Broderick Crawford
Pontificia Universidad Cat´olica de Valpara´ıso, Universidad T´ecnica Federico Santa Mar´ıa, Chile
Carlos Castro, Eric Monfroy
Universidad T´ecnica Federico Santa Mar´ıa, University of Nantes, France
Ant Colony Optimization (ACO), Constraint Programming (CP), Constraint Propagation, Arc Consistency
(AC), Set Partitioning Problem (SPP).
In this paper, we focus on the resolution of Set Partitioning Problem. We try to solve it with Ant Colony
Optimization algorithms and Hybridizations of Ant Colony Optimization with Constraint Programming tech-
niques. We recognize the difficulties of pure Ant Algorithms solving strongly constrained problems. There-
fore, we explore the addition of Constraint Programming mechanisms in the construction phase of the ants so
they can complete their solutions. Computational results solving some test instances are presented showing
the advantages to use this kind of hybridization.
There exist some problems for which the effective-
ness of Ant Colony Optimization (ACO) is lim-
ited, among them the strongly constrained problems.
Those are problems for which neighbourhoods con-
tain few solutions, or none at all, and local search
has a very limited use. Probably, the most signifi-
cant of those problems is the Set Partitioning Problem
(SPP) and a direct implementation of the basic ACO
framework is unable of obtaining feasible solutions
for many SPP standard tested instances (Maniezzo
and Milandri, 2002). The best performing meta-
heuristic for SPP is a genetic algorithm due to Chu
and Beasley (Chu and Beasley, 1998; Beasley and
Chu, 1996). There already exists some first ap-
proaches applying ACO to the Set Covering Problem
(SCP) (Alexandrov and Kochetov, 2000; Leguizam´on
and Michalewicz, 1999). More recent works (Hadji
et al., 2000; Lessing et al., 2004; Gandibleux et al.,
2004) apply Ant Systems to the SCP and related prob-
lems using techniques to remove redundant columns
and local search to improve solutions. In this pa-
per, we explore the addition of a lookahead mecha-
nism to the two main ACO algorithms: Ant System
(AS) and Ant Colony System (ACS). Trying to solve
larger instances of SPP with AS or ACS implemen-
tations derives in a lot of unfeasible labelling of vari-
ables and the ants can not obtain complete solutions
using the classic transition rule when they move in
their neighbourhood. We propose the addition of a
lookahead mechanism from Constraint Programming
(CP) in the construction phase of ACO thus only fea-
sible partial solutions are generated. The lookahead
mechanism allows the incorporation of information
about the instantiation of variables after the current
decision (Michel and Middendorf, 1998; Gagne et al.,
The Set Partitioning Problem is the NP-complete
problem of partitioning a given set into mutually in-
dependent subsets while minimizing a cost function
defined as the sum of the costs associated to each of
the eligible subsets. In the SPP matrix formulation
we are given a m× n matrix A = (a
) in which all the
matrix elements are either zero or one. Additionally,
each column is given a non-negative cost c
. We say
that a column j can cover a row i if a
= 1. Let J de-
notes the set of the columns and x
a binary variable
which is one if column j is chosen and zero otherwise.
The SPP can be defined formally as follows:
Crawford B., Castro C. and Monfroy E. (2008).
In Proceedings of the Third International Conference on Software and Data Technologies - PL/DPS/KE, pages 380-383
DOI: 10.5220/0001892403800383
Minimize f(x) =
Sub ject to
= 1; i = 1,...,m (2)
In this section, we briefly present ACO algorithms
and give a description of their use to solve SPP. More
details about ACO algorithmscan be foundin (Dorigo
et al., 1999; Dorigo and Gambardella, 1997). The ba-
sic idea of ACO algorithms comes from the capabil-
ity of real ants to find shortest paths between the nest
and food source. From a Combinatorial Optimization
point of view, the ants are looking for good solutions.
