Integrating TCP-Nets and CSPs:The Constrained TCP-Net (CTCP-Net) Model

Shu Zhang, Malek Mouhoub(B), and Samira Sadaoui

Department of Computer Science, University of Regina, Regina, Canada{mouhoubm,sadaouis}@uregina.ca

Abstract. In this paper, a new framework for constraint and preferencerepresentation and reasoning is proposed, including the related definitions,algorithms and implementations. A Conditional Preference Network (CP-Net) is a widely used graphical model for expressing the preferences amongvarious outcomes. While it allows users to describe their preferences overvariables values, the CP-Net does not express the preferences over the vari-ables themselves, thus making the orders of outcomes incomplete. Due tothis limitation, an extension of CP-Nets called Tradeoffs-enhanced Con-ditional Preference Networks (TCP-Nets) has been proposed to representthe relative importance between variables. Nonetheless, there is no rese-arch work reporting on the implementation of TCP-Nets as a solver. More-over, the TCP-Net only deals with preferences (soft constraints). Hardconstraints are not explicitly considered. This is a real limitation whendealing with a wide variety of real life problems including both constraintsand preferences. This has motivated us to propose a new model integratingTCP-Nets with the well known Constraint Satisfaction Problem (CSP)framework for constraint processing. The new model, called ConstrainedTCP-Net (CTCP-Net), has been implemented as a three-layer architec-ture system using Java and provides a GUI for users to freely describe theirproblem as a set of constraints and preferences. The system will then solvethe problem and returns the solutions in a reasonable time. Finally, thiswork provides precious information for other researchers who are inter-ested in CSPs and graphical models for preferences from the theoreticaland practical aspects.

Keywords: Constraint Satisfaction ·Qualitative preferences ·CP-Nets ·TCP-Nets

1 Introduction

Preference elicitation, representation and reasoning plays an important role inmany real life applications, such as collaborative filtering, product configuration,automated decision making system and recommender systems. In most cases,we cannot require the users to have enough patience and knowledge to make adecision or a choice. Thus, helping the users to make a decision efficiently andcorrectly is important as discussed by the researchers in the field.c© Springer International Publishing Switzerland 2015M. Ali et al. (Eds.): IEA/AIE 2015, LNAI 9101, pp. 201–211, 2015.DOI: 10.1007/978-3-319-19066-2 20

202 S. Zhang et al.

While past research work has focused on the quantitative representation ofpreferences through a utility function based on the well-known Multi-AttributeUtility Theory(MAUT) for example [1] or the C-semiring framework for pref-erences in temporal problems [2], it is more natural to describe preferencesin a qualitative way. Conditional Preference Networks(CP-Nets)[3] is a qual-itative graphical model for representing qualitative preference information toreflect the conditional dependency of preferences under ceteris paribus (allelse being equal) interpretation. It provides a convenient and intuitive toolfor specifying the problem, and in particular, the decision maker’s preferences.Tradeoffs-enhanced Conditional Preference Network(TCP-nets)[4] is introducedby extending CP-nets, allowing users to describe their relative importance onvariables, thus improving the limitations of CP-Nets. However, research on aTCP-Net is very scarce as this latter is a very new model initially proposed in2006[4]. Furthermore, to our best knowledge there is no implementation of TCP-Nets. In other words, we do not know the performance of TCP-Nets for real woldapplications. Moreover, hard constraints are not considered. In this paper, weextended the TCP-Net with constraints, producing a more powerful and novelmodel called Constrained TCP-Net (CTCP-Net) to address problems under con-straints and preferences. The CTCP-Net allows users to add their unconditionaland conditional constraints into a Condition-Constraint Table (CCT). The Con-straint Satisfaction Problem (CSP) framework [5] has been used to filter thevariables values which do not satisfy the CCT. This filter greatly reduces thesearch space, thus enhancing the efficiency of the solving algorithm. Our pro-posed model comes with a friendly GUI and can constitute the core of an onlineshopping system as long as it connects to the corresponding online shoppingdatabases, for electronic selections, air ticket selections and more.

