Elicitation strategies for fuzzy constraint problems with missing preferences: algorithms and experimental studies

Elicitation strategies f or fuzzy constraint pr oblems with missing pr efer ences: algorithms and experimental studies Mirco Gelain 1 , Maria Silvia Pini 1 , Francesca Rossi 1 , K. Brent V enable 1 , and T ob y W alsh 2 1 Dipartimento di Matematica Pura ed Applicata, Uni versit ` a di Pado va, Italy E-mail: { mgelain,mp ini,frossi,kvenable } @math.u nipd.it 2 NICT A and UNS W Sydney , Australia, Email: T oby . W alsh@nicta.com.au Abstract. Fuzzy constraints are a popular approach to handle preferences and ov er-constraine d problems in scenarios where one needs to be cautious, such as in medical or space applications. W e consider here f uzzy constraint problems where some of the preferences may be missing. This models, for examp le, set- tings where agents are distributed and have priv acy issues, or where there is an ongoing preference elicitation process. In this setting, we study ho w to find a solution which is optimal irrespectiv e of the missing preferences. In the process of finding such a solution, we may elicit preferences from the user if necessary . Ho wev er , our goal is to ask the user as little as possible. W e define a combined solving and preference elicitation scheme with a large number of different in- stantiations, each corresponding to a concrete algorithm which we compare ex- perimentally . W e compute both the number of elicited preferences and the ”user effo rt”, which may be larger , as it contains all the preference va lues the user has to compute to be able to respond to the elicitation requests. While the number of elicited preferences is important when the conce rn is t o commu nicate as lit tle in- formation as possible, the user ef fort measures also the hidden work the user has to do to be able to comm unicate the elicited p references. Ou r experimental results sho w that some of our algorithms are very good at fi nding a necessarily optimal solution while asking the user for only a very small fraction of the missing pref- erences. The user ef fort is a lso very small for the best algorithms. Finally , we test these algorithms on hard constraint problems with possibly missing constraints, where the aim is to find feasible solutions irrespecti ve of the missing co nstraints. 1 Intr oduction Constraint pr ogramm ing is a p owerful paradig m for solving schedu ling, plannin g, and resource allocation problem s. A problem is represen ted by a set of variables, each with a domain o f values, and a set of co nstraints. A solution is an assign ment of values to the variables which satisfies all constraints and which optionally maximizes/minimizes an objective f unction. Soft constraints are a way to model o ptimization problem s by allowing f or sev eral levels of satisfiability , m odelled by the use of prefere nce or cost values th at represent how mu ch we like an in stantiation of the v ariables of a constraint. It is usually assumed that the data (variables, domains, (soft) constraints) is com- pletely known before solving starts. This is often u nrealistic. In w eb ap plications an d multi-agen t systems, the data is fr equently on ly partially k nown an d may be ad ded to at a later date by , fo r exam ple, e licitation. Data m ay also come from different sou rces at d ifferent tim es. In multi-ag ent systems, agents may release data reluctantly du e to priv acy co ncerns. Incomp lete sof t co nstraint pr oblems can mode l such situa tions b y allowing some of the pr eferences to be m issing. An algo rithm has been p roposed an d tested to solve such inco mplete pro blems [7]. The go al is to find a solution th at is gu aranteed to be optimal irr espective of the missing p reference s, eliciting pr eferences if n ecessary u ntil such a solutio n exists. T wo notions of optimal solution are considered: p ossibly optimal solutions ar e assignm ents that are optimal in at least o ne way of revealing the un spec- ified p referen ces, while nec essarily o ptimal solutio ns ar e assignm ents tha t are optimal in all ways th at the unspecified preferences c an be re vealed. The set of possibly optimal solutions is never e mpty , while the set of necessarily optimal solutions can be empty . If there is no necessarily optimal solution, the algorithm proposed in [7 ] uses branch and boun d to find a ”promising solution” (specifically , a comp lete assignment in the best possible com pletion of the cu rrent problem) and elicits th e missing p reference s related to this assignment. This process is repeated till there is a necessarily optimal solution. Although this algo rithm beh aves reaso nably well, it make so me specific ch oices about solving and preference elicitation that may not be o ptimal in practice, as we shall see in this pap er . For example, the algorithm only elicits missing prefer ences after ru n- ning b ranch and b ound to exhaustion . As a seco nd examp le, the algorith m elicits all missing pref erences r elated to the candid ate solu tion. Many oth er strategies a re possi- ble. W e might elicit prefe rences a t the end o f every co mplete branch , o r even at e very node in the search tr ee. Also, when choosing th e v alue to assign