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Using Soft CSPs for Approximating Pareto-Optimal Solution Sets

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We consider constraint satisfaction problems where solutions must be optimized according to multiple criteria.
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Using Soft CSPs for Approximating Pareto-Optimal Solution Sets
Marc Torrens and Boi Faltings
Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland
Marc.Torrens@epfl.ch, Boi.Faltings@epfl.ch
Abstract
We consider constraint satisfaction problems where so-
lutions must be optimized according to multiple crite-
ria. When the relative importance of different criteria
cannot be quantified, there is no single optimal solution,
buta possibly very large set of Pareto-optimal solutions.
Computing this set completely is in general very costly
and often infeasible in practical applications.
We consider several methods that apply algorithms for
soft CSP to this problem. We report on experiments,
both on random and real problems, that show that such
algorithms can compute surprisingly good approxima-
tions of the Pareto-optimal set. We also derive variants
that further improve the performance.
Introduction
Constraint Satisfaction Problems (CSPs) (Tsang 1993; Ku-
mar 1992) are ubiquitous in applications like configuration,
planning, resource allocation, scheduling, timetabling and
many others. A CSP is specified by a set of variables and a
set of constraints among them. A solution to a CSP is a set
of value assignments to all variables such that all constraints
are satisfied.
In many applications of constraint satisfaction, the objec-
tive is not only to find a solution satisfying the constraints,
but also to optimize one or more preference criteria. Such
problems occur in resource allocation, scheduling and con-
figuration. As an example, we consider in particular elec-
tronic catalogs with configuration functionalities:
a hard constraint satisfaction problem defines the avail-
able product configurations, for example different fea-
tures of a PC.
the customer has different preference criteria that need to
be optimized, for example price, certain functions, speed,
etc.
More precisely, we assume that optimization criteria are
modeled as functions that map each solution into a numer-
ical value that indicates to what extent the criterion is vio-
lated; i.e. the lower the value, the better the solution.
Copyright
c
2002, American Association for Artificial Intelli-
gence (www.aaai.org). All rights reserved.
We call such problems Multi-criteria Constraint Opti-
mization Problems (MCOP), which we define formally as
follows:
Definition 1. A Multi-criteria Constraint Optimization
Problem (MCOP) is defined by a tuple
,
where
is a finite set of variables,
each associated with a domain of discrete values
, and a set of constraints .
Eachconstraint
is defined by a function on some subset of
variables. This subset is called the scope of the constraint.
A constraint over the variables
is a function
that defines whether the tuple is al-
lowed (0) or disallowed (
) in case of a hard constraint, or
the degree to which it is preferred in the case of a preference
criterion, with 0 being the most preferred and MAX-SOFT
1
.
What the best solution to a MCOP is depends strongly
on the relative importance of different criteria. This may
vary depending on the customer, the time, or the precise val-
ues that the criteria take. For example, in travel planning for
some people price may be more important than the schedule,
while for others it is just the other way around. People find it
very difficult to characterize the relative importance of their
preferences by numerical weights. Some research has be-
gun to address this problem by inferring constraint weight-
ing from the way people choose solutions (Biso, Rossi, &
Sperduti 2000).
Pareto-optimality
When relative importance of criteria is unknown, it is not
possible to identify a single best solution, but at least certain
solutions can be classified as certainly not optimal. This is
the case when there is another solution which is as good as
or better in all respects. We say that a solution
dominates
another solution
if for every constraint , the violation
cost in
is no greater than that in , and if for at least
one constraint,
has a lower cost than . This is defined
formally as follows:
Definition 2. Given a MCOP
with constraints
1
MAX-SOFT is a maximum value for soft constraints. By using
a specific maximum valuation for soft constraints, we can easily
differenciate between a hard violation and soft violation.
1
2 (dominated by 1)
3
4
5 (dominated by 3 and 4)
6
Figure 1: Example of solutions in a CSP with two preference cri-
teria. The two coordinates show the values indicating the degrees
to which criteria
(horizontal) and (vertical) are violated.
and two solutions and of :
dominates
and
The idea of Pareto-optimality (Pareto 1896 1987) is to
consider all solutions which are not dominated by another
one as potentially optimal:
Definition 3. Any solution which is not dominated by an-
other is called Pareto-optimal.
