A LOCAL SEARCH APPROACH

TO SOLVE INCOMPLETE FUZZY CSPs

Mirco Gelain, Maria Silvia Pini, Francesca Rossi, Kristen Brent Venable

Department of Pure and Applied Mathematics, University of Padova, 35121 Padova, Italy

Toby Walsh

NICTA and UNSW, Sydney, Australia

Keywords:

Local search, Preferences, Incompleteness, Elicitation.

Abstract:

We consider fuzzy constraint problems where some of the preferences may be unspeciﬁed. This models, for

example, settings where agents are distributed and have privacy issues, or where there is an ongoing preference

elicitation process. In this context, we study how to ﬁnd an optimal solution without having to wait for all the

preferences. In particular, we deﬁne local search algorithms that interleave search and preference elicitation,

with the goal to ﬁnd a solution which is ”necessarily optimal”, that is, optimal no matter what the missing

data are, while asking the user to reveal as few preferences as possible. While in the past this problem has

been tackled with a branch & bound approach, which was guaranteed to ﬁnd a solution with this property, we

now want to see whether a local search approach can solve such problems optimally, or obtain a good quality

solution, with fewer resources. At each step, our local search algorithm moves from the current solution to a

new one, which differs in the value of a variable. The variable to reassign and its new value are chosen so to

maximize the quality of the next solution. To compute this, we elicit some of the missing preferences in the

neighbor solutions. Experimental results on randomly generated fuzzy CSPs with missing preferences show

that our local search approach is promising, both in terms of percentage of elicited preferences and scaling

properties.

1 INTRODUCTION

Constraint programming (Rossi et al., 2006) is a pow-

erful paradigm for solving scheduling, planning, and

resource allocation problems. A problem is repre-

sented by a set of variables, each with a domain of val-

ues, and a set of constraints. A solution is an assign-

ment of values to the variables which satisﬁes all con-

straints and which optionally maximizes/minimizes

an objective function. Soft constraints (Bistarelli

et al., 1997) are a way to model optimization prob-

lems by allowing for several levels of satisﬁability,

modeled by the use of preference or cost values that

represent how much we like a certain way to instan-

tiate the variables of a constraint. Incomplete fuzzy

constraint problems (IFCSPs) (Gelain et al., 2010a;

Gelain et al., 2007; Gelain et al., 2008) can model sit-

uations in which some preferences are missing before

solving starts. In this scenario, we aim at ﬁnding a

solution which is necessarily optimal, that is, optimal

no matter what the missing preferences are, while ask-

ing the user to reveal as few preferences as possible.

While in the past this problem has been tackled

with a branch & bound approach, which was guar-

anteed to ﬁnd a solution with this property, we now

show that a local search method can solve such prob-

lems optimally, or obtain a good quality solution, with

fewer resources. Our local search algorithm starts

from a randomly chosen assignment to all the vari-

ables and, at each step, moves to the most promising

assignment in its neighborhood,obtained by changing

the value of one variable. The variable to reassign and

its new value are chosen so to maximize the quality of

the new assignment. To do this, we elicit some of the

preferences missing in the neighbor assignments. The

algorithm stops when a given limit on the number of

steps is reached. Experimental results on randomly

generated fuzzy CSPs with missing preferences show

that our local search approach is promising, in terms

of percentage of elicited preferences, solution quality,

582

Gelain M., Pini M., Rossi F., Brent Venable K. and Walsh T..

A LOCAL SEARCH APPROACH TO SOLVE INCOMPLETE FUZZY CSPs.

DOI: 10.5220/0003174505820585

In Proceedings of the 3rd International Conference on Agents and Artiﬁcial Intelligence (ICAART-2011), pages 582-585

ISBN: 978-989-8425-40-9

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

and scaling properties.

Incompleteness in soft constraints has also been

considered in (Fargier et al., 1996; Pini et al., 2010).

However, incomplete knowledge in these contexts is

represented by the presence of uncontrollable vari-

ables, i.e., variables such that the agent has not the

power to ﬁx their value, and not by the presence of

missing preferences.

A preliminary version of some results of the paper

is contained in (Gelain et al., 2010b)

2 BACKGROUND

We now give some basic notions about incomplete

fuzzy constraint problems and local search.

Incomplete Fuzzy Problems. Incomplete Fuzzy

constraint problems (IFCSPs) (Gelain et al.,

2010a) extend Fuzzy Constraint Problems (FC-

SPs) (Bistarelli et al., 1997) to deal with partial

information. In FCSPs preference values are between

0 and 1, the preference of a solution is the minimum

preference value contributed by the constraints, and

the optimal solutions are those with the highest value.

