This document summarizes ant colony optimization algorithms for solving the traveling salesman problem. It first discusses how real ant behavior inspired the development of artificial ant colony algorithms. It then describes how the ant colony optimization metaheuristic can be applied to solve combinatorial optimization problems like the traveling salesman problem. Finally, it reviews several specific ant colony algorithms for the traveling salesman problem, including Ant System, Elitist Ant System, Rank-Based Ant System, Min-Max Ant System, Ant Colony System, and Approximate Nondeterministic Tree Search.

1. Ant Colony Optimization Algorithms
for the Traveling Salesman Problem
ACO 3.1-3.5
Kristie Simpson
EE536: Advanced Artificial
Intelligence
Montana State University

2. 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.

3. 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).

4. 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

5. 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.

6. 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:

7. University of Heidelburg
NAME : att532
TYPE : TSP
COMMENT : 532-city problem
(Padberg/Rinaldi)
DIMENSION : 532
EDGE_WEIGHT_TYPE : ATT
NODE_COORD_SECTION
1 7810 6053
2 7798 5709
3 7264 5575
4 7324 5560
5 7547 5503
6 7744 5476
7 7821 5457
8 7883 5408
att532 : 27686
http://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/

8. 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.

9. 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).

10. Skeleton for ACO algorithm
 Set parameters, initialize pheromone trails.
 While termination condition not met
– ConstructAntSolutions
– ApplyLocalSearch
– UpdatePheromones
 Only solution construction and pheromone
updates considered.

11. 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

12. 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)

13. 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:

14. Ant System (AS)
 Pheromone update:

15. 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.

16. 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.

17. 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.

18. 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.

19. 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.

20. 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.

21. 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.

22. 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.

23. 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)

24. 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.

25. Hyper-cube Framework for ACO

26. 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.

27. Partially Asynchronous Parallel
Implementation (PAPI)
 Information exchanged at fixed intervals.
 Studies show it is better to exchange the best
solutions rather than all solutions.