to a variable, we might ask the user (wh o knows the missing pre ferences) for help. Finally , we might not elicit all the missing p referenc es r elated to the cur rent can didate solutio n. For example, we might just ask the user for the worst preference among the missing ones. In th is paper we consider a general algorithm scheme which greatly g eneralizes that propo sed in [7]. I t is based on three parameters: what to elicit, when to eli cit it, a nd w ho chooses the value to be assigned to the next variable. W e test all 16 possible d ifferent instances of the scheme (amon g which is the algo rithm in [7]) on r andomly g enerated fuzzy constraint pro blems. W e demo nstrate that som e o f th e alg orithms ar e very go od at finding nece ssarily optimal solution witho ut eliciting too many preferen ces. W e also test the algorithm s o n pr oblems with h ard constrain ts. Finally , we consider pr oblems with fuzzy tempor al constraints, where problem s ha ve more specific structure. In our experiments, we compute the elicited preferences, that is, the missing v alues that th e user has to provide to the system because they are req uested by the alg orithm. Providing th ese values usually h as a cost, either in terms of com putation effort, o r in terms of priv acy decrease, or also in terms of communicatio n band width. Th us know- ing how many preferences are elicited is i mpor tant if we care about any of these issues. Howe ver , we also compute a measure of the user’ s effort, which may be larger than t he number o f elicited pref erences, as it contains all the preferen ce values the user has to consider to b e ab le to respond to the elicitation requests. For examp le, we may ask the user for the worst prefer ence value am ong k missing ones: the user will commun icate only o ne value, but he will h av e to con sider all k of them. While knowing the n umber of elicited preferen ces is importan t when the co ncern is to commu nicate as little infor- mation as possible, the user effort measu res also the h idden work th e user has to do to be able to co mmunic ate the elicited pre ferences. This u ser’ s effort is therefore also an importan t measure. As a m otiv ating examp le, reco mmend er sy stems give sugg estions based on p ar- tial kn owledge of the user’ s pref erences. Our a pproach co uld impr ove perform ance by identifyin g so me key question s to ask b efore g iving r ecommen dations. Pr i vacy con- cerns regardin g t he p ercentage of elicited prefere nces are motiv ated by eavesdropping. User’ s effort is instead related to the b urden on the user . Our resu lts show th at the cho ice o f p referenc e elicitation strategy is cru cial fo r th e perfor mance o f th e solver . While the best algorithms n eed to elicit as little as 10 % of the m issing prefer ences, the worst one n eeds much m ore. The u ser’ s effort is also very small fo r the best algorithms. Th e perf ormance of the best algorith ms shows th at we only n eed to ask the user a very sma ll amount o f addition al infor mation to be ab le to solve problems with missing data. Sev eral other approach es have addressed similar issues. For examp le, op en CSPs [4,6] and in teractive CSPs [ 9] work with d omains th at can be par tially sp ecified. As a second examp le, in dy namic CSPs [2] variables, domains, and constrain ts may chan ge over time. Ho we ver , the incompleteness considere d in [6,5] is on domain values as well as on the ir prefer ences. W o rking un der this assumptio n m eans that the ag ent that p ro- vides ne w v alues/costs for a variable kn ows all possible costs, since they are capable of providing the best value first. I f th e co st com putation is expen si ve or time consumin g, then computing all such costs (in order to g iv e th e mo st preferred value) is not desir- able. W e assum e instead , as in [ 7], tha t all values ar e g i ven at the beginning , an d that only some prefe rences are missing. Because of this assumption, we don ’t need to elicit preferen ce values in order , as in [ 6]. 2 Backgr ound In this section we give a brief overview of th e f undamen tal n otions an d c oncepts o n Soft Constraints and Incomp lete Soft Constraints. Incomp lete Soft Constraints pr oblems ( ISCSPs) [7] extend Soft Co nstraint Prob- lems (SCSPs) [1] to d eal with p artial info rmation. W e will f ocus on a specific in stance of this framework in which the soft constraints are fuzzy . Giv en a set of v ariables V with finite domain D , an incomplete fuzzy constraint is a pair h idef , con i wh ere c on ⊆ V is th e scop e o f th e constrain t and idef : D | con | − → [0 , 1] ∪ { ? } is the pr eference func tion of the constraint associating to each tup le of assignments to the variables in con eith er a pr eference value rang ing between 0 and 1, or ? . All tuples mapp ed into ? by i def ar e called inco mplete tup les , meaning that their preferen ce is unspecified. A fuz zy constraint is an in complete fuzzy constrain t with no incomplete tuples. An incomp lete fuzzy con straint pr ob lem (IFCSP) is a pair h C, V , D i wh ere C is a set of incom plete fuzzy constrain ts over the v ariables in V with dom ain D . Given an IFCSP P , I T ( P ) denotes the set o f all in complete tuples in