Unifying and extending hybrid tractable classes of CSPs


 Griselda Jones
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1 Journal of Experimental & Theoretical Artificial Intelligence Vol. 00, No. 00, MonthMonth 200x, 1 16 Unifying and extending hybrid tractable classes of CSPs Wady Naanaa Faculty of sciences, University of Monastir, Tunisia (Received 00 Month 200x; In final form 00 Month 200x) Finding a solution to a Constraint Satisfaction Problem (CSP) is known to be an NPcomplete task. Much work have been concerned with identifying tractable classes of CSPs. Tractability is obtained by imposing specific problem structures, specific constraint relations or both. A tractable CSP class whose tractability is due to both structural and relational properties is said to be hybrid. In this paper, we present a hybrid tractable CSP class that brings together and generalizes many known hybrid tractable CSPs. The proposed class is characterized by means of simple but powerful notions from set theory. 1 Introduction Constraint satisfaction problems (CSP) is a suitable framework for modeling many artificial intelligence problems. A CSP is defined by a set of variables and a set of constraints over these variables. Each variable is associated with a domain containing the values that can be assigned to that variable. A solution is an assignment of a value to every variable that satisfies all the constraints. When a problem has, at least, one solution it is said to be consistent. Finding a solution to a CSP or proving that none exists is known to be an NPcomplete task. Nevertheless, constraint solvers, that are programs especially dedicated to CSP solving, use polynomial filtering subroutines that can introduce significant simplifications on the problems to be solved without changing their solution sets. The goal of such subroutines is to establish a limited form of (local) consistency that may, in some cases, guarantee the consistency of the whole problem. Problems that can be solved by simply establishing limited local consistencies are tractable, i.e., they can be solved by polynomial algorithms. Tractable problems can be arranged in tractable classes based on the specific properties that make them tractable. There are two main types of problem tractability: structural tractability and relational tractability. Structural tractability is obtained by restricting the hypergraph formed by the scopes of the constraints to have some specific feature. The class of problems whose underlying hypergraphs are hypertrees are certainly the most known structural tractable class (Dechter and Pearl, 1987). Relational tractability is obtained by restricting the allowable constraint relations. The CSP class involving maxclosed relations is one of the already identified relational tractable classes (Jeavons and Cooper, 1995). More recent works have studied a new kind of tractability, called hybrid tractability, which simultaneously exploits both structural and relational properties. This type of tractability allows to derive new tractable problems that are neither structural nor relational. As yet there is no unifying theory which provides a precise description of the hybrid tractable classes. Thus, this work, which is intended to unify and generalize many known hybrid tractable CSP classes. The paper is organized as follows: the next section introduces some definitions and notations related to the CSP Corresponding author. Journal of Experimental & Theoretical Artificial Intelligence ISSN X print/ ISSN online c 2006 Taylor & Francis Ltd DOI: / xxxxxxxxxxxx
2 2 framework. Section 3 is devoted to the set theoretic notions underlying our approach. The main notion proposed in this paper, namely, the rank of a CSP is detailed in Section 4. The rank of the dual of a CSP is presented in the same section. In Section 5, we present the derived notion of directional rank. In section 6, we show that the proposed CSP class is related to many known tractable CSP classes. Finally, Section 7 is a brief conclusion. 