Definition 4. Given a MCOP
, the Pareto-optimal set
2
is
the set of solutions which are not dominated by any other
one.
In Figure 1, the Pareto-optimal set is
, as solu-
tion 7 is dominated by 4 and 6, 5 is dominated by 3 and 4,
and 2 is dominated by 1.
Pareto-optimal solutions are hard to compute because un-
less preference criteria involve only few of the variables, the
dominance relation can not be evaluated on partial solutions.
Research on better algorithms for Pareto-optimality is still
ongoing (see, for example, Gavanelli (Gavanelli 2002)), but
since it cannot escape this fundamental limitation, generat-
ing all Pareto-optimal solutions is likely to always remain
computationally very hard.
Therefore, Pareto-optimality has so far found little use in
practice, despite the fact that it characterizes optimality in a
more realistic way. This is especially true when the Pareto-
optimal set must be computed very quickly, for example in
interactive configuration applications (e.g. electronic cata-
logs).
Another characteristic of the Pareto-optimal set is that it
usually contains many solutions; in fact, all solutions could
be Pareto-optimal. Thus, it will be necessary to have the end
2
also called the efficient frontier of
.
user, or another program that has the information about rela-
tive importance of constraints, pick the best solution among
the set that has been returned.
Soft CSPs
Given the intractability of computing all Pareto-optimal so-
lutions, the predominant approach in constraint satisfaction
has been to map multiple criteria into a single one and then
compute a single solution that is optimal according to this
criterion. The key question in this case is how to com-
bine the preference orders of the individual constraints into
a global preference order for picking the optimal solution.
For the scenario in Figure 1, some commonly used soft CSP
algorithms would give the following results:
1. in MAX-CSP (Freuder & Wallace 1992), we sum the val-
ues returnedby each criterion, possibly with a weight, and
pick the solution with the lowest sum as the optimal solu-
tion. In Figure 1, if we assume that both constraints carry
equal weight, this is solution
.
2. in fuzzy CSP (Fargier, Lang, & Schiex 1993), we char-
acterize each solution by the worst criteria violation, i.e.
by the maximum value returned by any criterion, and pick
the solution with the lowest result as the optimum. In Fig-
ure 1, this is solution
.
3. in hierarchical CSP (Borning, Freeman-Benson, & Wil-
son 1992), criteria have a weight expressing their degree
of importance and we order solutions according to the
lowest weight of all the constraints that are somehow vio-
lated. In Figure 1, if we assume that criterion
is more
important than criterion
, the optimum is solution . If
we consider
as more important than , then the opti-
mum is solution
.
MAX-CSP can be solved efficiently by branch and bound
techniques. Fuzzy and hierarchicalCSPs can be solvedmore
efficiently by incrementally relaxing constraints in increas-
ing order of importance until solutions are found. More re-
cently, it was observed that most soft CSP methods can be
seen as instances of the more general class of soft constraint
using c-semirings introduced by (Bistarelli et al. 1999). De-
pending on the way that the semiring operators are instan-
tiated, one obtains the different soft constraint formalisms.
Semiring-based CSPs are developed in detail in (Bistarelli
2001).
All these methods make the crucial assumption that vio-
lation costs for different criteria are comparable or can be
made comparable by known weighting metrics. This as-
sumption is also crucial because it ensures that there is a
single optimal solution which can be returned by the algo-
rithm. The weights have a crucial influence on the solution
that will be returned: in Figure 1, depending on the relative
weights, solutions 1, 4 or 6 could be the optimal ones.
Soft CSP with Multiple Solutions
Interestingly, most decision aids already return not the sin-
gle optimal solution, but an ordered list of the top-ranked
solutions. Thus, web search engines return hundreds of
documents deemed to be the best matches to a query, and
electronic catalogs return a list of possibilities that fit the
criteria in decreasing degree. In general, these solutions
have been calculated assuming a certain weight distribution
among constraints. It appears that listing a multitude of
nearly optimal solutions is intended to compensate for the
fact that these weights, and thus the optimality criterion, are
usually not accurate for the particular user.
