Nogood Recording for Valued Constraint Satisfaction Problems.

Transcription

1 Nogood Recording for Valued Constraint Satisfaction Problems. Pierre DAGO and Gérard VERFAIIE ONERA-CERT 2, avenue Edouard Belin, BP Toulouse Cedex, France fdago, Abstract In the frame of classical Constraint Satisfaction Problems (CSPs), the backtrack tree search, combined with learning methods, presents a double advantage : for static solving, it improves the search speed by avoiding redundant explorations; for dynamic solving (after a slight change of the problem), it reuses the previous searches to build a new solution quickly. Backtrack reasoning concludes the reject of certain combinatorial choices. Nogood Recording memorizes these choices in order not to reproduce. We aim to use NogoodRecording in the wider scope of the Valued CSP framework(vcsp) to enhance the branch and bound algorithm. Therefore, nogoods are used to increase the lower bound used by the branch and bound to prune the search. This issue leads to the definition of the Valued Nogoods and their use. This study focuses particularly on penalty and dynamic VCSPs which require special developments. However, our results give an extension of the Nogood Recording to the general VCSP framework. 1 Introduction. Complete solving algorithms of constraint satisfaction problems (CSPs) follow the frame of the basic backtrack tree search. Various mechanisms have enhanced this initial frame Forward-checking, Real Full ookahead[5], Backjumping[2, 7], Nogood Recording[2, 8]. Real problems often require an optimization capability when it is not possible to satisfy all the constraints, or when the constraints are preferential rather than absolute. The definition of valued constraint satisfaction problems (VCSPs)[3] and the corresponding complete solving tool[4, 3] based on the depth first branch and bound algorithm solves this point. The enhancements proposed for this basic algorithm (based on the comparison between an upper bound and a lower bound) focus on two points. The first consists in finding an upper bound as low as possible; related methods are heuristic and tend to construct solutions closer to the optimum earlier. The second issue is to produce a lower bound as hight as possible. Corresponding algorithm follow the principle of the lookahead[4, 3, 10]. The aim of this study is to construct a better upper bound using learning methods. More precisely, we are to extend the Nogood Recording in the scope of VCSPs. Section 2 is a recall of CSP and VCSP models. Sections 3 and 4 give a definition of the Valued Nogoods. Then, in the last two sections, we propose several ways to use them and discuss their performances for static and dynamic[8, 9] Penalty CSP[4] solving. 2 Classical CSP & valued CSP A CSP is a pair (V; C) where V is a set of variables and C a set of constraints. A finite set of values is associated with each variable. We denote x i either the value i of the variable x, or the assignment of the value i to the variable x (x = i). A constraint determines the forbidden combinations of values for a set of variables fx c1 ; x c2 ; : : : ; x cn g. It can be represented as : c = [(x c1 6= i 11 ) _ : : : _ (x cn 6= i n1 )] ^ [(x c1 6= i 12 ) _ : : : _ (x cn 6= i n2 )] : : : ^ [(x c1 6= i 1m ) _ : : : _ (x cn 6= i nm )] We note c = fx c1 ; x c2 ; : : : ; x cn g the set of variables involved in the constraint c, and C the set of variables involved in the constraints in the set C. We call partial assignment a subset of assigned variables A = fx k ik g. When all of the variables are assigned, the assignment is said to becomplete. We note à = fx k g the subset of variables which are assigned by A.