More formally, given a set of variables V with ﬁ-

nite domain D, an incomplete fuzzy constraint is a pair

hide f, coni where con ⊆ V is the scope of the con-

straint and idef : D

|con|

−→ ([0, 1] ∪ {?}) is the pref-

erence function of the constraint associating to each

tuple of assignments to the variables in con either a

preference value that belongs to the set [0, 1], or ?. All

tuples mapped into ? by idef are called incomplete tu-

ples, meaning that their preference is unspeciﬁed.

Given an assignment s to all the variables of

an IFCSP P, pref(P, s) is the preference of s in

P. More precisely, it is deﬁned as pref(P, s) =

min

<idef ,con>∈C|idef (s

↓con

)6=?

ide f(s

↓con

). In words, the

preference of an assignment s in an IFCSP is obtained

by taking the minimum of the known preferences as-

sociated to the projections of the assignment, that is,

of the appropriated sub-tuples in the constraints.

In the fuzzy context, a complete assignment of

values to all the variables is an optimal solution if its

preference is maximal. This optimality notion is gen-

eralized to IFCSPs via the notion of necessarily op-

timal solutions, that is, complete assignments which

are maximal no matter the value of the unknown pref-

erences.

Research partially supported by the Italian MIUR

PRIN project 20089M932N: “Innovative and multidisci-

plinary approaches for constraint and preference reason-

ing”.

In (Gelain et al., 2010a) several algorithms are

proposed to ﬁnd a necessarily optimal solution of

an IFCSP. All these algorithms follow a branch and

bound schema where search is interleaved with elici-

tation. Elicitation is needed since the given problem

may have an empty set of necessarily optimal solu-

tions. By eliciting more preferences, this set eventu-

ally becomes non-empty. Several elicitation strategies

are considered in (Gelain et al., 2010a) in the attempt

to elicit as little as possible before ﬁnding a necessar-

ily optimal solution.

Local Search. Local search (Hoos and Stutzle,

2004) is one of the fundamental paradigms for solving

computationally hard combinatorial problems. Given

a problem instance, the basic idea underlying local

search is to start from an initial search position in

the space of all possible assignments (typically a ran-

domly or heuristically generated assignment, which

may be infeasible, sub-optimal or incomplete), and to

improve iteratively this assignment by means of mi-

nor modiﬁcations. At each search step we move to a

new assignment selected from a local neighborhood,

chosen via a heuristic evaluation function. This pro-

cess is iterated until a termination criterion is satis-

ﬁed. The termination criterion is usually the fact that

a solution is found or that a predetermined number

of steps is reached. To ensure that the search process

does not stagnate, most local search methods make

use of random moves: at every step, with a certain

probability a random move is performed rather than

the usual move to the best neighbor. Another way to

prevent local search from locking in a local minima

is Tabu search (Glover and Laguna, 1997) that uses

a short term memory to prevent the search from re-

turning to recently visited assignments for a speciﬁed

amount of steps.

A local search approach has been deﬁned in

(Aglanda et al., 2004) for soft constraints. Here we

adapt it to deal with incompleteness.

3 LOCAL SEARCH ON IFCSPs

We will now present our local search algorithm for

IFCSPs that interleaves elicitation with search. We

basically follow the same algorithm as in (Codognet

and Diaz, 2001), except for the following.

To start, we randomly generate an assignment of

all the variables. To assess the quality of such an as-

signment, we compute its preference. However, since

some missing preferences may be involvedin the cho-

sen assignment, we ask the user to reveal them.

A LOCAL SEARCH APPROACH TO SOLVE INCOMPLETE FUZZY CSPs

583

In each step, when a variable is chosen, its local

preference is computed by setting all the missing pref-

erences to the preference value 1. To choose the new

value for the selected variable, we compute the pref-

erences of the assignments obtained by choosing the

other values for this variable. Since some preference

values may be missing, in computing the preference

of a new assignment we just consider the preferences

which are known at the current point. We then choose

the value which is associated to the best new assign-

ment. If two values are associated to assignmentswith

the same preference, we choose the one associated to

the assignment with the smaller number of incomplete

tuples. In this way, we aim at moving to a new assign-

ment which is better than the current one and has the

fewest missing preferences.

Since the new assignment, say s

′

, could have in-

complete tuples, we ask the user to reveal enough of

this data to compute the actual preference of s

′

. We

call ALL the elicitation strategy that elicits all the

missing preferences associated to the tuples obtained

projecting s

′

on the constraints, and we call WORST

the elicitation strategy that asks the user to reveal only

the worst preference among the missing ones, if it is

less than the worst known preference. This is enough

to compute the actual preference of s

′

since the prefer-

ence of an assignment coincides with the worst pref-

erence in its constraints.