P . When th ere are n o incom plete tuples, we will denote a fuzzy constraint problem by FSCP . Giv en an IFCSP P , a completion of P is an IFCSP P ′ obtained from P by asso- ciating to each in complete tu ple in every co nstraint an elemen t in [0 , 1] . A co mpletion is p artial if some prefer ence remains u nspecified. C ( P ) deno tes the set of all p ossible completion s of P and P C ( P ) den otes the set of all its partial completions. Giv en an assignment s to all the variables of an IFCSP P , pref ( P, s ) is the pref- erence of s in P , defined as pref ( P, s ) = min ∈ C | idef ( s ↓ con ) 6 =? idef ( s ↓ con ) . It is obtain ed by taking the m inimum am ong the kn own pre ferences associated to th e projection s of the as signment, that is, of the appro priated sub-tuples in the constraints. In the fuzzy context, a c omplete assignment of v alues to all the variables is an opti- mal solu tion if its p reference is maximal. The optim ality notion of FCSPs is generalize d to I FCSPs via the notions of necessarily and p ossibly optima l solution s , that is, com- plete assignmen ts which ar e maxima l in all or some co mpletions. Giv en an IFCSP P , we de note by N O S ( P ) (resp., P O S ( P ) ) the set of necessarily (resp ., p ossibly) o pti- mal solutions of P . Notice that N O S ( P ) ⊆ P O S ( P ) . Mo reover , wh ile P OS ( P ) is never empty , N OS ( P ) m ay be empty . In p articular, N O S ( P ) is empty whenever the revealed p referen ces d o not fix the relationship between one assignment and all others. In [7] an alg orithm is pro posed to find a nec essarily o ptimal solution of an IFCSP based on a characterization of N OS ( P ) and P OS ( P ) . Th is character ization uses th e preferen ces of the o ptimal solu tions of two special co mpletions of P , namely the 0 - completion o f P , den oted by P 0 , o btained f rom P by associatin g p referenc e 0 to each tuple of I T ( P ) , and th e 1 -completio n of P , d enoted by P 1 , o btained fro m P by as- sociating pref erence 1 to each tuple of I T ( P ) . No tice that, by mo notonic ity of min , we have that p ref 0 ≤ pref 1 . When pr ef 0 = pref 1 , N O S ( P ) = O pt ( P 0 ) ; thus, any optimal solution of P 0 is a ne cessary optimal solution. Otherwise, N OS ( P ) is em pty and P OS ( P ) is a set of s olution s with preference between pr ef 0 and pref 1 in P 1 . The algorithm prop osed in [7] finds a necessarily op timal so lution of the given IFCSP by interleaving the computation o f pref 0 and pref 1 with preferen ce elicitation steps, until the two values c oincide. M oreover , the p reference elicitation is g uided b y the fact that only solu tions in P OS ( P ) can become ne cessarily optima l. Thus, th e alg orithm only elicits preferen ces related to optimal solutions of P 1 . 3 A general solver scheme W e now pro pose a more g eneral schema for solv ing I FCSPs based on interlea ving branch and bo und ( BB) search with elicitatio n. This schem a generalizes the con crete solver presented in [7], but h as several other in stantiations tha t we will consider a nd compare e xperime ntally in this p aper . The schem e uses branch and bound . This consid- ers the variables in some ord er , choosing a value for each variable, and pruning branches based on an up per bo und (assuming the goal is to maxim ize) on the preferen ce value of any c ompletion of the curren t partial assign ment. T o deal with missing prefere nces, branch and bound is applied to both the 0-completion and the 1 -completion o f the prob- lem. If they have th e same solution, th is is a nece ssarily op timal so lution and we c an stop. If not, we elicit some of the missing prefer ences and con tinue br anch an d b ound on the new 1 -completion. Preferences can be elicited a fter each run o f branc h and bound (as in [ 7]) or du ring a BB ru n wh ile pr eserving the corr ectness of the appr oach. For examp le, we can elicit preferen ces at the end of e very complete branch (that is, regarding preferen ces of e very complete assignmen t considered in the b ranch and b ound algorithm ), or at ev ery node in the sear ch tree (thus co nsidering every partial assignm ent). Moreover, when cho osing the value for the next variable to be assigned, we can ask the u ser (who knows the missing pref erences) for help. Finally , ra ther than eliciting a ll the missing pref erences in the possibly optimal solution, or the complete or partial assignment under considera tion, we can elicit just o ne of th e missing p referen ces. For example, with fuzzy constraint problem s, eliciting just the worst preference among the missing ones is sufficient since only th e worst value is impor tant to the comp utation of the overall prefer ence v alue. More precisely , the algorith m schema we prop ose is based o n th e following parameter s: 1. Who cho oses the value of a variable: the algo rithm can choose the values in de- creasing order either w .r . t. their p referenc e values in th e 1 -completio n (Who=dp ) or in th e 0 -comp letion (Who=dpi). Otherwise, the user can sug gest this choice. T o do this, he can consid er all the preference s (revealed or not) for the values o f th e current variable ( lazy user , Who=lu for sh ort); or h e co nsiders also th e p referenc e values in constraints between this variable an d the past variables in the sear ch order ( smart user , Who=su for short). 