Ant Colony Optimization Algorithms for the Traveling Salesman Problem

ACO 3.1-3.5Kristie SimpsonEE536: Advanced Artificial IntelligenceMontana State University

ACO Review

Chapter 1: From Real to Artificial Ants (Dr. Paxton)– Looked at real ants and the double bridge

experiment.– Defined a stochastic model for real ants, and then

modified the definition for artificial ants.– Discussed the Simple-ACO algorithm.

ACO Review

Chapter 2: The ACO Metaheuristic (Chris, Shen)– Introduced combinatorial optimization problems.– Discussed exact and approximate solutions to

NP-hard problems.– Discussed the ACO Metaheuristic and example

applications (TSP presented in section 2.3.1).

Chapter 3: ACO Algorithms for TSP

“But you’re sixty years old. They can’t expect you to keep traveling every week.” –Linda in act I, scene I of Death of a Salesman, Authur Miller, 1949

Why use TSP?

NP-Hard (permutation problem, N!). Easy application of ACO. Easy to understand. Ant System (the first ACO alogrithm) was

tested on TSP. Solutions tend to be most efficient for other

applications.

What is TSP?

Starting from his hometown, a salesman wants to find a shortest tour that takes him through a given set of customer cities and then back home, visiting each customer city exactly once.

Represented by a weighted graph G = (N,A). The goal in TSP is to find a minimum length

Hamiltonian circuit of the graph. An optimal solution is:

University of Heidelburg

NAME : att532TYPE : TSPCOMMENT : 532-city problem

(Padberg/Rinaldi)DIMENSION : 532EDGE_WEIGHT_TYPE : ATTNODE_COORD_SECTION1 7810 60532 7798 57093 7264 55754 7324 55605 7547 55036 7744 54767 7821 54578 7883 5408

att532 : 27686 http://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/

ACO Algorithms for the TSP

G = (C, L) is equal to G = (N, A). All cities have to be visited and that each city

is visited at most once. Pheromone trail: the desirability of visiting

city j directly after i. Heuristic: inversely proportional to the

distance between two cities i and j.

Tour Construction

1) Choose a start city.

2) Use pheromone and heuristic values to add cites until all have been visited.

3) Go back to the initial city.

Note: Tour may be improved with a local search (section 3.7).

Skeleton for ACO algorithm

Set parameters, initialize pheromone trails. While termination condition not met

– ConstructAntSolutions– ApplyLocalSearch– UpdatePheromones

Only solution construction and pheromone updates considered.

ACO Algorithms

Ant System (AS) Elitist Ant System (EAS) Rank-Based Ant System (ASrank) Min-Max Ant System (MMAS) Ant Colony System (ACS) Approximate Nondeterministic Tree Search

(ANTS) Hyper-Cube Framework for ACO

Ant System (AS)

m ants concurrently build tour. Pheromone initialized to m/Cnn. Ants initially in randomly chosen sites. Random proportional rule used to decide which city

to visit next. (see Box 3.1 for good parameter values)

Ant System (AS)

Each ant k maintains a memory Mk for its neighborhood.

After all ants have constructed their tours, the pheromone trails are updated.

Pheromone evaporation:

Ant System (AS)

Pheromone update:

Elitist Ant System (EAS)

First improvement on AS. Provide strong additional reinforcement to the arcs

belonging to the best tour found since the start of the algorithm.

Rank-Based Ant System (ASrank)

Another improvement over AS. Each ant deposits an amount of pheromone that

decreases with its rank. In each iteration, only the best (w-1) ranked ants and

the best-so-far ant are allowed to deposit pheromone.

Min-Max Ant System (MMAS)

Four modifications with respect to AS.– Strongly exploits the best tours found.

This may lead to stagnation. So…

– Limits the possible range of pheromone values.– Pheromone values initialized to upper limit.– Pheromone values are reinitialized when system

approaches stagnation.

Min-Max Ant System (MMAS)

After all ants construct a solution, pheromone values are updated. (Evaporation is the same as in AS)

Lower and upper limits on pheromones limit the probability of selecting a city.

Initial pheromone values are set to the upper limit, resulting in initial exploration.

Occasionally pheromones are reinitialized.

Ant Colony System (ACS)

Uses ideas not included in the original AS. Differs from AS in three main points:

– Exploits the accumulated search experience more strongly than AS.

– Pheromone evaporation and deposit take place only on the best-so-far tour.

– Each time an ant uses an arc, some pheromone is removed from the arc.

Ant Colony System (ACS)

Pseudorandom proportional rule used to decide which city to visit next.

Only best-so-far ant adds pheromone after each iteration. Evaporation and deposit only apply to best-so-far.

Ant Colony System (ACS)

The previous pheromone update was global. Each ant in ACS also uses a local update that is applied after crossing an arc.

Makes arc less desirable for following ants, increasing exploration.

Approximate Nondeterministic Tree Search (ANTS)

Uses ideas not included in the original AS. Not applied to TSP. Computes lower bounds on the completion of

a partial solution to define the heuristic information that is used by each ant during the solution construction.

Creates a dynamic heuristic where the lower the estimate the more attractive the path.

Approximate Nondeterministic Tree Search (ANTS)

Two modifications with respect to AS:– Use of a novel action choice rule.

– Modified pheromone trail update rule. (No explicit pheromone evaporation)

Hyper-cube Framework for ACO

Uses ideas not included in the original AS. Not applied to TSP. Automatically rescales the pheromone values for

them to lie always in the interval [0,1]. Decision variables {0, 1} typically correspond to the

components used by the ants for construction. A solution problem then corresponds to one corner

of the n-dimensional hyper-cube, where n is the number of decision variables.

Hyper-cube Framework for ACO

Parallel Implementation

Fine-grained – few individuals per processor, frequent information exchange.– Can lead to major communication overhead.

Coarse-grained – larger subpopulations per processor, information exchange is rare.– Much more promising for ACO.– p colonies on p processors.

Partially Asynchronous Parallel Implementation (PAPI)

Information exchanged at fixed intervals. Studies show it is better to exchange the best

solutions rather than all solutions.