2 et A be a partial assignment and x be a variable in V? Ã, the assignment A [ fx i g is called extension of A on the variable x with the value x i. et A 0 be a partial assignment such that à 0 = V? Ã, A [ A 0 is called a complete extension of A. A solution of the CSP is a complete assignment that satisfies every constraint. A partial assignment A which satisfies all of the constraints it assigns (constraints of which all variables belong to the assignment) is defined as locally consistent : 8c 2 C= c Ã; A j= c. A partial assignment A such that there exists a complete extension of A, which is a solution, is said to be globally consistent. The VCSP framework defined in [3] does not aim at satisfying all constraints but at minimizing a valuation depending on the satisfaction of the constraints. More precisely, we need : a set of valuations E (to evaluate constraints and assignments), with a total order (to compare valuations), a minimum element? (expressing the satisfaction of a constraint or the consistency of an assignment) and a maximum element > (expressing an unacceptable violation). an application from C to E such that (c) expresses the importance of a constraint c. an operator to aggregate valuations, which is commutative, associative, monotonic with, and such that? be a neutrality, and > an absorbing element. So, the valuation of a complete assignment is the aggregation of the valuations of the unsatisfied constraints. A solution is a complete assignment whose valuation is lower than >. It is an optimal solution if there is no other solution with a lower valuation. We call the valuation of a problem the valuation of its optimal solutions. The local valuation of a partial assignment is the aggregation of the valuations of the constraints assigned by A ( c Ã) and violated (A j= :c): v(a) = (c). c2c=aj=:c We call global valuation of an assignment A the minimum of the valuations of the complete extensions of A. Because of the monotonicity of the aggregation operator, the local valuation of any partial assignment is less than or equal to its global valuation. A partial assignment is said to be locally (resp. globally) consistent if its local (resp. global) valuation is lower than >. Several classes of VCSPs belong to this frame. We can quote some of them : possibilistic VCSP, (E; ;?; >; ) = ([0; 1]; <; 0; 1; max). penalty VCSP, (E; ;?; >; ) = (N [ f+1g; <; 0; +1; +). Partial CSP[4] is the subset of penalty VCSP with 8c; (c) = 1. classical CSP, (E; ;?; >; ) =(f0; 1g; <; 0; 1; _). 0 means satisfaction, 1 violation. 3 Valued nogoods. 3.1 Definitions. Definition 3.1 A classical nogood is a partial assignment which is globally inconsistent. A valued nogood is a pair (A; v), where A is a partial assignment and v a valuation such that every complete extension of A has a valuation greater than or equal to v. (v is less than or equal to the global valuation of A). CSPs are peculiar instances of VCSPs and a classical nogood A corresponds to the valued nogood (A; >). We present some basic properties of valued nogoods. Properties If (A; v) is a nogood, v 0 v and A 0 A, then (A; v 0 ) is a nogood. (A 0 ; v) is a nogood. Proof : (3.0.1) The first point is obvious : if v is less than or equal to the global valuation of A, v 0 v too. For the second, let A 00 be a complete extension of A 0. It is also a complete extension of A, then v(a 00 ) v. So, some nogoods can be defined from others. We will focus on the most relevant nogoods, with a high valuation and based on a small number of variables. 3.2 Building nogoods. The aim of the two properties presented below is to build nogoods with the information collected during the search, as partial assignments are evaluated. Property 3.1 8A; (A; v(a)) is a valued nogood. Proof : (3.1) The local valuation of any partial assignment A is less than or equal to its global valuation. The next unites of a peculiar set of nogoods in order to produce a more relevant nogood. Property 3.2 et A be a partial assignment and x 62 à be a variable. et A 1,: : :,A n be all the possibles extensions of A with each possible value x 1,: : :,x n for x. If (A 1 ; v 1 ),: : :,(A n ; v n ) are nogoods, then (A; min(v i )) is a nogood. i