2 Definitions and Notations Definition 2.1 A constraint satisfaction problem (CSP) is defined by a triple (X, D, C) where: X is a finite set of variables. D is a set of value domains containing a value domain for each variable. C is a set of constraints. Every constraint is a pair (σ, ρ) where σ X is the scope of the constraint and ρ is a σ ary relation specifying the σ tuples permitted by the constraint. The arity of a constraint is the size of its scope. The arity of a problem is the maximum arity over its constraints. A binary CSP is a CSP having arity two. A variable x X should be assigned a value only from its domain, which will be denoted by D x. A partial instantiation is an assignment of values to some of the variables. A unary instantiation is a partial instantiation that assigns a value to a single variable while a complete instantiation is one which assigns a value to every variable of the CSP. The scope of instantiation t, denoted by σ(t), is the subset of variables instantiated by t. An instantiation t satisfies a constraint (σ, ρ) if and only if σ σ(t) and the tuple t σ is in ρ, where t σ denotes the projection of t over the variables of σ. An instantiation t is consistent if and only if it satisfies all the constraints whose scopes are subsets of σ(t). A solution is a complete instantiation that satisfies all the constraints. If a problem has, at least, one solution, it is said to be consistent otherwise it is inconsistent. The constraint satisfaction problem is generally NPcomplete, which means that there is no polynomial algorithm that ensures solution finding, unless P=NP. One of the existing means for accelerating problem solving consists in removing some combinations of values whose removal does not affect the solution set of the problem. The goal of such removals is to achieve a limited form of consistency, called local consistency, which allows an easy calculation of consistent instantiations of a certain size. There are many levels of local consistency; they can be distinguished by a single parameter as follows (Freuder, 1982): Let P = (X, D, C) be a CSP instance and let k, 1 k X be an integer. A consistent instantiation t of any k 1 variables of P is said to be kconsistent relative to an additional variable x if and only if there exists v D x such that the tuple t, (x, v) is consistent, where (x, v) denotes a unary instantiation. The whole problem is said to be kconsistent if and only if every consistent instantiation of any k 1 variables is kconsistent relative to every additional variable. If P is kconsistent for some k, 1 k X then it is said to be locally consistent and if it is iconsistent for all i : 1,..., k it is said to be strong kconsistent. Finally, if P is strong X consistent then it is called globally consistent. The most used levels of local consistency are arc and pathconsistencies, which stand respectively for 2 and 3consistency. Enforcing kconsistency, for some k, on P is achieved by removing some value combinations from the domains and/or the relations defining the constraints. This may introduce new (k 1)ary constraints and then the structure of the resulting CSP, which is called the kconsistency closure of P, may change. The time complexity of enforcing kconsistency is O(n k d k ) where n is the number of variables and d is the size
3 3 of the largest value domain (Cooper, 1989). In practice, the process of establishing kconsistency can be lightened by considering a weak form of kconsistency called directional kconsistency (Dechter and Pearl, 1987). Let P = (X, D, C) be a CSP, whose variables are totally ordered by and let k, 1 k X be an integer. P is directional kconsistent relative to if and only if every consistent instantiation t of any k 1 variables is k consistent relative to every variable that succeeds, w.r.t., all the variables instantiated by t. P is strong directional kconsistent if and only if it is directional iconsistent for all i : 1,..., k. As mentioned above, directional kconsistency is a weak form of (full) kconsistency in the sense that k consistency entails directional kconsistency but the converse is not true. 3 Independent sets In set theory, two sets are said to be independent if none of them is a subset of the other, or equivalently, if their intersection is strictly included in both of them. We can generalize this notion to cope with many sets at a time as follows Definition 3.1 Let E be a finite set and let {E i } i I be a finite family of subsets of E. The family {E i } i I is said to be independent if and only if for all J I J I j J E j i I E i (1) {E i } i I is said to be dependent otherwise. We assume that j E j = E. It is easy to verify that, for any finite family of finite subsets, if the two strict inclusions in (1) are replaced by large ones, the resulting implication holds trivially. Hence, an equivalent definition of dependence stating that a finite family of finite subsets {E i } i I is dependent if and only if J I such that j J E j = i I E i (2) Note that any finite family of subsets of a finite set E that contains E itself is necessarily dependent. Example 3.2 Figure 1 depicts three families of subsets, each of which is composed of three subsets and all subsets are built from a