeighted graph G = (N, A)Each arc has a weight dij distanceProblem:Find minimum length Hamiltonian circuit

TSP: construction graphIdentical to the problem graphC = NL = Astates = set of all possible partial tours

TSP: constraintsAll cities have to be visitedEach city only onceEnforcing allow ants only to go to new nodes

TSP: pheromone trailsDesirability of visiting city j directly after i

TSP: heuristic informationij = 1 / dijUsed in most ACO for TSP

TSP: solution constructionSelect random start cityAdd unvisited cities iterativelyUntil a complete tour is built

ACO algorithms for TSPAnt SystemElitist Ant SystemRank-based Ant SystemAnt Colony SystemMAX-MIN Ant System

Ant System: Pheromone initializationPheromone initializationij = m / Cnn, where:m number of antsCnn path length of nearest-neighbor algorithm

Ant System: Tour constructionAnt k is located in city i is the neighborhood of city iProbability to go to city :

Tour construction: comprehension = 0 greedy algorithm = 0 only pheromone is at workquickly leads to stagnation

Ant System: update pheromone trails evaporationEvaporation for all connections(i, j) L:ij (1 ) ij, [0, 1] evaporation ratePrevents convergence to suboptimal solutions

Ant System: update pheromone trails depositTk path of ant kCk length of path TkAnts deposit pheromone on visited arcs:

Elitist Ant SystemBest-so-far ant deposits pheromone on each iteration:

Rank-based Ant SystemRank all antsEach ant deposits amounts of pheromone proportional to its rank

MAX-MIN Ant SystemOnly iteration-best or best-so-far ant deposits pheromonePheromone trails are limited to the interval [min, max]

Ant Colony SystemDiffers from Ant System in three points:More aggressive tour construction ruleOnly best ant evaporates and deposits pheromoneLocal pheromone update

Ant Colony SystemTour Construction

Local pheromone update:ij (1 )ij + 0,

Comparing Ant System variants

State of the art in TSPCONCORDE http://www.tsp.gatech.edu/concorde.html Solved an instance of 85900 citiesComputation took 286-2719 CPU days!

Current ACO research activityNew applicationsTheoretical proofs

Further readingM. Dorigo, T. Sttzle. Ant Colony Optimization. MIT Press, 2004.http://iridia.ulb.ac.be/~mdorigo/ACO/

Next timeSome proofs of ACO convergence

Thank you!

Any questions?

This presentation is available at:http://rain.ifmo.ru/~chivdan/presentations/Daniel Chivilikhin [mailto: chivdan@gmail.com]

Used resourceshttp://teamaltman.com/wp-content/uploads/2011/06/Uncertainty-Ant-Apple-1024x1024.jpg http://myrealestatecoach.files.wordpress.com/2012/04/ant.jpghttp://ars.els-cdn.com/content/image/1-s2.0-S1568494613000264-gr3.jpghttp://www.theorie.physik.uni-goettingen.de/forschung/ha/talks/stuetzle.pdf http://www.buyingandsellingwebsites.com/wp-content/uploads/2011/12/Ant-150.jpghttp://moodle2.gilbertschools.net/moodle/file.php/1040/Event-100_Days/Ant_Hormiga.gifhttp://4.bp.blogspot.com/_SPAe2p8Y-kg/TK3hQ8BUMtI/AAAAAAAAAHA/p75K-GcT_oo/s1600/ant+vision.GIFhttp://1.bp.blogspot.com/-tXKJQ4nSqOY/UQZ3vabiy_I/AAAAAAAAAjA/IK8jtlhElqk/s1600/16590492-illustration-of-an-ant-on-a-white-background.jpg