2. What is elicited: we can e licit the preferenc es of all th e incomplete tuples of the current assignment (What=a ll) or o nly the worst pr eference in the curr ent assign- ment, if it is worse than the known ones (What=worst); 3. When elicitation takes place: we can elicit pr eferences at the end of the bra nch and bo und search (When=tre e), or dur ing the search , when we have a c omplete assignment to all variables (When=branch) or whenev er a ne w v alue is assigned to a variable (When=node). By choosing a value for each of the three above parameters in a consistent way , we obtain in total 16 different algor ithms, as summarize d in Figure 1, where the circled instance is the concrete solver used in [7]. Figures 2 and 3 sh ow th e pseu do-cod e o f the gener al scheme f or solvin g IFCSPs. There are thre e algorithm s: ISCSP-SCHEME, BBE and BB. ISCSP-SCHEME takes as input an IFCSP P and the values for the three parameter s: Who, What and When. It r eturns a p artial com pletion of P that has som e necessarily op timal solution s, one of these necessarily op timal solu tions, and its prefe rence value. It starts by com puting via br anch and bo und (algor ithm BB) an o ptimal solution o f P 0 , say s max , and its preferen ce pre f max . Next, pro cedure B B E is called . If B B E succeeds, it returns a partial completion of P , say Q , on e o f its ne cessarily o ptimal solutions, say s 1 , and its associated pre ference pr ef 1 . Otherwise, it retur ns a solutio n eq ual to nil . In the first case the output o f IFCSP-SC HEME coincides with th at of BBE, otherwise IFCSP- SCHEME return s P 0 , one of its optima l s olution s, and its preference. Procedur e BBE takes as input the same v alues as IFCS P-SCHEME and, in addition, a solu tion sol a nd a pref erence l b represen ting the current lower bo und o n th e op timal Fig. 1. Instan ces of the general s cheme. IFCSP-SCHEME ( P , W ho , W hat , W hen ) Q ← P 0 s max , pr ef max ← B B ( P 0 , − ) Q ′ , s 1 , pr ef 1 ← B B E ( P, 0 , W ho, W hat, W hen, s max , pr ef max ) If ( s 1 6 = nil ) s max ← s 1 , pr ef max ← pr ef 1 , Q ← Q ′ Return Q , s max , pr ef max Fig. 2. Algo rithm IFCSP-SCHEME. preferen ce value. Fun ction ne xtV ari abl e , applied to the 1 -comp letion of the IFCSP , returns the n ext variable to be assigned. T he alg orithm then assigns a value to this variable. I f the Boolean f unction nextV al u e returns tru e (if there is a value in the domain) , we select a value for curre ntV ar accord ing to the v alue of parameter W ho . Function U pp erB ound co mputes an up per boun d on the prefer ence of any com- pletion of the current partial assignment: the minimum over the pr eferences of the con- straints in volving only variables that have already been instantiated. If When=tre e, elicitation is han dled by pr ocedur e E l icit @ tree , and takes place on ly at the end of the search over the 1 - completion . The user is not inv olved in the value assignment steps within the sear ch. At the en d of th e searc h, if a solu tion is foun d, the user is asked either to reveal all th e preferenc es o f the incomplete tuples in the s olution (if What=all), or on ly the worst one amon g them (if What=worst). If such a preferen ce is better than the best found so far , BB E is called recursively with the ne w best so lution and prefer ence. If When=bran ch, B B is pe rformed only on ce. The user may be asked to ch oose the next value for the curr ent variable being instantiated. Prefer ence elicitation , which is handled by func tion E l ici t @ br anch , takes place during search, whenever all variables have be en instantiated and the user can be asked either to reveal the pr eferences of all the incomplete tuples in the assignment (What=all), or the worst preference amo ng tho se of the in complete tuples of the assignment (What=worst). In both cases the information gathered is suf ficient to test such a prefere nce v a lue against the current lo wer bound. If When=no de, preferen ces are elicited every time a new value is assigned to a variable an d it is handled by proced ure E l ici t @ node . Th e tu ples to be considered for elicitation are those inv olving the value which ha s just be en assigned a nd belo nging to constraints between th e current variable and alre ady instantiated v ariables. If What=all, the u ser is a sked to provide the pref erences of all the inc omplete tu ples inv olving the new assign ment. Otherwise if What=worst, the u ser provides only the preference of t he worst tuple. BBE ( P , nI nst V ar , W ho , W ha t , W hen , sol , lb ) sol ′ ← sol , p r ef ′ ← l b curr entV ar ← nextV ar iabl e ( P 1 ) While ( nextV al ue ( cur r entV ar, W ho ) ) If ( W hen = node ) P, pr ef ← E licit @ N ode ( W hat, P , cu r r entV ar, lb ) ub ← U pper B ound ( P 1 , cur r entV ar ) If ( ub > l b ) If ( nI nstv ar = number of v ar iables in P ) If ( W hen = br a nch ) P, pr ef ← E licit @ br a nch ( W hat, P, lb ) If ( pr ef > l b ) sol ← g etS olution ( P 1 ) lb ← p r ef ( P 1 , sol ) else B B E ( P, nI nstV a r + 1 , W ho, W hat, W hen, sol, l b ) If ( W hen = tre e and nI