The rest of the paper is organized as follows. Section 2 reviews the relatedresearch work on constraints and preferences representation and reasoning. Basicconcepts, CSPs, CP-Nets and TCP-Nets are discussed in details in this section.In Section 3, our proposed CTCP-Net model is defined. The solving algorithmis then presented in section 4. Finally, section 5 concludes this research worksand prospects possible future works.

2 Background

2.1 CSPs

A Constraint Network (CN) includes a finite set of variables with finite domains,and a finite set of constraints restricting the possible combinations of variablevalues [5]. Given a CN, a Constraint Satisfaction Problem (CSP) consists offinding a set of assigned values to variables that satisfy all the constraints. ACSP is known to be an NP-hard problem in general1, and is solved with abacktrack search algorithm of exponential time cost. In order to reduce this

1 There are special cases where CSPs are solved in polynomial time, for instance, thecase where a CSP network is a tree [6,7].

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cost in practice, constraint propagation techniques have been proposed [5,6,8].The idea here is to reduce the size of the search space before and during thebacktrack search. In the past four decades the CSP framework, with its solvingtechniques, has demonstrated its ability to efficiently model and solve a large sizereal-life applications, such as scheduling and planning problems, configuration,bioinformatics, vehicle routing and scene analysis [9].

2.2 CP-Nets and TCP-Nets

A Conditional Preference Network (CP-Net) is a graph model for representingand reasoning on conditional ceteris paribus preferences in a compact, intuitiveand structural manner. This model allows users to express their preferences ina qualitative way, which is more natural and comfortable for users comparedto quantitative descriptions. Non-conditional preferential independence is usedto represent the fact that customers’ preference relation over values of a givenfeature is the same regardless of the values given to other features. The definition[4] is shown as follows.

Definition 1. [4] Let x1, x2 ∈ D(X) for some X ⊆ V and y1, y2 ∈ D(Y ),where Y = V − X. We say that X is preferentially independent of Y iff, for allx1, x2, y1, y2 we have that x1y1 � x2y1 ⇔ x1y2 � x2y2

In reality, the domains of features and customers’ preference are much morecomplex. In most cases, the preferential independence relies on a certain valueof other features, hence we call it conditionally preferentially independent andthe corresponding definition [4] is as follows.

Definition 2. [4] Let X,Y and Z be a partition of V and let z ∈ D(Z). X isconditionally preferentially independent of Y given z iff, for all x1, x2, y1, y2 wehave that x1y1z � x2y1z ⇐⇒ x1y2z � x2y2z

Moreover, we can say that X is conditionally preferentially independent ofY under Z if the above formula is satisfied for every value of Z.

In order to address the limitations of CP-Nets, tradeoffs-enhanced CP-nets(TCP-Nets) have been proposed by extending the relative preferences to thevariables themselves through the non-conditional and conditional relative impor-tance properties [4] defined below.

Definition 3. [4] Let a pair of variables X and Y be mutually preferentiallyindependent given W = V − X,Y . We say that X is more important than Y ,denoted by X � Y , if for every assignment w ∈ D(W ) and for every xi, xj ∈D(X), ya, yb ∈ D(Y ), such that xi � xj given w, we have that: xiyaw � xjybw.

The above definition also works if yb � ya given w.In general, customers describe their preference under certain conditions,

hence the conditional relative importance is more commonly used.

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Definition 4. [4] Let X and Y be a pair of variables from V , and let Z ⊆W = V − X,Y . We say that X is more important than Y given z ∈ D(Z) iff,for every assignment w′ on W ′ = V −({X,Y }⋃

Z) we have: xiyazw′ � xjybzw′

whenever xi � xj given zw′. We denote this relation by X �Z Y . Finally, if forsome z ∈ D(Z) we have either X �Z Y , or Y �Z X, then we say that the relativeimportance of X and Y is conditioned on Z, and write RI(X,Y |Z).