set of three elements (represented by dots). By referring to (1) or (2), one can deduce that the two families of subsets in Figure 1(a) and (b) are dependent while the one depicted in Figure 1(c) is independent. This is because any two subsets of the family of Figure 1(c) have a nonempty intersection, while the intersection of the three subsets is empty. In fact, the latter family of subsets is the only independent family composed of three subsets that can be built from a set of three elements. Indeed, by observing that the area corresponding to the union of the three subsets of Figure 1(c) is partitioned into seven regions, one can verify that if any of the elements is taken out of the region where it is, the resulting family of subsets will be dependent. PROPOSITION 3.3 If {E i } i I is an independent family of subsets of a finite set E, then {E j } j J is independent for all J I. Proof If J = I then the statement holds trivially. Then assume that there exists J I such that {E j } j J is dependent and proceed to get a contradiction. Since {E j } j J is dependent there must exit J J such that j J E j = j J E j. It follows that
4 4 (a) (b) (c) Figure 1. Three families of subsets built from a set having three elements. Only the third family, the one on the right of the figure, is independent. E i = i I E i i I\J j J = E i = i I\J i (I\J) J E i E j j J E j which implies that {E i } i I is dependent, since (I \ J) J I. Thus a contradiction. Proposition 3.3 suggests that determining whether a finite family of subsets {E i } i I of a finite set E is independent or not can be done by checking statement (1) only for J I such that J = I 1. This can be achieved in O( I 2 E ) steps by computing O( I ) intersections, involving each O( I ) subsets of a set containing O( E ) elements. This supposes, however, that the elements of E are encoded as integers in {0,..., E 1}. PROPOSITION 3.4 Let E be a finite set and let {E i } i I be a finite family of subsets of E. If {E i } i I is independent then I min i I E i + 1 (3) Proof Assume, without loss of generality, that I = {1,..., I }. Since {E i } i I is independent, we must have i I\{j} E i i I E i, j : 1,..., I Then, there must exist e kj This is equivalent to E such that e kj i I\{j} E i e kj / i I E i j : 1,..., I e kj i I\{j} E i e kj / E j j : 1,..., I (4) For (4) to hold, all the e kj, j : 1,..., I must be different. It follows that I 1 E i for all i, 1 i I. In particular, I 1 min i I E i. Hence the result. A consequence of Proposition 3.4 is that for any independent family of finite subsets {E i } i I, we have I E (5)
5 5 Because min i I E i + 1 cannot exceed E except if E i = E, for all i I. But in this latter case, {E i } i I will be dependent. 4 Ranked CSPs Let P = (X, D, C) be a CSP instance and let (σ, ρ) C be a rary constraint whose scope σ contains a variable x and let t be any instantiation of the r 1 remaining variables of σ. Denote by ρ x (t) the subset of D x that extends t to form a tuple of ρ, that is ρ x (t) = {v D x t, (x, v) ρ} (6) The subset ρ x (t) is called the extension of tuple t to variable x w.r.t. relation ρ. In what follows, I will designate a subset of the constraint index set, that is, I {1,..., C }. Let {(σ i, ρ i )} i I be a subset of constraints all having a variable x in their scopes and let {t i } i I be a set of tuples with t i being an instantiation of the variables in σ i \ {x}. The set {ρ x i (t i)} i I is, therefore, a family of extentions to x. Such a family of extentions is said to be consistent if and only if the tuple i I t i is consistent, where denotes the natural join operator. On the other hand, since {ρ x i (t i)} i I is a finite family of subsets of D x, it can either be dependent or independent depending on whether or not it verifies (1). Using notation {ρ x i (t i)} i I to designate a family of extensions assumes that the ρ i s are all different since I is a set and not a multiset. This is not a limitation, because only consistent and independent families of extensions are relevant to our approach, while allowing the same relation to appear more than once yields an inconsistent or a dependent family of extensions. With some abuse of terminology, we define the rank of a variable in a CSP as follows. Definition 4.1 Let P be a CSP and let x be a variable of P. The rank of x is defined as the size of the largest independent and consistent family of extensions to x. Definition 4.2 The rank of a CSP is the maximum rank over all its variables. Example 4.3 Consider the CSP depicted in Figure 2, where all variables are assumed to have the same domain, which is {1, 2, 3}. The instance contains