nstV ar = 0 ) If( sol = ni l ) sol ← sol ′ , pr ef ← pr e f ′ else P, pr ef ← E licit @ tr ee ( W hat, P, sol, lb ) If( pr ef > pr ef ′ ) B B E ( P, 0 , W ho, W hat, W hen, sol, pref ) else B B E ( P , 0 , W ho, W hat, W hen, sol ′ , pr ef ′ ) Fig. 3. Algorithm BBE. Theorem 1. Given an IFCSP P a nd a co nsistent set of values fo r parameters Wh en, What and Who , Algorithm IFCSP-SCHEME a lways terminates, and r eturns an I FCSP Q ∈ P C ( P ) , a n assignment s ∈ N OS ( Q ) , and its pr efer ence in Q . Pr oo f. Let u s fir st n otice that, as far as correctn ess and termination c oncern, th e value of parameter Who is irrelev ant. W e con sider tw o separate cases, i.e., When=tre e and and W hen=b ranch or node. Case 1: When =tree. Clearly I FCSP-SCHEME ter minates if and only if BBE terminates. If we consider th e pseudoco de of p rocedu re BBE shown in Algorithm 3, we see that if When = tree, BBE terminates wh en sol = nil . Th is hap pens o nly wh en the search fails to find a solu tion of the curren t pro blem with a p reference strictly greater th an th e cu rrent lower boun d. Let us den ote with Q i and Q i +1 respectively the IFCSPs g i ven in in put to the i -th and i + 1 -th recursive c all of BBE. First we no tice that only proced ure E l ic it @ tr ee modifies the IFCSP in input by possibly adding new elicited preferences. Moreover , wh atev er the value of parameter What is, the returned IFCSP is either the same as the on e in input or it is a (possibly partial) completion of the one in input. Thus we have Q i +1 ∈ P C ( Q i ) and Q i ∈ P C ( P ) . Since the search is alw ays performed on the 1 - completion of the curren t IFCSP , we can con clude that for every solution s , pr ef ( Q i +1 , s ) ≤ pr ef ( Q i , s ) . Let us now den ote with l b i and l b i +1 the lower boun ds given in input r espectively to the i -th and i + 1 - th recur si ve call of BBE. It is easy to see that l b i +1 ≥ l b i . Thus, since at e very iteration we h av e that the pr eferences of solutions can o nly get lower , an d the boun d can only get higher, an d sinc e we have a finite num ber of solutions, we can conclude that BBE always terminates. The reasoning that follows relies on the fact th at value pref returned b y f unction E li cit @ tr ee is the final p referenc e after e licitation of assign ment sol g iv en in input. This is true since either Wh at = a ll and th us all p referen ces have been elicited and the overall preferenc e of sol can be co mputed or only the worst p referen ce has been elicited but in a f uzzy con text where the overall prefer ence coincide with the worst on e. If called with When = tree IFCSP-SCHEME exits when the last br anch and bou nd search h as ended retur ning sol = nil . In such a case sol and pre f are upd ated to co ntain the best solution an d associated prefere nce found so far , i.e., sol ′ and pr ef ′ . Th en, the algorithm returns the current IFCSP , say Q , and s ol and pre f . Follo wing the same reasoning as above done for Q i we can conclu de that Q ∈ P C ( P ) . At the end of every wh ile loop execution, assignmen t sol either co ntains an opti- mal so lution sol of the 1 - completion o f th e cu rrent I FCSP o r sol = nil . sol = nil iff there is n o as signmen t with preferenc e hig her th an lb in the 1 -completion of the current IFCSP . In th is situatio n, sol ′ and pref ′ are an optimal solution and prefere nce of the 1 -comp letion of the cu rrent IFCSP . Howev er, since th e pr eference of s ol ′ , pr ef ′ is inde- penden t of unknown preferences and since due to mo notonicity the optimal p referen ce value of the 1 -co mpletion is always greater than or equal to that of th e 0 -comple tion we ha ve that sol ′ and pref ′ are an optimal solution and preference of the 0 -completion of the current IFCSP as well. By Theorem s 1 and 2 of [7] we can co nclude th at N O S ( Q ) is not empty . If pref = 0 , then N O S ( Q ) contain s all the assignments and thus also sol . Th e alg orithm cor- rectly returns the same I FCSP given in inpu t, assignm ent sol and its preferen ce pr ef . If instead 0 < pr ef , again the algorithm is c orrect, since by Theorem 1 of [7] we know that N O S ( Q ) = O pt ( Q 0 ) , and we have shown that sol ∈ Op t ( Q 0 ) . Case 2: When=branch or node. In order to prove that the a lgorithm terminates, it is sufficient to show tha t B B E ter- minates. Since th e doma ins are finite, the labeling phase pr oduces a num ber of finite choices at every level o f the search tree. More over , sinc e the num ber of variables is limited, then, we h ave also a finite number of lev els in the tree. Hence, B B E considers at most all the possible assignm ents, that ar e a finite nu mber . At th e en d of the exe- cution o f I FCSP-SCHEME, sol , with pref erence pr ef is on e of the o ptimal solutio ns of the current P 1 Thus, for every a ssignment s ′ , pr ef ( P 1 , s ′ ) ≤ pr ef ( P 1 , sol ) . More- over , for ev ery com pletion Q ′ ∈ C ( P ) an d for every assignment s ′ , p ref ( Q ′ , s ′ ) ≤ pre f ( P 1 , s ′ ) . Hen ce, f or every assignmen t s ′ and for every Q ′ ∈ C ( P ) , we hav e th at pre f ( Q ′ , s ′ ) ≤ pre f ( P 1 , sol ) . In o rder to p rove that sol ∈ N OS ( P ) , now it is suffi- cient to p rove that for every Q ′ ∈ C ( P ) , pre f ( P 1 , sol ) = pr ef ( Q ′ , sol ) . Th is is true, since sol has a prefer ence that is indep endent from th e missing prefe rences of P , both when eliciting all the missing prefer ences, and when eliciting only the worst one either at branch or node le vel. In fact, in both cases, the preference of sol is the same in e very completion . Q.E.D. If Whe n=tree, th en we elicit a fter each BB run , an d it is p roven in [7] th at I FCSP- SCHEME never elicits p referen ces inv olved in solutio ns which are n ot p ossibly op- timal. This is a desirable pro perty , since o nly possibly optimal solution s can b ecome necessarily op timal. Howe ver , the experiments will show that solvers satisfyin g such a desirable property are often out-per formed in practice. 