For example, in Figure 1, if we assume that constraints
haveequal weight, the order of solutions would be
, and the top four according to this
weighting are also the Pareto-optimal ones.
The questions we address in this paper follow from this
observation:
how closely do the top-ranked solutions generated by
a scheme with known constraint weights, in particular
MAX-CSP, approximate the Pareto-optimal set, and
can we derive variations that cover this set better while
maintaining efficiency?
We have performed experiments in the domain of config-
uration problems that indicate that MAX-CSP can indeed
provide a surprisingly close approximation of the Pareto-
optimal set both in real settings and in randomly gener-
ated problems, and derive improvements to the methods that
could be applied in general settings.
Using Soft CSP Algorithms for Approximating
Pareto-optimal Sets
To approximate the set of Pareto-optimal solutions, the sim-
plest solution is to simply map the MCOP into an opti-
mization problem with a single criterion obtained by a fixed
weighting of the different criteria, called a weighted con-
strained optimization problem (WCOP):
Definition 5. A WCOP is an MCOP with an associated
weight vector
, . The optimal
solution
to a COP is a tuple that minimizes the valuation
function
The best solutions to a WCOP is the set of the solutions
with the lowest cost.
is called the valuation of . We
call feasible solutions to a WCOP those solutions which do
not violate any hard constraint.
Note that when the weight vector consists of all 1s,
WCOP is equivalent to MAX-CSP and is also an instanti-
ation of the semiring CSP framework. WCOP can be solved
by branch-and-bound search algorithms. These algorithms
can be easily adapted to returnnot only the best solution, but
an ordered list of the
best solutions. In our work we use
Partial Forward Checking (Freuder & Wallace 1992) (PFC),
which is a branch and bound algorithm with propagation.
Pareto-optimality of WCOP Solutions
As mentioned before, in practice it turns out that among
the
best solutions to a WCOP, many are also Pareto-
optimal. Theorem 1 shows indeed that the optimal solution
of a WCOP is always Pareto-optimal, and that furthermore
among the
best solutions all those which arenotdominated
by another one are Pareto-optimal for the whole problem:
Theorem1. Let be the set of the best solutions ob-
tained by optimizing with weights
. If and is not dominated by any
then is Pareto-optimal.
Proof. Assume that
is not Pareto-optimal. Then, there is
a solution
which dominates solution , and by
Definition 2:
and
As a consequence, we also have:
i.e. must be better than according to the weighted
optimization function. But this contradicts the fact that
.
This justifies the use of soft CSP to find not just one, but
a larger set of Pareto-optimal solutions. In particular, by
filtering the
best solutions returned by a WCOP algorithm
to eliminate the ones which are dominated by another one
in the set, we find only solutions which are Pareto-optimal
for the entire problem. We can thus bypass the costly step of
proving non-dominance on the entire solution set.
The first algorithm is thus to find a subset of the Pareto
set by modeling it as a WCOP with a single weight vector,
generating the
best solutions, and filtering them to retain
only those which are not dominated (Algorithm 1).
Algorithm 1: Method for approximating the Pareto-
optimal set of a MCOP using a single WCOP using PFC
(Partial Forward Checking).
Input: : MCOP.
: the maximal number of solutions to com-
pute.
Output:
: an approximation of the Pareto-optimal
set.
PFC (WCOP (P, ), )
eliminateDominatedSolutions ( )
The Weighted-sums Method
The above method has the weakness that it generates solu-
tions that are optimal with respect to a certain weight vector
and thus likely to be very similar to one another. The iter-
ated weighted-sums approach, as described for example by
Steuer in (Steuer 1986), attempts to overcome this weak-
ness by calling a WCOP method several times with different
weight vectors. Each WCOP will give us a different sub-
set of Pareto-optimal solutions, and a good distribution of
weight vectors should give us a good approximation of the
Pareto-optimal set.