five binary constraints: Those whose scopes are {x 1, x 2 }, {x 1, x 3 } and {x 3, x 4 } are specified by relation ρ = {(1, 2), (1, 3), (2, 1), (3, 1)}, while the remaining two are defined by ϱ = {(1, 1), (2, 2), (2, 3), (3, 3)}. By examining the ordered value pairs contained in ρ and ϱ, one can see that the various extensions involved in this example can be one of the following subsets: {1}, {2}, {3} and {2, 3}. The extensions, ρ x1 (x 2, 2), ϱ x2 (x 3, 2), ϱ x3 (x 2, 3) and ρ x2 (x 1, 1), for instance, are respectively equal to {1}, {2}, {3} and {2, 3}. As it has been already outlined in Example 3.2, these subsets cannot form an independent family of extensions with more than two members. Thus, the problem has rank two or less. On the other hand, ρ x1 (x 2, 1) and ϱ x1 (x 4, 1), which are two extensions to variable x 1, are independent since they are equal to {2, 3} and {1}, respectively. Furthermore, (x 2, 1), (x 4, 1) is consistent since there is no constraint between x 2 and x 4. Thus, {ρ x1 (x 2, 1), ϱ x1 (x 3, 1)} is a consistent and independent family of extensions to x 1. The problem has, therefore, rank two. Next, we show that the class of CSP with bounded arity and bounded rank can be polynomially identified. Assume that r and κ, which stand respectively for the arity and the rank of the CSP are constants. According to Definition 4.2, determining the rank of a CSP requires the computation of the rank of each of its variables. The rank of the whole CSP is, therefore, obtained by taking the maximum over these ranks. According to Definition 4.1, determining the rank of a variable x requires determining the size of the largest independent and consistent family of extensions to x. This can be done by looking for independent and consistent families of
6 6 x ρ x o ρ o x 4 ρ x 3 3 ρ o Figure 2. The constraint graph of a binary CSP on four variables (left) and the relations defining the constraints (middle and right). extensions to x with increasing sizes. Thus, one can proceed by looking for an independent and consistent family of extensions to x whose size is s = 1. If such a family exists then the rank of the CSP instance is, at least, one and the process continues by looking for a consistent and independent family of extensions to x whose size is s = 2. If the search is successful again then the rank of the CSP is, at least, two, and so on. Hence, if the rank of x is κ x, it is necessary to repeat the process κ x + 1 times. The goal of the last iteration, i.e. s = κ x + 1, is to prove that all the consistent families of extensions to x of size κ x + 1 are dependent. The number of families of extensions to x involving s distinct constraints is, in the worst case, equal to ( e s) d (r 1)s, where e is the number of constraints and d is the size of the largest value domain. Determining whether an extension set of size s is independent or not can be achieved in s 2 d steps. Thus, the overall number of operations needed to compute the rank of a variable is bounded by d κ+1 s=1 s2( e s) d (r 1)s yielding a time complexity of O(e κ+1 d (r 1)(κ+1)+1 ) for a single variable and then an overall complexity of O(ne κ+1 d (r 1)(κ+1)+1 ) for the whole CSP. Because of the assumptions on the arity and the rank of the problems studied, the latter function is polynomial. Hence, the class of CSP whose arity and rank are both bounded by a constant can be identified in polynomial time. The main theoretical result of this paper relates the level of local consistency ensuring global consistency to the rank of the considered CSP. THEOREM 4.4 Let P be a rary CSP instance whose rank is κ. If P is strongly (κ(r 1) + 1)consistent, then it is globally consistent. Proof We assume that P is strongly (κ(r 1) + k)consistent for any k 1 and we prove that it is also strongly (κ(r 1) + k + 1)consistent. We must, therefore, show that any (κ(r 1) + k)ary consistent instantiation t of P can be extended with a unary instantiation of an additional variable, say x. Denote by I the subset of {1,..., C } such that σ i contains x and t instantiates all the variables in σ i \ {x} for all i I and denote by t i the projection of t on σ i \ {x}. At this stage, we distinguish two cases: (i) I κ. Let t = i I t i. We, therefore, have t = i I σ i \ {x}, where t denotes the number of variables instantiated by t. We obtain t I (r 1) since x σ i and σ i r for all i I. It follows that t κ(r 1)+k 1 since I κ and k 1. On the other hand, P is assumed to be strongly (κ(r 1) + k)consistent. Then, t can be consistently extended with a unary instantiation of variable x, say (x, v), yielding the consistent instantiation t, (x, v). Furthermore, the remaining variables instantiated by t, i.e, the variables in σ(t) \ σ(t ) cannot form, with some of the variables in σ(t ) {x}, the complete scope of a constraint wich includes x, because otherwise the definition of I will be violated. It follows that t, (x, v) is also a (κ(r 1) + k + 1)consistent instantiation of P. (ii) I > κ. First, notice that {ρ x i (t i)} i I is a consistent family of extensions to x since t i t, for all i I and t is consistent. On the other hand, P has rank κ, so every subset of {ρ x i (t i)} i I whose size exceeds κ is dependent. So, there must exist J I, J κ such that