Accordingly, if for some z ∈ Z, we can also find Y �Z X, then we can concludethat the relative importance of X and Y relies on Z, denoted as RI(X,Y | Z).

3 Constrained TCP-Nets (CTCP-Nets)

3.1 Definition of the CTCP-Net

Following the definition of the TCP-Nets in [4] we define the CTCP-Net T as atuple < G, cc, cp, i, ci, cct, cpt, cit >, where:

1 G is the set of nodes corresponding to a set of problem variables{X1,X2, . . . , Xn}.

2 cc is the set of directed cc − arcs{α1, α2, . . . , αn} corresponding to condi-tional constraints. A cc − arc[

−−−−→Xi,Xj ] means that Xj is restricted by a given

condition on Xi’s values.3 cp is the set of directed cp−arcs{β1, β2, . . . , βn} corresponding to conditional

preferences. A cp−arc <−−−−→Xi,Xj > means that the preferences over the values

of Xj depend on the actual value of Xi.4 i is the set of directed i-arcs {γ1, γ2, . . . , γn} corresponding to non-conditional

relative importance relations. An i-arc (−−−−→Xi,Xj) corresponds to the following

relative importance: Xi � Xj .5 ci is the set of undirected ci−arc{λ1, λ2, . . . , λ3}, where ci stands for condi-

tional relative importance. A ci − arc( ̂Xi,Xj) is in Tiff there isRI(Xi,Xj |Z) for some Z ⊆ G − {Xi,Xj}.

6 cct associates with conditional constraint table with every node X ∈ G. Acct is from D(Cd(X)) (ie..assignments to X’s conditional nodes) to constraintvalue over D(X). Where Cd(X) is X ′s corresponding conditional variable.

7 cpt associates a Conditional Preference Table (CPT) with every nodeX ∈ G. CPT (X) is a mapping from D(Pa(X)) (ie., assignments to X’sparents nodes) to strict partial order over D(X). Pa(X) is X ′s conditional(dependent) variable.

8 cit associates with every ci−arc γ = ( ̂Xi,Xj), a (possibly partial) mappingCIT (γ) from D(S(Xi,Xj)) to orders over the set {Xi,Xj}.

Note that the sub tuple < cp, i, ci, cpt, cit > corresponds to a TCP-Net.Moreover, if the sets i and ci are empty then we have a CP-Net.

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3.2 Constraint Propagation for CTCP-Nets

We define two types of constraints: non-conditional and conditional and we prop-agate them respectively using our node and arc consistency algorithms in thepreprocessing stage as well as the backtrack search of our solving algorithm.

A non-conditional constraint is a constraint where the condition clause isnull. We define it as follows: NULL =⇒ rel. rel denotes a relationship betweenan attribute value and a constant and is defined as X rel a, where X is anattribute and a is a constant. There are 5 relationships in our model,that isrel ∈ {=, =,≺,�,�,�}. For this type of constraints, we remove the unsatisfiedvalues according to each constraint. After enforcing node consistency, all theunsatisfied values will be removed from the corresponding domains. In otherwords, the remaining values satisfy all the non-conditional constraints.

A conditional constraint cci is defined as follows.

cci = (rel1) and|or (rel2) and|or . . . (relm) =⇒ rel (1)

We denote by varsCd(cci) the set of variables in the condition of cci and byvarCc(cci) the variable present in the conclusion of cci.

Here rel is dependent on the condition (rel1) and \ or (rel2)and \ or , ..., and \ or (relm). If this latter condition is true then rel takeseffect. eg. “I do not want to buy a black laptop if the brand is Sony”. The“brand is Sony” is the condition here while no black laptop is the constraint,which can be noted as (Brand = Sony) =⇒ Color = black. Note that m = 0corresponds to a non conditional constraint.

The following is the general procedure we use to manage non conditional andconditional constraints.

– m=0. This is a non-conditional constraint. We process it as a unary con-straint in the pre-processing step to restrict the values that each attributecan take. This is done by removing any attribute value that is inconsistentwith the unary constraint.