Algorithm 2: Weighted-sums method for approximating
the Pareto-optimal set of a MCOP. The
weight vectors
are generated to give an adequate distribution of solu-
tions.
Input: : MCOP.
: the maximal number of solutions to com-
pute.
: is a collection of
weight vectors.
Output:
: an approximation of the Pareto-optimal
set.
PFC (WCOP (P, ), )
foreach
do
PFC (WCOP (P, ), )
eliminateDominatedSolutions ( )
Basically, the proposed main method (Algorithm 2) con-
sists of performing several runs over WCOPs with differ-
ent weight vectors for the soft constraints and one run over
the vector
. The method has two parameters: ,
which is the maximal number of solutions to be found, and
which is the collection of weight
vectors. Algorithm 2 performs
iterations, one for
each different WCOP with weight vector
, and one for a
WCOP with the weight vector
. At each iteration,
it computes the best
solutions to the correspond-
ing WCOP. At the end of each iteration, dominated solutions
are filtered out, so by Theorem 1, the resulting set of solu-
tions are Pareto-optimal.
Empirical Results on Random Problems
We have tested several different instances and variances of
the described methods on randomly generated problems:
Method 1: consits of one search using Algorithm 1.
Method 2: Algorithm 2 with iterations using
randomly generated weight vectors. In our experiments
varied from 3 to 19 in steps of 2.
Method 3: Algorithm 2 with
iterations, one for each
constraint. The iteration for the constraint
, is per-
formed with the weightvector
, where
and .
Method 4: Algorithm 2 with
iterations, one for
each cuple of constraints. The iteration for the constraints
and , is performed with the weight vector
, where , and
.
Method 5: Algorithm 2 with
iterations, one
for each constraint and one for each cuple of constraints.
This method is a mixed method between method 3 and 4.
It takes the vectors from both methods.
Method 6: Using the Lexicographic Fuzzy CSP
(Dubois, Fargier, & Prade 1996) approach for obtaining
the
solutions.
The advantage of working with Method 1 is that it uses
standard well-known algorithms. The disadvantage is that
it tends to find solutions which are in the same area of the
solution space. To increase the diversity of the resulting set
of Pareto-optimal solutions, we propose to perform several
iterations with random weight vectors (Method 2). The intu-
ition behind method 3 is to remove the effect of one criteria
at each iteration in order to avoid strong dominance of some
of the constraints. The same idea is behind methods 4 and 5.
The motivation for method 6 is to compare how well other
instantiations of the semiring CSP framework apply to this
problem. Lexicographic Fuzzy CSPs are the most power-
ful version of Fuzzy CSP methods which are interesting be-
cause they admit more efficient search algorithms (Bistarelli
et al. 1999).
Random Soft Constraint Satisfaction Problems
Generation
The topology of a random soft CSP is defined by:
, where is the numberof vari-
ables in the problem and
the size of their domains. is
the graph density in percentage for unary and binary hard
constraints.
is the tightness in percentage for disallowed
tuples in unary and binary hard constraints.
and are
the graph density and tightness in percentage for unary and
binary soft constraints. Valuations for soft constraints can
take values from 0 to
. For simplicity, hard and soft
constraints are separated and we are not considering mixed
constraints, therefore
. For building random
instances of soft CSPs, wechoose the variablesforeach con-
straint following a uniform probabilistic distribution. In the
same way, we choose the tuples in constraints. Valuations
for soft tuples are randomly generated between
and
and valuationsfor hardtuples are represented by a maximum
valuation (
).
The algorithmshavebeen tested with differentset of prob-
lems of soft CSPs with 5 or 10 variables and 10 values for
each variable. Hard unary/binary constraint density
has
been varied from 20% to 80% in steps of 20, and the tight-
ness for hard constraints
varies also from 20% to 80% in
steps of 20. Soft unary/binary constraint density
has been
varied from 20% to 80% in steps of 10, with tightness fixed
at
. In the case of 5 variables, in total there could
be
soft constraints (5 unary constraints
and 10 binary constraints). In the case of 10 variables, in
total there could be
soft constraints
(10 unary constraints and 45 binary soft constraints).