7 7 j J ρ x j (t j ) = i I ρ x i (t i ) This means that the constraints of {(σ i, ρ i )} i I do not rule out any value from D x which is not ruled out by the constraints of {(σ j, ρ j )} j J given the partial instantiation t. Thus, in the context of instantiation t, the constraints of {(σ i, ρ i )} i I\J are irrelevant and can therefore be removed from the set of relations constraining variable x. Thus, J κ constraints constraining x are left, in which case we return to (i) to deduce that t can be consistently extended to variable x. In what follows, we show that the rank of a CSP does not increase if the value domains of the CSP are reduced. Definition 4.5 A CSP P = (X, D, C ) is said to be a reduction of a CSP P = (X, D, C) if and only if X = X D x D x for every x X C = {(σ, ρ ) (σ, ρ) C ρ = ρ x σ D x}. where denotes the nary Cartesian product operator. PROPOSITION 4.6 The rank of a CSP cannot be smaller than that of one of its reductions. Proof Given a CSP instance P = (X, D, C) and one of its reductions P = (X, D, C ), we first establish that, for all (σ, ρ ) C, we have Notice that t is also in ρ (σ \ {x}) since ρ ρ. ρ x (t) = ρ x (t) D x, for all t ρ (σ \ {x}) (7) Let t be in ρ (σ \ {x}). Since ρ (σ \ {x}) y σ\{x} D y we obtain t y σ\{x} D y and then t, (x, v) y σ D y, for all v D x. It follows that ρ x (t) = {v D x t, (x, v) ρ } = {v D x t, (x, v) ρ y σ D y} = {v D x t, (x, v) ρ} = {v D x t, (x, v) ρ} D x = ρ x (t) D x Thus, (7) holds.
8 8 Next, assume that κ, the rank of P, is greater than κ and proceed to get a contradiction. This assumption implies that there must exist x X such that κ x > κ, and then, there must exist an independent family of extensions to x in P whose size is κ x. Denote by {ρ x i (t i)} i I such a family of extensions. By (1), we must have J I, j J ρ x j (t j ) ρ x i (t i ) i I Since ρ x i (t i) = ρ x i (t i) D x, i : 1,..., κ x, it follows that J I, j J ρ x j (t j ) D x ( ) ρ x i (t i ) D x i I and since trivially holds, we precisely have J I, J I, ρ x j (t j ) ρ x i (t i ) j J i I ρ x j (t j ) ρ x i (t i ) j J i I This implies that {ρ x i (ti)} i I is an independent and consistent family of extensions of P whose size, κ x, is greater than κ, the rank of P. Thus, a contradiction. Finally, it is easy to see that the reduction of a CSP instance can have a much smaller rank than that of the original instance. This can occur, for instance, when the reduction is obtained by keeping a single value in the domain of every variable. In which case and in accordance with (5) the rank of the resulting reduction is one whatever the rank of the original instance is. In what follows, we study the rank of the dual of a CSP. The dual of a (primal) CSP is a binary CSP, where a variable is associated to every constraint of the primal problem. The value domain of each of these variables ranges over all tuples permitted by the corresponding constraint of the primal problem. Whenever a pair of constraints in the primal problem have some common variables in their scopes, the corresponding pair of variables in the dual problem must be related by a binary constraint. Such a binary constraint enforces the shared variables to be assigned to the same values. More precisely, in the dual problem, a variable is associated with every constraint c = (σ, ρ) of the primal problem. The value domain of this dual variable is ρ. If in the primal problem, there is a pair of constraints c = (σ, ρ) and c = (σ, ρ ) such that σ and σ share some variables, then the dual problem contains the binary constraint ( c, c, ϱ), where ϱ = {(t, t ) ρ ρ t σ σ = t σ σ } (8) All the constraints involved in a dual problem are defined through the specific relation given by (8). It follows that the expression of the extensions associated with the dual constraints simplifies to ϱ c (t) = {t ρ t σ σ = t σ σ } (9) Clearly, if we have a solution to the dual problem, we can easily deduce a solution to the primal one by selecting the values appearing in the tuples assigned to the dual variables.