3 Proof : (3.2) et A 0 be a complete extension of A. A 0 is complete so a value x k is assigned to x. A 0 is therefore a complete extension (vi). of Ak = A [ fx k g. Then v(a 0 ) vk min i 4 Nogoods for penalty & dynamic VCSP. In classical CSPs, it is interesting to consider the nogoods that are produced as new constraints, and to include them in the problem. A tuple (or partial assignment) that is forbidden by a nogood is then added to the list of the forbidden tuples of the constraint that links the variables of the nogood (if necessary, a new constraint is created). Adding constraints induced by the CSP necessarily produces redundancies in the violations. This method can only be valid if the aggregation operator is idempotent (8v; v v = v). Otherwise, duplicating a constraint with a valuation v would come to replace its cost by v v. More generally, each redundancy can increase the assignment valuations and therefore change the problem definition. Now, the aggregation of valuations is all the more important in the non idempotent cases : a constraint or a nogood does not usually produce an inconsistency without being added to other constraints. Therefore, we tend to keep in nogoods relevant information about the constraints upon which they are based, in order to avoid any duplication of valuations. Moreover, as a nogood is a consequence of the CSP constraints, any relaxation of the CSP jeopardizes the collected information. We must be able to distinguish the nogoods which are still valid from those which are no longer justified. To take into account the dynamic problems, it is necessary to justify a nogood by the constraints it is based on. Therefore, two reasons lead us to give a justification to the nogoods : the ability to deal with dynamic CSPs, and the aggregation of nogoods with a non idempotent operator. It is then necessary to state a more complete definition of the valued nogoods. The following still takes place in the general scope of the VCSP, although it is only useful for dynamic or non idempotent problems (penalty VCSP for instance). 4.1 Definition. We note A #V the projection of a partial assignment A on the subset of variables V : A #V = fx i 2 A=x 2 Vg. We denote v C (A) the valuation of a partial assignment A on the problem restricted to the subset of constraints C : v C (A) = (c). Note that v C (A) = v C (A # C ). c2c=aj=:c Definition 4.1 A valued nogood is a triple (A; v; C), where A is a partial assignment, v a valuation and C a set of constraints (called justification ) such that v is less than or equal to the global valuation of A on the problem restricted to the constraints in C. (8A 0 complete extension of A, v v C (A 0 ).) A classical nogood A corresponds to the valued nogood (A; >; C). Properties (sorting) et (A; v; C) be a nogood, A 0 A, v 0 v and C 0 C. (A 0 ; v; C), (A; v 0 ; C) and (A; v; C 0 ) are nogoods. Proof : (4.0.1) The first two nogoods follow from properties applied to the problem restricted to C. The third follows from the monotonicity of : let A 00 be a complete extension of A, C C 0 =) v C 0(A 00 ) v C(A 00 ) v. This properties implicitly define the more relevant nogoods, based on few variables and constraints and with a high valuation. In practice, a compromise between these three criteria is necessary. The set of constraints justifying the nogood allows us to state an additional theorem to remove useless values from the forbidden assignment. Theorem 4.1 (projection) et (A; v; C) be a nogood. (A # C ; v; C) is a nogood. Proof : (4.1) et A 0 be a complete extension of A # C. et A 00 be the complete extension of A with A 0 #V?Ã. We have A00 = A # C 0 # C. Then v C(A 0 ) = v C(A 0 # C) = v C(A 00 ) = # C ) vc(a00 v. The following theorems are translations of the building properties 3.1 and 3.2 of the previous section. Theorem 4.2 (building) 8A; C, (A; v C (A); C) is a nogood. A corollary is that, if C is the set of constraints unsatisfied by A, (A; v(a); C) is a nogood. (v(a) = v C (A)) Proof : (4.2) This is a consequence of property 3.1 applied to the problem restricted to C Theorem 4.3 (building) et A be a partial assignment and x 62 Ã be a variable. et A 1,: : :,A n be all the possible extensions of A with each possible value x 1,: : :,x n for x. (A; min S (v i ); i i If (A 1 ; v 1 ; C 1 ),: : :,(A n ; v n ; C n ) are nogoods, then C i ) is a nogood. Proof : (4.3) Using property 4.0.1,we can state that(ak; vk; S C i) are nogoods. The result S is obtained by applying property 3.2 to the problem restricted to C i. i 4.2 Adding nogoods. The new definition of nogoods allows us to aggregate several nogoods in a new one, with a higher valuation. i