– m=1. This is conditional constraint involving one relation in the premise ofthe condition. We process it as a form of binary constraint using Algorithm1 in the pre-processing step to remove the inconsistent values. Note thatAlgorithm 1 enforces directional arc consistency between the variable in thepremise and the one in the conclusion of the conditional constraint. Forinstance, if the conditional constraint is X = a =⇒ Y = b and value ahas been removed from the domain of X then b has to be removed from thedomain of Y .

As well, the conditional constraint is also used to propagate the effect ofan assignment during the backtrack search following the look ahead strategy.For instance, if the conditional constraint is X = a =⇒ Y ≥ b and thecurrent assignment is X = a then all values from the domain of Y that areless than b should be removed.

– m > 1. This is a conditional constraint involving more than one relationin the condition of the conditional constraint. We process it similarly to the

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case where m = 1 but using generalized directional arc consistency as we aredealing with a form of n-ary constraint in this particular case. For instance,let us assume we have the following conditional constraint: A = a or B =b =⇒ C < c then if a is removed from the domain of A or b is removed fromthe domain of B then all the values from C’s domain that are greater or equalto c should be removed. This propagation can happen in the pre processingstage as well as the search phase. For example, if during the backtrack searchA (or B) is assigned a value other than a (or b) then C cannot be assigneda value that is greater or equal to c.

Algorithm CTCP-GAC1. Given a CTCP-Net T =< G, cc, cp, i, ci, cct, cpt, cit >

(X: set of variables, C: set of constraints between variables)2. Q ← {(i, j) | i ∈ cc ∧ j ∈ varCc(i)}3. While Q �= Nil Do4. Q ← Q − {(i, j)}5. If REV ISE(i, j) Then6. If Domain(j) = Nil Then return false7. Q ← Q � {(k, l) | k ∈ cc ∧ j ∈ varsCd(k)8. ∧ l ∈ varCc(k) ∧ k �= i ∧ j �= l}9. End-If10. End-While11. Return true

Function REV ISE(i, j)(REVISE for bound consistency)1. domainSize ← |Domain(j)|2. While |Domain(j)| > 0

∧¬seekSupportArc(i, j,min(j)) Do3. remove min(j) from Domain(j)4. End-While5. While |Domain(j)| > 1

∧¬seekSupportArc(i, j,max(j)) Do6. remove max(j) from Domain(j)7. End-While8. Return domainSize �= |Domain(j)|

Function REV ISE(i, j)(REVISE for arc consistency)1. REV ISE ← false2. nbElts ← |Domain(j)|3. For each value a ∈ Domain(j) Do4. If ¬seekSupport(i, j, a) Then5. remove a from Domain(j)6. REV ISE ← true7. End-If8. End-For9. Return Revise

Fig. 1. CTCP-GAC algorithm for CTCP-Nets, updated from [10]

Arc consistency is enforced with an arc consistency algorithm [5,8]. Since weare dealing with n-ary constraints, we use an adapted version of the GeneralizedArc Consistency (GAC) algorithm presented in [10]. This latter is a revisedversion of the original GAC algorithm proposed in [11] as well as a modifiedversion of the bound consistency algorithm for discrete CSPs in the case ofinequality relations [12]. More precisely, bounds consistency is first used throughinequality relations to reduce the bounds of the different domains of variables.The adapted GAC is then used to further reduce the domains of the variables.Let us describe now the details of our method.