For every different class of problems, 50 different in-
stances were generated, and each instance has been tested
with the different proposed methods. The methods have
been tested varying the number of total solutions to be com-
puted from 30 to 530 in steps of 50, from 530 to
in
steps of 100, from
to in steps of and
from
to in steps of .
For the differentproblem topologies the average of the re-
sults for each instance are evaluated in the followingsection.
Results
Firstly, we are interested in knowing how many Pareto-
optimal solutions there are in a problem depending on the
0
1000
2000
3000
4000
5000
6000
7000
8000
3 4 5 6 7 8 9 10 11 12
number of pareto optimal solutions
number of criteria (soft constraints)
Random problem with n=5, d=10, hc=20%
hard tightness = 20%
hard tightness = 40%
hard tightness = 60%
hard tightness = 80%
Figure 2: Number of Pareto-optimal solutions depending on
howmanysoft constraintswe considerfor random generated
problems with 5 variables, 10 values per domain and 20%
of hard unary/binary constraint density. The number of of
solutions in average for the generated problems are:
for hard tightness = 20%, for hard tightness = 40%,
for hard tightness = 60% and 778 for hard tightness =
80%.
number of criteria (soft constraints). In Figure 2, it is shown
that the number of Pareto-optimalsolutions clearly increases
when the number of criteria increases. The same phenom-
ena applies for instances with 5 and 10 variables. On the
other hand, we have observed that even if the number of
Pareto-optimal solutions decreases when the problem gets
more constrained (less feasible solutions) the percentage in
respect to the total number of solutions increases. Thus, the
proportion of the Pareto-optimal solutions is more important
when the problem gets more constrained.
We have evaluated the proposed methods for each type
of generated problems. Figure 3 shows the proportion in
avarage of Pareto-optimal solutions found by the different
methods for problems with 6 soft constrains. We emphasize
the results of the methods up to 530 solutions because in
real applications it could not be feasible to compute a larger
set of solutions. When computing up to
solutions,
the behavior of the different methods does not change sig-
nificantly. The 50 randomly generated problems used for
Figure 3 had in avarage
feasible solutions (satisfy-
ing hard constraints) and
Pareto-optimal solutions. The
iterative methods perform better than the single search al-
gorithm (Method 1) in respect to the total number of solu-
tions computed. It is worth to note that the iterative meth-
ods based on Algorithm 2 find more Pareto-optimal solu-
tions when the number of iterations increase. Lexicographic
Fuzzy method (Method 6) results in finding a very low per-
centage of Pareto-optimal solutions (less than
). With
Method 6, Theorem 1 does not apply, thus the percentage
shown of Pareto-optimal solutions is computed a posteriori
by filtering out the Pareto-optimal solutions that were not
really Pareto-optimal for the entire problem.
Another way of comparing the different methods is to
0
5
10
15
20
25
30
35
30 130 230 330 430 530
% of Pareto optimal solutions found
Total number of computed solutions
Evaluation of different methods
1 iteration
3 iterations (2 random)
5 iterations (4 random)
7 iterations (6 random)
11 iterations (10 random)
15 iterations (14 random)
19 iterations (18 random)
6 iterations, 1 criteria left out
16 iterations, 2 criteria left out
22 iterations, mixed method
Lexicographic Fuzzy
Figure 3: Pareto-optimal solutions found by the different
proposed methods (in %). Methods are applied to 50 ran-
domly generated problems with 10 variables, 10 values per
domain, 40% of density of hard unary/binary constraints
with 40% of hard tightness and 6 criteria (soft constraints).
The number of total computed solutions for each method
varies from 30 to 530 in steps of 100.