4 Pr oblem generator and experimental design T o test the per forman ce of th ese different a lgorithms, we cr eated IFCSPs u sing a gen - erator which i s a simple extension of the standard random mo del for har d constraints to soft and incomplete constraints. The generato r has the following parameters: – n : numb er of v ariables; – m : cardina lity of the v ariable domains; – d : density , that is, the perc entage of binary constraints present in the problem w .r .t. the total numb er of possible binary constrain ts t hat can be defin ed on n variables; – t : tig htness, that is, the per centage of tup les with prefere nce 0 in eac h constraint and in each doma in w .r .t. the total num ber of tu ples ( m 2 for the constrain ts, since we have o nly binary constraints, and m in the domains); – i : incom pleteness, that is, the per centage o f in complete tuples (that is, tu ples with preferen ce ? ) in each constrain t and in each domain. Giv en values for these par ameters, we gener ate IFCSPs as follows. W e first generate n variables and th en d % of the n ( n − 1) / 2 p ossible co nstraints. The n, for every domain and for every constraint, we gener ate a random preferen ce value in (0 , 1] fo r each of the tuples (that are m fo r the domain s, and m 2 for the con straints); we rando mly set t % of these preferen ces to 0 ; and we rand omly set i % of the preferences as incomp lete. Our experiments measure the per cen tage of elicited prefer ence s (over all the m issing preferen ces) as th e gener ation pa rameters vary . Since some of the algor ithm instances require the u ser to suggest th e value fo r the next variable, we also s how the user’ s ef fort in the various solvers, fo rmally d efined as the nu mber o f missing pref erences the user has to consider to giv e the required help. Besides the 16 instan ces of the sch eme describe d above, we also considered a ”base- line” algor ithm that elicits p reference s of rando mly cho sen tuple s e very time branch and bound en ds. All algo rithms are named by means o f the th ree param eters. For ex- ample, algor ithm DPI .WORST .BRANCH has p arameters Who=dpi, What=worst, and When=bran ch. F or the baseline algorithm, we use the name DPI.RANDOM.TREE. For ev ery choice of par ameter values, 100 pr oblem instan ces are g enerated. The results shown are the average over the 100 instances. Also, when it is not specified otherwise, we set n = 10 and m = 5 . Howev er, we have sim ilar results (alth ough not shown in th is p aper for lack of space) for n = 5 , 8, 11, 14 , 17 , and 2 0. All o ur experiments have been perfor med on an AMD Athlon 6 4x2 2800+, with 1 Gb RAM, Linux operating system, and using JVM 6.0.1 . 5 Results In this section we summar ize and discuss our experimental comparison of the different algorithm s. W e fir st fo cus on in complete f uzzy CSPs. W e then consider two special cases: in complete CSPs whe re all constraints ar e har d, and incomp lete fuzzy tem poral problem s. In all the experimental resu lts, the associatio n between an algorithm name and a line symbol is shown below . 5.1 Incomplete fuzzy CSPs Figure 4 sh ows the perc entage of elicited p referen ces when we vary the incom pleteness, the density , and the tightne ss respecti vely . For reason s of space, we s how only the results for specific v a lues of the p arameters. Howev er, the trends observed here ho ld in general. It is easy to see that the best algo rithms ar e those that elicit at the br anch lev el. In particular, algorithm SU.W ORST .BRANCH elicits a very small per centage of missing preferen ces (less th an 5%), no m atter the amoun t o f incomp leteness in the p roblem , and also independ ently of the density and the tightness. This algorithm outperfor ms all others, but relies on h elp from the user . The b est algor ithm that do es not need such help is DPI.WORST .BRANCH. T his nev er elicits more than about 10% of the missing preferen ces. Notice that the baseline algorithm is alw ays the worst one, and need s nearly all the missing prefer ences before it finds a necessarily optimal solution. Notice also that the alg orithms with W hat=worst ar e almo st always better than tho se with What=all, an d that When=bran ch is almost always better than When=node or When=tree. Figure 5 (a) shows the u ser’ s effort as in completene ss varies. As could be pr e- dicted, the effort grows slightly with th e incomp leteness level, and it is eq ual to the percentag