ACO can be applied in a very straightforward way
to SPP. The columns are chosen as the solution com-
ponents and have associated a cost and a pheromone
trail (Dorigo and Stutzle, 2004). Each column can be
visited by an ant only once and then a final solution
has to cover all rows. A walk of an ant over the graph
representation corresponds to the iterative addition of
columns to the partial solution obtained so far. Each
ant starts with an empty solution and adds columns
until a cover is completed. A pheromone trail τ
a heuristic information η
are associated to each el-
igible column j. A column to be added is chosen
with a probability that depends of pheromonetrail and
the heuristic information. The most common form of
the ACO decision policy (Transition Rule Probabil-
ity) when ants work with components is:
(t) =
if j / S
where S
is the partial solution of the ant k. The
β parameter controls how important is η in the prob-
abilistic decision (Dorigo and Stutzle, 2004; Lessing
et al., 2004). In this work the pheromonetrail τ
is put
on the problems component (each eligible column j)
and we use a dynamic heuristic information defined
as η
, where e
is the so called cover value,
that is, the number of additional rows covered when
adding column j to the current partial solution, and c
is the cost of column j. We use two instances of ACO:
Ant System (AS) and Ant Colony System (ACS) al-
gorithms (Dorigo and Stutzle, 2004). Trying to solve
larger instances of SPP with the original AS or ACS
implementation derives in a lot of unfeasible labelling
of variables, and the ants can not obtain complete so-
lutions. A direct implementation of the basic ACO
framework is incapable of obtaining feasible solution
for many SPP instances. Then we explore the addition
of a lookahead mechanism in the construction phase
of ACO thus only feasible solutions are generated.
The two basic techniques of Constraint Programming
to solve combinatorial problems are Constraint Prop-
agation and Constraint Distribution (Apt, 2003). Con-
straint Propagation embeds any reasoning which con-
sists in explicitly forbidding values or combinations
of values for some variables of a problem because a
given subset of its constraints cannot be satisfied oth-
erwise. The algorithm proceeds as follows: when a
value is assigned to a variable, the algorithm recom-
putes the variable domains and assigned values of all
its dependent variables (variable that belongs to the
same constraint). This process continues recursively
until no more changes can be done. It causes the algo-
rithm to recompute the values for further downstream
variables. In the case of binary variables the con-
straint propagation works very fast in strongly con-
strained problems like SPP.
The problem cannot be solved using Constraint
Propagation alone, Constraint Distribution or Search
is required to reduce the search space until Constraint
Propagation is able to determine the solution. Con-
straint Distribution splits a problem into complemen-
tary cases once Constraint Propagation cannot ad-
vance further. By iterating propagation and distribu-
tion, propagation will eventually determine the solu-
tions of a problem (Apt, 2003).
Recently, some efforts have been done in order to in-
tegrate Constraint Programming techniques to ACO
algorithms (Meyer and Ernst, 2004; Focacci et al.,
2002). In this work, ACO useCP in the variableselec-
tion (when adding columns to partial solution). The
CP algorithm used in this paper is Forward Check-
ing with Backtracking. The algorithm is a combina-
tion of Arc Consistency Technique and Chronological
Backtracking (Dechter and Frost, 2002). It performs
Arc Consistency between pairs of a not yet instanti-
ated variable and an instantiated variable, i.e., when a
value is assigned to the current variable, any value in
the domain of a future variable which conflicts with
this assignment is removed from the domain. For-
ward Checking search procedure guarantees that at
each step of the search, all the constraints between
already assigned variables and not yet assigned vari-
ables are arc consistency. This reduces the search tree
and the overall amount of computational work done.
The cost is that Arc consistency enforcing always in-
creases the information available on each variable la-
Table 3 presents the results when adding Forward
Checking to the basic ACO algorithmsfor solving test
instances taken from the OR-Library (Beasley, 1990).
Table 1 presents the problem code, the number of
rows (constraints), the number of columns (decision
variables), the best known cost value for each instance
(IP optimal), and the density (percentage of ones in
the constraint matrix) respectively. Table 2 presents
the results obtained by better performing metaheuris-
tics with respect to SPP, Genetic Algorithm of Chu
and Beasley (Chu and Beasley, 1998), Genetic Algo-
rithm of Levine et al. (Levine, 1994) and the most
recent algorithm by Kotecha et al. (Kotecha et al.,
2004). An entry of ”X” in the table means no feasi-
ble solution was found. The algorithms have been run
with the following parameters settings: influence of
pheromone (alpha)=1.0, influence of heuristic infor-
mation (beta)=0.5 and evaporation rate (rho)=0.4 as
suggested in (Leguizam´on and Michalewicz, 1999;
Lessing et al., 2004; Dorigo and Stutzle, 2004). The
number of ants has been set to 120 and the maximum
number of iterations to 160, so that the numberof gen-
erated candidate solutions is limited to 19.200. For
ACS the list size was 500 and Qo=0.5. Algorithms
were implemented using ANSI C, GCC 3.3.6, under
Microsoft Windows XP Professional version 2002.