0
10
20
30
40
50
60
70
80
90
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
% of Pareto optimal solutions found
Time in seconds
Evaluation of different methods
1 iteration
3 iterations (2 random)
5 iterations (4 random)
7 iterations (6 random)
11 iterations (10 random)
15 iterations (14 random)
19 iterations (18 random)
6 iterations, 1 criteria left out
16 iterations, 2 criteria left out
22 iterations, mixed method
Figure 4: Number of Pareto-optimal solutions found by the
different proposed methods with respect to the computing
time. For this plot, the problemshave 10 variables, 10 values
per domain with 40% of hard unary/binary constraints with
40% of hard tightness and 6 criteria (soft constraints).
compare the number of Pareto-optimal solutions found with
respect to the computing time (Figure 4). Using this com-
parison, Method 1 performs the best. The performance of
the variants of Method 2 decreases when the number of it-
erations increases. Method 3 performs better than method 4
which performs better than method 5 in terms of computing
time.
In general, we can observe that when the number of it-
erations of the methods increases the performance regarding
the total numberof computedsolutionsalso increases butthe
performance regarding the computing time decreases. This
is due to the fact that the computingtime of findingthe
best
solutions with PFC is not linear with respect of finding the
best solutions with iterations ( solutions per itera-
tion). For example, computing
solutions with one it-
eration takes
seconds and computing solutions
with 7 iterations (of
solutions) takes seconds.
Even if the tests based on Algorithm 2 takes more time
than Algorithm 1 for getting the same percentage of Pareto-
optimal solutions, they are likely to produce a more repre-
sentative set of the Pareto-optimal set.
Using a brute force algorithm that computes all the feasi-
ble solutionsand filter out those which are dominatedtook in
avarage
seconds for the same problemsas in the above
figures. This demonstrates the interest of using approxima-
tive methods for computing Pareto-optimal solutions, espe-
cially for interactive configuration applications (e.g. elec-
tronic catalogs).
Empirical Results in a Real Application
The Air Travel Configurator
The problem of arranging trips is here modeled as a soft
CSP (see (Torrens, Faltings, & Pu 2002) for a detailed de-
scription of our travel configurator). An itinerary is a set of
legs, where each leg is represented by a set of origins, a set
of destinations, and a set of possible dates. Leg variables
represent the legs of the itinerary and their domains are the
possible flights for the associated leg. Another variable rep-
resents the set of possible fares
3
applicable to the itinerary.
The same itinerary can have several different fares depend-
ing on the cabin class, airline, schedule and so on. Usu-
ally, for each leg there can be about 60 flights, and for each
itinerary, there can be about 40 fares. Therefore, the size of
the search space for a round trip is
and
for a three leg trip is
. Constraint sat-
isfaction techniques are well-suited for modeling the travel
planning problem. In our model, two types of configuration
constraints (hard constraints) guarantee that:
1. a flight
for a leg arrives before a flight for a leg
takes off, and
2. a fare can be really applicable to a set of flights (depend-
ing on the fare rules).
Users normally have preferences about the itinerary they
are willing to plan. They can have preferences about the
schedule of the itinerary, the airlines, the class of service,
3
In travel industry, the fare applicable to an itinerary is not the
sum over the fares for each flight.
0
10
20
30
40
50
60
70
80
90
100
0 10 20 30 40 50 60 70
% of 50 Pareto-optimal Solutions
Number of Solutions
Pareto-optimal Solutions for the travel configurator
50 Pareto-optimal Solutions
Figure 5: How many solutions we need for getting a cer-
tain number of Pareto-optimal solutions ? This example is
based on a round trip and showsthat, for instance, 50Pareto-
optimal solutions can be found out of about less than 70 so-
lutions.
and so on. Such preferences are modeled as soft constraints.
Thus, this problem can be naturally modeled as a MCOP
with hard constraints for ensuring the feasibility of the so-
lution and soft constraints for taking into account the users
preferences.
Tests on the Travel Configurator Application
Method 1 has been tested with our travel configurator. We
have generated 68 instances of itineraries: 58 round trips, 5
3-leg trips, 1 6-leg trip, 3 5-leg trips and 1 7-leg trip. These
instances were tested with 5 unary soft constraints simulat-
ing user’s preferences. For this application, the goal is to
find a set of Pareto-optimal solutions to be shown to the
user. Thus, the problem is not to find all Pareto-optimal so-
lutions but a relatively small set of Pareto-optimal solutions.