The modified GAC algorithm that we call CTCP − GAC is described infigure 1. This algorithm enforces arc consistency on all variables domains.CTCP − GAC starts with all possible pairs (i, j) where j is a variable involvedby the constraint i. Each pair is then processed, through the function REV ISEas follows. Each value v of the domain of j should have a value supporting it(such that the constraint j is satisfied) on the domain on every variable involved

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by i otherwise v will be removed. If there is a change in the domain of j (afterremoving values without support) after calling the function REV ISE then thischange should be propagated to all the other variables sharing a constraint withj. When used as a bound consistency algorithm, cc involves inequality relationsand the REV ISE function (the function that does the actual revision of thedomains) is defined as shown in figure 1 [12]. In the other case, the REV ISEfunction is defined as shown in the bottom right of figure 1 [11]. In the func-tion REV ISE (for bound consistency) of figure 1, the function seekSupportArc(respectively the function seekSupport of REV ISE for semantic constraints infigure 1) is called to find a support for a given variable with a particular value.For instance when called in line 2 of the function REV ISE for bound consis-tency, the function seekSupportArc looks, starting from the lower bound of j’sdomain, for the first value that has a support in i’s domain. When doing so, anyvalue not supported will be removed.

We also propose a Condition-Constraint Table (CCT) in our model to clearlypresent all the unconditional and non-conditional constraints. It consists of twocolumns, conditions and constraints, and each consists of the following parts:variables, relations, value and connectives (if applicable). The fist column canbe empty (Null). Each row is a full description of constraints, which must becompletely satisfied. For a certain attribute Xi, its CCT is noted as CCT (Xi).The CCT including all the constraints is called CCT (U), where U is set of allvariables.

An example is presented in Table 1. After enforcing node and arc consistencyin the preprocessing stage of our proposed solving method, we run a backracksearch algorithm with a look ahead strategy [5] to find the Pareto optimal solu-tions of a given CTCP-Net. In order to improve the time performance of thebacktrack search, variables are first ordered following the most constrained vari-ables first heuristic [13]. Some of these variables will then be reordered accordingto the dependencies imposed by the CTCP-Net. In this regard, variables needto be sorted after their respective parents in the corresponding conditional con-straint, conditional preference or unconditional relative importance relation. Inaddition to this static variable ordering that occurs before the backtrack search,some variables are rearranged dynamically during the backtrack search accord-ing to the conditional relative importance relations (anytime the variable, therelative importance relies on, is assigned a particular value). Variables values areordered according to the CPTs. Note that, like for variable ordering, some ofthese orders depend on values assigned to some other variables and this is donedynamically during the backtrack search.

We adopt the Forward Check strategy [6] as the constraint propagation tech-nique during the backtrack search. Anytime a variable (that we call currentvariable) is assigned a value during the search, we propagate this decision tothe non assigned variables using our CTCP-GAC algorithm as described abovein our general procedure. In addition to reducing the size of the search space,this propagation will also detect earlier later failure. For instance, if one of thedomains of the non assigned variables becomes empty then we assign anothervalue to the current variable or backtrack to the previously assigned variable

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if there are no more values to assign to the current one. This backtrack searchmethod will continue until all the variables are assigned in which case we obtainan optimal solution or the search space is exhausted. In this latter case theCTCP-Net is inconsistent.

Table 1. Condition-Constraint table

Conditions ConstraintsNull Color �= black

Brand = sony Price ≤ 1000(Color = red) ∧ (Price > 1000) Brand = sony

(Brand = dell) ∨ (Brand = sony) Color = silver

4 Experimentation

In order to evaluate the time performance of our solving method, we conductedseveral experiments on real data selected from Kjiji.ca2. The experiments areconducted on a PC with the following specifications: Inter(R) Core(TM)i7-4500UCPU @1.8GHz and 16GB RAM; and running Windows 8 64-bit operating sys-tem. The test platform is MyEclipse 8.5. The application corresponds here topurchasing a vehicle online. 50 products are used for the experiments and for eachproduct the following attributes are considered: brand, model, year, engine size,color, milage, price, transmission, body type and seller name. The constraintsand preferences are represented in our model as shown below.