4 Property 4.1 (Adding) et A and A 0 be two subsets of the same partial assignment and, (A; v; C) and (A 0 ; v 0 ; C 0 ) be two nogoods such that C \C 0 = ;. Then (A[A 0 ; v v 0 ; C [ C 0 ) is a nogood. Proof : (4.1) et A 00 be a complete extension of A [ A 0. v C[C 0(A 00 ) = But C \ C 0 = ;, thus v C[C 0(A 00 ) = c2c[c 0 =A 00 j=:c c2c=a 00 j=:c (c)! (c) c2c 0 =A 00 j=:c! (c) v C[C 0(A 00 ) = v C(A 00 ) v C 0(A 00 ) v C[C 0(A 00 ) v v 0 If we are to add nogoods with overlapping justifications, it is necessary to subtract redundant valuations. Therefore, we define the equivalent of a subtraction operator. We must lay down an additional condition on the valuation structure : let (E; ;?; >; ) be a valuation structure, 8y x, a valuation z 2 E such that x = y z must exist. Definition 4.2 et (E; ;?; >; ) be a valid valuation structure, the subtraction operator is defined by : if y x, x y = min(z= x = y z): otherwise x y =?: Some properties are deduced from this definition : Properties et x and y be two valuations, (x y) y x. et x and y be two valuations, (x y) y x. is monotonic : let x, y and z be three valuations, if x y then x z y z. Proof : (4.1.1) The first point is obvious : if y x, (x (x y) y = x by definition. y) y = y x; else The second is also obvious : necessarily, y x y then, y, is the minimum valuation z such that by definition, (x y) z y = x y. (x y) y is therefore less than x. et us prove the third. If z y, y z =? and the property is satisfied. Otherwise, let umin = y z = min(u=u z = y). If x y, with the first property, (x z) z y. We have the choice between two cases, either (x z) z = y then x z umin = y z because umin is a lower bound, or (x z)z y = uminz and we conclude with the monotonicity of that x z y z. In peculiar cases : x y is equal to x if y x, and 0 otherwise in possibilistic VCSPs. x y = max(0; x? y) in penalty VCSPs. This definition allows us to state two last theorems. The first reduces the set of constraints in a justification. The most important objective is to remove duplicated constraints in the justifications of two nogoods in order to satisfy the hypothesis of the property 4.1 in any case; this is the second theorem. The first theorem would also be useful to remove constraints whose penalty have been decreased by a dynamic change. Theorem 4.4 (subtraction) et (A; v; C) be a nogood and C 0 be a subset of C. (A; v ; C? C 0 ) is a nogood. c2c 0 (c) Proof : (4.4) et A 0 be a complete extension of A. Because of the monotonicity? of the aggregation operator, (c) v c2c 0 C 0(A 0 ) ) v C?C 0(A 0 )? (c) v c2c 0 C?C 0(A 0 ) v C 0(A 0 ) ) v C?C 0(A 0 )? (c) v c2c C(A 0 ) v 0 Finally, properties4.1.1 lead to the conclusion v C?C 0(A 0 ) v (c). c2c 0 Theorem 4.5 (adding) et (A; v; C) and (A 0 ; v 0 ; C 0 ) be two nogoods. (A [ A 0 ; (v v 0 ) ; C [ C 0 ) is a nogood. c2c\c 0 (c) Proof : (4.5) Following theorem 4.4, (A 0 ; v 0 c2c\c 0 (c) ; C 0? (C \ C 0 )) is a nogood. Adding (A; v; C) with property 4.1 shows that 1 (A [ A 0 ;(v v 0 ) 5 Using nogoods. 5.1 A basic algorithm. (c) c2c\c 0 ; C [ C 0 ) is a nogood. We describe here, for the purpose of explanation, a basic way to use valued nogoods during a branch and bound tree search. Three functions are useful : the first, Evaluate, computes a lower bound on the valuations of the complete extensions of a partial assignment and returns the corresponding nogood. The second, Union, unites nogoods when the failure of all the values of a variable occurs. The third, Store, records relevant nogoods. et (V; C) be a VCSP, and an upper bound initially set to >. A generic algorithm is given by function STEP (figure 1) called with A = ; STEP(;) returns a nogood (;; v min ; C min ) where v min is the problem valuation, and C min is a sufficient subset of constraints which implies that no complete assignment 1 (v 0 v 00 ) v 00 v 0 ) v (v 0 v 00 ) v 00 (v v 0 ) ) v (v 0 v 00 ) (v v 0 ) v 00.