e of elicited pr eferences only when Wha t=all and Who =dp or dp i. For exam- ple, when Wha t=worst, e ven if Who=dp or dpi, the user has to consider m ore prefer- ences than those elicited, since to id entify th e worst preferen ce value the user nee ds to check all o f them (that is, tho se in volved in a partial or complete a ssignment). DPI.WORST .BRANCH require s the user to loo k at 60% o f the missing p referenc es at most, even wh en incompleteness is 100%. Figure 5 (b) sho ws the user’ s effort as density varies. Als o in this case, as expected, the effort grows slightly with the den sity level. In this case DPI.WORST .BRANCH requires the user to look at most 40% of th e missing pre ferences, even when the density is 80%. All these alg orithms have a useful an ytime property , since they can be stopped e ven before their ter mination o btaining a possibly optimal solution with preferen ce value equal to the b est solutio n consider ed up to that poin t. Figure 6 shows how fast the var- ious algorithms reach optimality . The y axis re presents the solution quality during exe- cution, n ormalized to allow for com parison amon g different prob lems. The alg orithms that p erform best in ter ms of elicited p referenc es, such as DPI.WORST .BRANCH, ar e also those that approach optimality fastest. W e can therefore s top such algorithms early and still obtain a solution of good quality in all comple tions. Figure 7 ( a) shows the per centage of elicited pr eferences over a ll the pr eferences (white bar s) and the u ser’ s effort (black b ars), as well as the pe rcentage of prefere nces present at the begin ning (grey bars) for DPI.WORST . BRANCH. Even with high levels of inco mpleteness, th is algor ithm elicits only a very small fraction o f th e pr eferences, while asking the user to consider at most half of the missing preferenc es. Figure 7 (b) shows results fo r L U.WORST .BRANCH, w here the u ser is inv olved in the choice of the value for the n ext variable. Compared to DPI.WORST .BRANCH, this algo rithm is b etter both in term s of e licited pr eferences and user’ s effort (while SU.WORST .BRANCH is better only for th e elicited p referen ces). W e conje cture that the he lp the user gives in cho osing the next value gu ides the search tow ards better solutions, thus resulting in an overall decrease of the numb er of elicited prefere nces. Although we are mainly inter ested in the amount of elicitation, w e also c omputed the time to ru n the 16 algorithms. Ignorin g the time taken to ask the user for miss- ing pr eferences, the best algorithms need ab out 20 0 m s to fin d the ne cessarily o ptimal solution f or problems with 10 v ariables and 5 elements in th e do mains, no matter the amount of incomp leteness. Most of the algorith ms need l ess than 500 ms. 5.2 Incomplete hard CSPs W e also te sted the se algo rithms on hard CSPs. In this case, p referenc es are o nly 0 and 1, an d n ecessarily optim al solution s are comp lete assignments which are feasible in all completion s. The pro blem ge nerator is adapted according ly . The param eter What now has a specific meanin g: What=worst means asking if there is a 0 in th e m issing preferen ces. If there is no 0, we can infer that all the missing prefere nces are 1s. Figure 8 sh ows th e per centage of elicited pref erences for h ard CSPs in term s of amount of incomp leteness, density , an d tigh tness. Notice that the scale on th e y axis varies to inc lude only the highest values. The best algo rithms are those with W hat=worst, where the inf erence explained above about missing preferen ces can be perfor med. It is easy to see a phase transition at about 35% tightne ss, which is when problem s pass fro m being so lvable to having no solution s. Howe ver , the p ercentage of elicited prefe rences is below 20% for all algorith ms e ven at the peak. Figure 9 ( a) shows the user’ s effort in terms of amo unt of incom pleteness and Figur e 9 ( b) shows the user’ s effort in terms of density for th e case o f hard CSPs. Overall, the 0 10 20 30 40 50 60 70 80 90 100 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 elicited preferences incompleteness (a) d=50%, t=10% 10 20 30 40 50 60 70 80 90 100 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 elicited preferences density (b) t=35%, i=30% 10 20 30 40 50 60 70 80 90 100 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 elicited preferences tightness (c) d=50%, i=30% Fig. 4. Percen tage of elicited preferences in incomplete fuzzy CS Ps. 10 20 30 40 50 60 70 80 90 100 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 user’s effort incompleteness (a) d=50%, t=10% 10 20 30 40 50 60 70 80 90 100 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 user’s effort density (b) t=10%, i=30% Fig. 5. Inc omplete fuzzy CSP s: user’ s effort 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 20 30 40 50 60 70 80 90 100 solution quality elicited preferences (a) solution quality Fig. 6. Inc omplete fuzzy CSP s: solutio n quality . (a) d=50%, t=10% (b) d=50%, t=10% Fig. 7. Inc omplete fuzzy CSP s: best alg orithms. 