As in many classical local search algorithms, to

avoid stagnation in local minima, we employ tabu

search and random moves. Our algorithm has two

parameters: p, which is the probability of a random

move, and t, which is the tabu tenure. When we have

to choose a variable to re-assign, the variable is either

randomly chosen with probability p or, with proba-

bility (1-p) and we perform the procedure described

above. Also, if no improving moveis possible, i.e., all

new assignments in the neighborhood are worse than

or equal to the current one, then the chosen variable is

marked as tabu and not used for t steps.

During search, the algorithm maintains the best

solution found so far, which is returned when the

maximum number of allowed steps is exceeded. We

will show later that, even when the returned solution

is a necessarily optimal solution, its quality is not very

far from that of the necessarily optimal solutions.

4 EXPERIMENTAL RESULTS

The test sets for IFCSPs are created using a gener-

ator that has the following parameters: n: number

of variables; m: cardinality of the variable domains;

d: density, i.e., the percentage of binary constraints

present in the problem w.r.t. the total number of pos-

sible binary constraints that can be deﬁned on n vari-

ables; t: tightness, i.e., the percentage of tuples with

preference 0 in each constraint and in each domain

w.r.t. the total number of tuples; i: incompleteness,

i.e., the percentage of incomplete tuples (i.e., tuples

with preference ?) in each constraint and in each do-

main. Our experiments measure the percentage of

elicited preferences (overall the missing preferences),

the solution quality (as the normalized distance from

the quality of necessarily optimal solutions), and the

execution time, as the generation parameters vary.

We ﬁrst considered the quality of the returned so-

lution. To do this, we computed the distance be-

tween the preference of the returned solution and that

of the necessarily optimal solution returned by algo-

rithm FBB (which stands for fuzzy branch and bound)

which is one of the best algorithms in (Gelain et al.,

2010a). In (Gelain et al., 2010a), this algorithm cor-

responds to the one called DPI.WORST.BRANCH.

Such a distance is measured as the percentage over

the whole range of preference values. For example, if

the preference of the solution returned is 0.4 and the

one of the solution given by FBB is 0.5, the prefer-

ence error reported is 10%. A higher error denotes a

lower solution quality.

Figures 1(a) and 1(b) show the preference er-

ror when the number of variables and tightness vary

(please notice that the y-axis ranges from 0% to 10%).

We can see that the error is always very small and its

maximum value is 3.5% when we consider problems

with 20 variables. In most of the other cases, it is be-

low 1.5%. We also can notice that the solution quality

is practically the same for both elicitation strategies.

If we look at the percentage of elicited prefer-

ences (Figures 1(c) and 1(d)), we can see that the

WORST strategy elicits always less preferences than

ALL, eliciting only 20% of incomplete preferences in

most of the cases. The FBB algorithm elicits about

half as many preferences as WORST. Thus, with 10

variables, FBB is better than our local search ap-

proach, since it guarantees to ﬁnd a necessarily opti-

mal solution while eliciting a smaller number of pref-

erences.

We also tested the WORST strategies varying the

number of variables from 10 to 100. In Figure 1(f) we

show how the elicitation varies up to 100 variables. It

is easy to notice that with more than 70 variables the

percentage of elicited preferences decreases. This is

because the probabilityof a complete assignmentwith

a 0 preference arises (since density remains the same).

Moreover, we can see how the local search algorithms

can scale better than the branch and bound approach.

In Figure 1(e) the FBB reaches a time limit of 10 min-

ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence

584

5 8 11 14 17 20

preference error

number of variables

ALL

WORST

(a) Solution quality varying n.

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80

preference error

tightness

ALL

WORST

(b) Solution quality varying t.

100

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80

percentage of elicited preferences

density

ALL

WORST

FBB

100

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

percentage of elicited preferences

incompleteness

ALL

WORST

FBB

(d) Elicited preferences varying i.

100000

200000

300000

400000

500000

600000

10 20 30 40 50 60 70 80 90 100

execution time (msec.)

number of variables

WORST

FBB

(e) Execution time varying n.

100

10 20 30 40 50 60 70 80 90 100

percentage of elicited preferences

number of variables

WORST

(f) Elicited preferences varying n.

Figure 1: Step limit 100000, tabu steps 1000,random walk probability 0.2, average on 100 problems, default values: d=50%

(for Figures (a), (b), and (d)), d=35% (for Figures (e) and (f)), i=30%, t=10%, n=10, m = 5.

utes with just 25 variables, while the WORST algo-

rithm needs the same time to solve instances of size

100.

5 CONCLUSIONS

We developed and tested a local search algorithm

to solve incomplete fuzzy CSPs. We tested differ-

ent elicitation strategies and our best strategies have

shown good results compared with the branch and

bound solver described in (Gelain et al., 2010a). More

precisely, our local search approach shows a very

good solution quality when compared with complete

algorithms. In addition it shows better scaling prop-

erties than such a complete method.

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