5 STEP(A) If à = V then A is a solution with a valuation lower than. v(a) return (A; v(a); C) Else et x be variable in V? Ã, N = ; For each possible value i for x et n Evaluate(A [ fx i g) If the valuation of n then N N [ fng Else N N [ STEP(A [ fx i g) et n union Union(N ) Store(A; n union) return n union Stored Nogoods Tree-search diagram Evaluation (lower bound < upper bound) Evaluation (lower bound upper bound) Nogood returned from Evaluation Nogoods Union and Recording Nogood returned from Union Figure 1. Algorithm. has a valuation less than v min (the optimal valuation of the problem restricted to the constraints in C min is also v min ). Evaluate(A) aims to produce a nogood based on the assignment A. This nogood being used to prune the search, the aim is to maximize its valuation (the nogood gives a lower bound of A s global valuation. If this valuation is not acceptable ( ), it is useless to develop the branch). Evaluate is based on the backward checking and builds a nogood V = (A; v(a); C), using theorem 4.2. This nogood V corresponds to the lower bound used by an ordinary branch and bound search. In order to improve it, Evaluate checks whether there exists a stored nogood (A i ; v i ; C i ) which is appropriate (A i A) and whose valuation is greater than v(a). The nogood with the highest valuation is returned. The aim of the function Union(N ) is to synthesize the information contained in a set of nogoods. It takes place when all the possible values for a variable have been rejected. Then, we have a set of nogoods N. Each one gives a lower bound of the global valuation of the extension of the current assignment with a value of the explored variable. Then, the function produces a new nogood with the building theorem 4.3. Store(A; n) records a nogood n built during the attempt to extend the assignment A. When a nogood is produced, all the variables in the current partial assignment are involved in the assignment of the nogood. As a tree search never explores the same partial assignment twice, we must reduce the assignment of the nogood. The most natural way to do this, used in classical nogood recording, is to use the projection theorem 4.1. It does not change the valuation nor the justification of the nogood but removes useless values. Then, if some values have been eliminated, the nogood will be stored Example. X1 a b c X2 a b c C1 : X1>X3 C1 : X2>X3 X3 a b C1 : X4>X3 C1 : X5>X3 Figure 2. Example. X4 a b C1 : X5>X4 X5 a b Consider the problem in figure 2. Figure 3 illustrate the tree explored by a standard branch and bound and by our basic algorithm (in grey bold). We also represented the nogood stored during the search (in bold those s that are useful to prune the search). The production of the first interesting nogood occurs while trying to extend the partial assignment A = fx 1 b ; X3 a ; X4 a g. The extension with X 5 a and X 5 b returns the nogoods (fx 1 b ; X3 a ; X4 a ; X5 a g; 4; fc 1 ; C 3 ;

6 {X1=a,X3=a},2,{C1,C3,C4,C5} {X1=a},2,{C1,C3,C4,C5} {X3=a},1,{C3,C4,C5} {X2=a},1,{C2} {X1=a,X3=b},2,{C1,C3} {X2=a,X3=a},2,{C1,C3,C4,C5} {X1=b},1,{C1,C3,C4,C5} {X2=b},1,{C2,C3,C4,C5},1,{C3,C4,C5} {x3=b},1,{c3} {x3=a},1,{c3,c4,c5} a a b c a b c a b c a b a b a b a b a b a b a b a b a b b,1,{c3,c4,c5} c {x3=b},1,{c3} X1 X2 X3 C1 : X3<X1 C2 : X3<X2 {x3=a,x4=b},1,{c4,c5} a b a b a b a b a b a b a b a b a b a b a b X4 C3 : X4>X3 a b a b a b a b a b a b a b a b a b a b a b {X1=a,X3=a, X4=a},2,{C1,C3,C4,C5} {X1=a,X3=a, X4=b},2,{C1,C4,C5} {X2=a,X3=a, X4=b},2,{C1,C4,C5} {X3=a, X4=b},1,{C4,C5} X5 C4 : X5>X3 C5 : X5>X4 α Relevant Nogood Recording Basic Nogood Recording Branch & Bound Figure 3. Tree search. C 4 ; C 5 g) and (fx 1 b ; X3 a ; X4 a ; X b 5 g; 2; fc 1 ; C 3 g). Their union (fx 1 b ; X3 a ; X4 a g; 2; fc 1 ; C 3 ; C 4 ; C 5 g) is then carried out. The projection of this nogood shows that X 2 is not involved in the justification. This value is therefore removed and the resulting nogood stored. The projection has the same effect when backtracking on X 3 and a new nogood (fx 1 a ; X3 a g; 2; fc 1 ; C 3 ; C 4 ; C 5 g) is stored; this one will be useful. While trying to extend fx 1 c ; X3 a g the backward checking gives a lower bound of 1. But the nogood n can be used and gives a higher lower bound, high enough to prune the search. There are two main criticisms of this basic version : Although several nogoods are produced, only a few of them are useful. Even after a projection, the assignment is much too big to be reproduced during the search. The number of nogoods grows (in the worst case) exponentially with the size of the problem. A practical algorithm will have to elect the nogoods that should be stored. 