The effectiveness of Constraint Programming is
showed to solve SPP, because the SPP is so strongly
constrained the stochastic behaviour of ACO can be
improved with lookahead techniques in the construc-
tion phase, so that almost only feasible partial solu-
tions are induced. In the original ACO implemen-
tation the SPP solving derives in a lot of unfeasible
labelling of variables, and the ants can not complete
solutions. With respect to the computational results
this is not surprising, because ACO metaheuristics
are general purpose tools that will usually be outper-
formed when customized algorithms for a problem
Table 1: Benchmark instances.
Problem Rows Columns Optimum Density
sppnw06 50 6774 7810 18.17
sppnw08 24 434 35894 22.39
sppnw09 40 3103 67760 16.20
sppnw10 24 853 68271 21.18
sppnw12 27 626 14118 20.00
sppnw15 31 467 67743 19.55
sppnw19 40 2879 10898 21.88
sppnw23 19 711 12534 24.80
sppnw26 23 771 6796 23.77
sppnw32 19 294 14877 24.29
sppnw34 20 899 10488 28.06
sppnw39 25 677 10080 26.55
sppnw41 17 197 11307 22.10
Table 2: Better performing Metaheuristics results.
Problem Beasley Levine Kotecha
sppnw06 7810 - -
sppnw08 35894 37078 36068
sppnw09 67760 - -
sppnw10 68271 X 68271
sppnw12 14118 15110 14474
sppnw15 67743 - -
sppnw19 10898 11060 11944
sppnw23 12534 12534 12534
sppnw26 6796 6796 6804
sppnw32 14877 14877 14877
sppnw34 10488 10488 10488
sppnw39 10080 10080 10080
sppnw41 11307 11307 11307
Table 3: ACO and ACO+FC results.
sppnw06 9200 9788 8160 8038
sppnw08 X X 35894 36682
sppnw09 70462 X 70222 69332
sppnw10 X X X X
sppnw12 15406 16060 14466 14252
sppnw15 67755 67746 67743 67743
sppnw19 11678 12350 11060 11858
sppnw23 14304 14604 13932 12880
sppnw26 6976 6956 6880 6880
sppnw32 14877 14886 14877 14877
sppnw34 13341 11289 10713 10797
sppnw39 11670 10758 11322 10545
sppnw41 11307 11307 11307 11307
Our main contribution is the study of the combina-
tion of Constraint Programming and Ant Colony Op-
timization solving benchmarks of the Airline Crew
Pairing Problem formulated as a Set Partitioning
Problem. The main conclusion from this work is
ICSOFT 2008 - International Conference on Software and Data Technologies
that we can improve ACO with CP. Computational
results also indicated that our hybridization is capa-
ble of generating optimal or near optimal solutions
for many problems. The concept of Arc Consistency
plays an essential role in Constraint Programming as
a problem simplification operation and as a tree prun-
ing technique during search through the detection of
local inconsistencies among the uninstantiated vari-
ables. We have shown that it is possible to add Arc
Consistency to any ACO algorithms and the compu-
tational results confirm that the performance of ACO
can be improved with this type of hybridisation. Any-
way, a complexity analysis should be done in order
to evaluate the cost we are adding with this kind of
integration. We strongly believe that this kind of inte-
gration between complete and incomplete techniques
should be studied deeply.
Future versions of the algorithm will study the
pheromone treatment representation and the incorpo-
ration of available techniques in order to reduce the
input problem (Pre Processing) and improve the so-
lutions given by the ants (Post Processing). The ants
solutions may contain expensive components which
can be eliminated by a fine tuning heuristic after the
solution, then we will explore Post Processing proce-
dures, which consists in the identification and replace-
ment of the columns of the ACO solution in each it-
eration by more effective columns. Besides, the ants
solutions can be improvedby other local search meth-
ods like Hill Climbing, Simulated Annealing or Tabu
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