In order to achieve this, we have applied branch and bound
algorithm with propagation (PFC) to discover how many
solutions we need to obtain a certain number of Pareto-
optimal solutions. Precisely, we study how many solutions
are needed to find 50 Pareto-optimal solutions.
Evaluation on the Travel Configurator
Figure 5 shows the test results for a round trip (3 variables,
with domain sizes 40, 60 and 60) with 5 unary soft con-
straints (expressing users’ preferences). We observe that
for getting a certain number of Pareto-optimal solutions in
this kind of problems, the number of solutions to compute
is very reasonable. Indeed, the method is shown very us-
able for interactive configuration applications, and specifi-
cally for electronic catalogs.
The plot shown in Figure 5 has been generated for all 68
instances of the problems previously described. For all the
examples we get similar results. By computing 90 solutions
to these problems, we always get 50 Pareto-optimal solu-
tions for all the examples tried.
In electronic catalogs and similar applications, it is useful
to find a certain number of Pareto-optimal solutions, even
if this set only represents a small fraction of all the Pareto-
optimal solutions. Actually, we consider that the number of
total solutions that can be shown to the user must be small
because of the limitations of the current graphical user inter-
faces.
Related Work
The most commonly used approach for solving a Multi-
criteria Optimization Problem is to convert a MCOP into
several COPs which can be solved using standard mono-
criteria optimization techniques. Each COP will give
then a Pareto-optimal solution to the problem. Steuer’s
book (Steuer 1986) gives a deep study on different ways
to translate a MCOP to a set of COPs. The most used
strategy is to optimize by one linear function of all crite-
ria with positive weights. The drawback of the method is
that some Pareto-optimal solutions cannot be found if the
efficient frontier is not concave
4
. Our methods are based on
this approach.
Gavanelli (Gavanelli 2002; 2001) addresses the problem
of multi-critera optimization in constraint problems directly.
His method is based in a branch and bound schema where
the Paerto dominance is checked against a set of previously
found solutions using Point Quad-Trees. Point Quad-Trees
are useful for efficiently bounding the search. However, the
algorithm can be very costly if the number of criteria or if
the numberof Pareto-optimal solutions are high. Gavanelli’s
method significantly improvesthe approach of Wassenhove-
Geders (Wassenhove & Gelders 1980). The Wassenhove-
Geder’s method basically consists of performing several
search processes, one for each criteria. Each iteration takes
the previous solution and tries to improve it by optimizing
another criteria. Using this method, each search produces
one Pareto-optimal solution, so a lot of search process must
be done in order to approximate the Pareto-optimal set.
The Global Criterion Method tries to solve a MCOP as a
COP where the criteria to optimize is a minimization func-
tion of a distance function to an ideal solution. The ideal
solution is precomputed by optimizing each criteria inde-
pendently (Salukvadze 1974).
Incomplete methods have also been developed for
solving multi-criteria optimization, basically: genetic
algorithms (Deb 1999) and methods based on tabu
search (Hansen 1997).
Conclusions
This paper deals with a very well-studied topic, Pareto-
optimality in multi-criteria optimization. It has been com-
monly understood that Pareto-optimality is intractable to
compute, and therefore has not been studied further. In-
stead, many applications have simply mapped multi-criteria
search into a single criterion with a particular weighting and
returned a list of the
best solutions rather than a single
best one. This solution allows leveraging the well-developed
4
in the case that the optimization function is a minimization
function, convex if the optimization function is a maximization
function.
framework of soft CSPs to Multi-criteriaOptimization Prob-
lems.
Our contribution is to have shown empirically that this
procedure, if combined with a filtering that eliminates dom-
inated solutions from the results of the optimization proce-
dure, results indeed a surprisingly good approximations of
the Pareto-optimal set. Based on this observation, we have
shown that the performancecan be improved at a small price
in cost by running the same procedure with different random
vectors.
We have implemented this method with great success in
a commercial travel planning tool, and believe that it would
apply well to many other applications as well.
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