5.2 Improvements. The nogoods, as they are built into the basic algorithm, give a lower bound of the subtree extending the current assignment. But the backward checking (the nogood V ) already provides a part of this information. In other words, the nogoods can be redundant with the definition of the problem and learn some information obviously accessible with a simple backward checking. For instance, the first nogood stored, (fx 1 a ; X3 a ; X4 a g; 2; fc 1 ; C 3 ; C 4 ; C 5 g) is redundant with the constraint C 1 : if the nogood can be used, then, at least, X 1 a and X 3 a are included in the partial assignment. Therefore, it is obvious to see that the constraint C 1 is not satisfied. The only information in the nogood that should be stored is the over-cost of 1 justified by the other constraints. The functions are then redefined as follow : Store(A; n) - let C be the set of constraints in the justification of n violated by the assignment A (given by the backward checking). - subtract C from n and project (theorems 4.4 and 4.1) Evaluate(A) - construct V = (A; v(a); C), C being the set of constraints that appear violated with a backward checking on A. - add V to every stored nogood (theorem 4.5). - return the nogood with the highest valuation. This operation (subtracting and adding) does not lose any information while reducing the justification. The projection is then more efficient, thus the nogoods more useful. et us reconsider the example and figure 3. a a a While exploring the partial assignment fx 1 ; X2 ; X3 ; X b a a a 4 g with a current nogood V = (fx 1 ; X2 ; X3 ; b X4 g; 2; fc 1 ; C 2 g), the extensions with X 5 return the nogood a a a (fx 1 ; X2 ; X3 ; b X4 3; fc 1 ; C 2 ; C 4 ; C 5 g). The violations of C 1 and C 2 are already included in V. their subtraction leads to the smaller nogood (fx 1 ; X2 ; X3 ; b X4 g; 1; a a a fc 4 ; C 5 g) that can be reduced by projection and then stored. The number of nogoods stored is still exponential. As we

7 do not know how to select the interesting nogoods nor how to avoid redundancies in the set of stored nogoods, we will apply the strategy used in the classical Nogood Recording : the nogoods will not be stored if its arity (cardinality of its assignment) exceed a fixed constant. 6 Experiments. 6.1 Results. Experiments have been carried out with several bounds on the arity of the recorded nogoods : 0, 2, 4 and 6 (corresponding to NR 0, NR 2,...). A comparison is made with the standard branch and bound algorithm ( B & B ). The algorithm uses the enhancement of the conflict directed backjumping because the nogoods built all the information useful to know where to backtrack. Although unnecessary, the algorithms keep a static order on the variables (less domains first) and their values. We measure their capability to record information for a dynamic use by a second run that keeps the nogood recorded but do not store any more nogoods. Results show the number of nodes visited and the run time needed to reach an optimal solution and to prove its optimality. It produces relevant information about the search effort and the overhead of the algorithm. The number n of nogood stored is also given Famous problems We first present results on standard problems[6] (Zebra, SEND+MORE=MONEY) considered as Partial CSPs : the aim is to minimize the number of constraints unsatisfied. When the time limit of 1800s is exceeded, a sign # is used, followed by the distance between the best valuation obtained and the optimum (instead of the CPU time). In this case, the node number counts correspond to the number of nodes necessary to reach the best valuation found. Zebra algorithm Static run Dynamic run time nodes n time nodes B & B 62s NR 2 14s s NR 4 24s s NR 6 46s s Send algorithm Static run Dynamic run time nodes n time nodes B & B 318s NR 2 576s s NR 4 # 1 # >1000 # 1 # Random problems The following experiments use random binary Partial CSPs, with 20 variables and a random number of possible values between 2 and 8 for each variable. The connectivity of the constraint graph, and the tightness of the constraints (ratio between the number of forbidden pairs of values and the number of possible pairs) are set at 40%. A logarithmic mean (due to the exponential behavior of CSP algorithms) over 90 samples is computed. Random VCSPs algorithm Static run Dynamic run time nodes n time nodes B & B 15s NR 2 24s s NR 4 25s s NR 6 47s s Real problems We now present results on the daily management problems for an earth observation satellite[1, 6] : given a set of photographs of various importance (expressed by a weight) which can be taken the next day; given material constraints (non overlapping between two successive photographs on the same instrument and data flow limitation); the aim is to select a feasible subset of photographs which maximizes the sum of their weights. These problems can be cast as penalty VCSPs : a variable is associated to each