0 1 2 3 4 5 6 7 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 elicited preferences (%) incompleteness (a) d=50%, t=10% 0 1 2 3 4 5 6 7 8 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 elicited preferences (%) density (b) t=10%, i=30% 0 2 4 6 8 10 12 14 16 18 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 elicited preferences (%) tightness (c) d=50%, i=30% Fig. 8. Elicited pre ferences in incomplete CSP s. 10 20 30 40 50 60 70 80 90 100 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 user’s effort incompleteness (a) d=50%, t=10% 10 20 30 40 50 60 70 80 90 100 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 user’s effort density (b) t=10%, i=30% Fig. 9. Incomp lete CSPs : user’ s ef fort (a) d=50%, t=10% Fig. 10 . Incomplete CSPs: best algorithm. best algorith m is again DPI.WORST .BRANCH. Figur e 10 gives the elicited preferences and user effort for this algorith m. 5.3 Incomplete temporal fuzzy CSPs W e also p erform ed some experiments on fu zzy simple temporal pr oblems [8]. T hese problem s ha ve c onstraints of the form a ≤ x − y ≤ b mod elling allo wed time intervals for durations and distances of e vents, and fuzzy preferen ces associated to each element of an interval. W e have generated classes of su ch pro blems following th e appro ach in [8], ad apted to co nsider inco mpleteness. Wh ile the class of problem s g enerated in [ 8] is tractab le, the pr esence of inco mpleteness makes them intractable in gen eral. Figure 11 sho ws that in this specialized domain it is also possible to find a necessarily optimal solution b y asking about 10% of the m issing p referen ces, for examp le via algorith m DPI.WORST .BRANCH. 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 elicited preferences incompleteness Fig. 11 . Percentage of elicited preference s in inco mplete fuzzy temporal CSPs. 6 Futur e work In the p roblems considered in this p apers, we have no inf ormation about the missing preferen ces. W e ar e cu rrently conside ring settings in wh ich each missing pre ference is associated to a r ange of possible values, that may b e smaller th an th e wh ole r ange of preferen ce values. F or such problems, we intend to define se veral notions of optimality , among which necessarily and possibly op timal solution s are just two examples, and to develop specific elicitation strategies for each of them. W e are also studying sof t constraint problem s wh en no preference is miss ing, but some of them are unstable, and have associated a range of possible alternative values. T o model fuzzy CSPs, we h av e no t used tradition al fuzzy set theor y [3], but sof t CSPs [1], since we intend to apply our work also to non-fu zzy CSPs. I n fact, we plan to co nsider inco mplete weighted co nstraint pro blems as well as d ifferent heuristics for choosing the next variable during the search. All algor ithms with W hat=all are no t tied to fuzzy CSPs an d are reasona bly ef ficient. Moreover , we intend to build solvers based on loc al search and variable elimination method s. Finally , we want to add elicitatio n costs and to use them also to guide the search, as done in [10] for hard CSPs. Acknowledgeme nts This work h as been partially supp orted b y Italian MIUR PRIN pro ject “Con straints and Pr eferences” (n . 20 0501 5491 ) . Th e last autho r is fun ded by th e Departmen t o f Broadban d, Comm unications and the Digital Econ omy , and the Australian Research Council. Refer ences 1. S. B istarelli, U. Montanari, and F . Rossi. Semiri ng-based constraint solving and optimiza- tion. J ACM , 4 4(2):201–236 , mar 1997. 2. R. Dechter and A. Dechter . Belief maintenance in dynamic constraint networks. In AAAI , pages 37–42, 1988. 3. D. Dubois and H. Prade. Fuzzy sets and Systems - Theory a nd App lications . Academic Press, 1980. 4. B. Faltings and S . Macho-Gonzalez. Open constraint satisfaction. In CP , volume 2470 of LNCS , pages 356–370 . Springer , 2002 . 5. B. Faltings and S. Macho-Gonzalez. Open constraint optimization. In CP , volume 2833 of LNCS , pages 303–317 . Springer , 2003 . 6. B. Faltings and S . Macho-Gonzalez. Open constraint programming. AI Journ al , 161(1- 2):181–20 8, 2005. 7. M. Gelain, M. S . Pini, F . Rossi, and K. B. V enable. Dealing with incomplete preferences in soft constraint problems. In Proc. CP’07 , volume 4741 of LNCS , pages 286–300. Springer , 2007. 8. L. Khatib, P . Morris, R. Morris, F . Rossi, A. S perduti, and K. Brent V enable. Solving and learning a tractable class of soft temporal prob lems: theoretical and experimental results. AI Communications , 20(3), 2007. 9. E. L amma, P . Mello, M. Milano, R. Cucchiara, M. Gavane lli, and M. P iccardi. C onstraint propagation and value acquisition: Why we should do it interactively . In IJCAI , pages 468– 477, 1999. 10. N. W ilson, D. Gri mes, and E. C. F reuder . A cost-based model and algorithms for interleaving solving and elicitation of csps. In Proc. CP’07 , volume 4741 of L NCS , pages 666–680. Springer , 2007. idef2(c) = 0.7 idef2(m) = 0.9 idef1(p)=0.8 idef1(sh) = ? D idef3(r, c) = 0.3 idef3(su, c) = ? idef3(b, c) = ? idef3(r, m) = ? idef3(b, m) = 0.2 idef3(su, m) = ? idef(p,m) = 0.7 idef(sh,c) = 0.8 idef(sh,m) = 0.1 idef(p, c) = ? T A 0 20 40 60 80 100 10 20 30 40 50 60 70 80 elicited preferences density 0 200 400 600 800 1000 1200 1400 1600 1800 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 time (msec) incompleteness