– Non-conditional constraints (NCC):(ncc1) Null → Saleby �= capital(ncc2) Null → Saleby �= roadway(ncc3) Null → Kilometers ≤ 150000(ncc4) Null → Brand �= Kia(ncc5) Null → Y ear ≥ 2002

– Conditional constraints (CC):(cc1) Saleby = nelson → Bodytype �= hatchback(cc2) Bodytype = hatchback → Saleby �= owner(cc3) (Brand = Honda)or(Bodytype = suv) → Color �= red(cc4) (Kilometer ≥ 13)and(Transmission = manual) → Brand �= Ford(cc4) (Price ≥ 8000) → Kilometers ≤ 120000

– Non-Conditional preferences (NCP):(ncp1) Null → Y ear(descendingorder)(ncp2) Null → Price(ascendingorder)(ncp3) Null → Kilometers(ascendingorder)(ncp4) Null → Transmission(auto � manumatic � manual)

– Conditional preferences (CP):(cp1) (Y ear ≥ 2005)and(Kilometers ≤ 15) → Brand(Toyota � Pontiac � Nissan

� Mazda � Kia succHyundai � Honda � Ford � Dogde � Chrysler �Chevrolet � Buick � Benz � BMW � Audi); otherwise : Brand(Audi� BMW � Benz � Buick � Chevrolet � Chrysler � Dodge � Ford �Honda � Hyundai � Kia � Mazda � Nissan � Pontiac � Toyota)

2 http://www.kijiji.ca/b-cars-vehicles/regina-area/c27l1700194

Integrating TCP-Nets and CSPs 209

(cp2) (Saleby �= owner)or(Transmission = manual) → Color(yellow � white �silver � red � grey � green � gold � brown � blue � black); otherwise :Color(black � blue � brown � gold � green � grey � red � silver �white � yellow)

(cp3) Brand = Honda → Bodytye(convertible � wagon � suv � coupe � sedan� truck � van succhatchback); otherwise : Bodytye(hatchback � coupe �sedan � suv � truck � van � wagon � convertible)

(cp4) Price ≤ 10000 → Saleby(owner � nelson � capital � roadway); otherwise :Saleby(nelson � capital � owner � roadway)

– Non-conditional relative importance(NCIR):(ncri1) Null → Price � Y ear(ncri2) Null → Price � Saleby(ncri3) Null → Y ear � Brand(ncri4) Null → Brand � Bodytype

– Conditional Relative importance(CIR):(cri1) Saleby �= owner → Y ear � Kilometers; otherwise : Kilometers � Y ear(cri2) Transmission = auto → Color � Bodytype; otherwise : Bodytype � Color

Fig. 2. Execution time(s) vs number of products

Figure 2 reports the running time required in seconds to return the optimalsolution when varying the number of products from 20 to 100. For each exper-iment, 30 run are conducted and the average running time is taken. As we cansee from the figure our proposed method is capable of provide an answer in lessthan 2 seconds even when the number of products is 100.

5 Conclusion and Future Work

Constraints and preferences handling is a complex but interesting problem thatis related to a wide variety of real world applications. Our proposed CTCP-Nets

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has the ability to represent and solve these constraint problems under preferencesby returning one or more solutions satisfying all the constraints and maximizingthe preferences. This process can be done in a very efficient running time, thanksto the constraint propagation techniques that we propose.

The proposed CTCP-Net has been implemented with a generic design thatoffers the flexibility for future maintenance and extensibility. It will be there-fore possible in the future to add other modules dealing with new features andproperties such as the case of cyclic CTCP-Nets.

In the near future we intend to deal with dynamic CTCP-Nets in the case ofthe addition or retraction of constraints and preferences. The case of constraintof preference addition can be relevant when looking for all k Pareto solutions. kcan however be very large and not practical. In this particular situation, insteadof returning all these optimal solutions we can ask the user to input more infor-mation in order to distinguish between them. These information can be in theform of new constraints or preferences. The system should then use these newinformation to search in an incremental way for new Pareto optimal solutions. Onthe other hand, the retraction of constraints can happen when the CTCP-Net isinconsistent and solving it will return no solutions. In this case we need to thinkabout relaxing some constraints in an incremental way in order to obtain theconsistency of the network. We have previously proposed constraint propagationas well as search algorithms for managing constraints in a dynamic environment[14,15] and are planning to adapt these techniques for the CTCP-Net.

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