photograph; its domain contains one value for each mean of achieving the photograph and one value to express that the photograph is not selected. Then, a unary constraint is set to forbid the rejection value with a valuation equal to the weight of the photograph. The material constraints are translated into binary and ternary imperative constraints (valuation equal to >). We ran the branch and bound algorithm and our nogood recording (bound on the arity of the nogoods equal to 0) on 5 problems of 100 variables. The following table shows only the time needed to reach an optimal solution and to prove it s optimality, and the average number of nogoods produced. When the time limit of 1800s is exceeded, a sign # is used, followed by the distance between the best valuation obtained and the optimum. Satellite scheduling problems algorithm problem number n B & B # 2 # 4 # 3 # 0 # 2 NR 0 5.4s 3.1s 15.2s 1.7s 4.4s 57

8 6.2 Discussion The results on random VCSPs show that the nogood recording is penalised by an important overhead. In producing very few nogoods, the search time is multiplied by 2. The second part of the overhead is directly related to the maximum arity of the nogoods it stores : as the latter grows, the number of stored nogoods increases and as does the time necessary to make the additions and the comparisons in the function Evaluate. The reduction of the search space can be significant, particularly on dynamic runs, and also increases with the number of stored nogoods, but not always enough to save time. It appears from these experiments alone that the selection of relevant nogoods (or the efficient use of a large number of nogoods) can be a very important issue. However, on non random problems, the nogood recording algorithm seems to be able to find and store interesting nogoods. By comparison with random experiments, the stored nogoods are much more useful, with a comparable number of nogoods memorized. Nogood recording matches exceptionally well the satellite scheduling problems, probably because the structure of its constraint graph (chain like) offers very interesting nogoods with a small arity. The Zebra problem contains a set of nogoods of small arity which corresponds to an interesting compromise between their quantity and their usefulness. On the contrary, for the Cross problem, when the number of nogoods is large enough to imply a significant pruning, it also destructs the CPU time efficiency. It follows from those experiments that the nogood recording efficiently increases the lower bound of a branch and bound, particularly for dynamic runs. However, with a time criterion, it can be very inefficient if the nogood selection do not match the problem (or when there are no useful nogoods). References [1] J. Agnese, N. Bataille, E. Bensana, D. Blumstein, and G. Verfaillie. Exact and approximate methods for the daily management of an earth observation satellite. In Based Systems for Space, [2] R. Dechter. Enhancement schemes for constraint processing : Backjumping, learning and cutset decomposition. Artificial Intelligence, 41(3): , [3] H. Fargier, T. Schiex, and G. Verfaillie. Valued constraint satisfaction problems. In IJCAI 95, pages , [4] E. Freuder and R. Wallace. Partial constraint satisfaction. Artificial Intelligence, 58:21 70, [5] R. Haralick and G. Elliot. Increasing tree search efficiency for constraint satisfaction problems. Artificial Intelligence, 14(3): , [6] //ftp.cert.fr/pub/ftp/pub/lemaitre/lvcsp/pbs. ftp site. [7] P. Prosser. Hybrid algorithms for the constraint satisfaction problems. Computational Intelligence, 9(3): , [8] T. Schiex and G. Verfaillie. Nogood recording for static and dynamic constraint satisfaction problems. International Journal on Artificial Intelligence Tools, 3(2): , [9] G. Verfaillie and T. Schiex. Maintien de solution dans les problèmes dynamiques de satisfaction de contraintes : bilan de quelques approches. Revue d Intelligence Artificielle, 95. [10] R. Wallace. Enhancements of branch and bound methods for the maximal constraint satisfaction problem. CP 95 workshop on Over Constrained System, sep Cassis, France. 6.3 Conclusion We have showed the feasibility of Nogood Recording in the VCSP scope. The algorithms we have proposed lead to interesting improvements for static or dynamic Partial CSP solving. They represent a few possibilities among the various ways of using valued nogoods as they are only partial answers to the questions : how to extract the best lower bound from a set of nogoods and how to select nogoods that should be recorded within reasonable time overhead and memory space use. The concept of valued nogoods paves the way for the extensions of all the classical enhancements based on the adjunction of deduced constraints (Path-consistency, Dynamic Backtracking, etc.).