9 9 THEOREM 4.7 The dual of a rary CSP has rank r or less and this bound is tight. Proof We show that every independent and consistent family of extensions in the dual of a rary CSP has r members or less. Let {c i = (σ i, ρ i )} i I be a subset of primal constraints all sharing some variables with a supplementary constraint c = (σ, ρ). In the dual problem, we, therefore, obtain the dual constraints ({c i, c}, ϱ i ), i I. Note that all these constraints share the dual variable c. Assume, for simplicity, that I = {1,..., I }. First, we show that if {ϱ c i (t i)} i I is an independent and consistent family of extensions then the following must hold σ i σ 1 j i 1 σ j for all i I (10) Suppose the converse of (10) is true and proceed to get a contradiction. So, there must exist ι I such that σ ι σ 1 j ι 1 Since {ϱ c i (t i)} i I is assumed to be consistent, and then so is i I t i, we must have σ j (11) In particular, we have t ι σ ι σ σ j = t j σ ι σ σ j, for all j I t ι σ ι σ σ j = t j σ ι σ σ j, for all j, 1 j ι 1 It follows that t ι σ ι σ σ j = ( 1 j ι 1 t j ) σ ι σ 1 j ι 1 1 j ι 1 σ j By (11), we obtain t ι σ ι σ = ( 1 j ι 1 t j ) (σ ι σ) (12) We have therefore ϱ c j(t j ) = {t ρ t σ j σ = t j σ j σ} 1 j ι 1 1 j ι 1 = {t ρ t (σ σ j ) = ( 1 j ι 1 t j ) (σ σ j )} 1 j ι 1 1 j ι 1 {t ρ t σ ι σ = ( 1 j ι 1 t j ) σ ι σ} {t ρ t σ ι σ = t ι σ ι σ} ϱ c ι(t ι ) The three inclusions are obtained by (11), (12), and (9), respectively. It follows that
10 10 1 j ι 1 ϱ c j(t j ) = 1 j ι ϱ c j(t j ) which means that {ϱ c j (t j)} 1 j ι is dependent. By proposition 3.3, we deduce that {ϱ c i (t i)} i I is dependent as well, thus contradicting the hypothesis. Hence, (10) is a necessary condition for the independence of {ϱ c i (t i)} i I. But, for (10) to hold, we must have i I, x i σ, such that x i σ i x i / 1 j i 1 Put in words, every σ i, i I, must contain a variable, x i σ, which is not in any σ j, j : 1,..., i 1. For this to hold, we must have I σ, and since σ r, we deduce that the size of the largest independent and consistent family of extensions in the dual problem cannot exceed r. Thus, the rank of the dual of a rary CSP is, at most, r. Next, we show that the dual of a rary CSP may have rank r. That is, it may contain a sizer family of independent and consistent extensions to the same variable. Suppose that the primal problem contains a subset {c i = (σ i, D r)} i I of r rary alldifferent constraints over value domain D = {1,..., r + 1}, where I = {1,..., r}. Suppose also that the primal problem contains a supplementary rary alldifferent constraint c = (σ, D r) such that σ = {x 1,..., x r }, σ i σ = {x i }, i I, and σ i σ j =, i, j I, i j. Moving to the dual problem, these requirements imply that every c i, i I will be connected to c by a binary constraint, that will be denoted by ({c i, c}, ϱ i ). Moreover, there will be no constraints between the c i s since these constraints do not share any variable in the primal problem. Let us select, from the value domain of each c i, i I, a rtuple t i such that t i [x i ] = i, where t i [x i ] denotes the value assigned to x i by t i. Consequently, {ϱ c i (t i)} i I is a family of consistent extensions of the dual problem, because i I t i is consistent since there are no constraints between the c i s. Furthermore, since ϱ c i(t i ) = {t D r t[i] = i}, for all i I σ j for any j I, we have 1 1,..., j 1, r + 1, j + 1,..., r i I\{j} ϱ c i(t i ) while 1,..., j 1, r + 1, j + 1,..., r / ϱ c j(t j ) because j r + 1. It follows that j I, i I\{j} ϱ c i(t i ) i I ϱ c i(t i ) By Proposition 3.3, this means that {ϱ c i (t i)} i I is an independent family of extensions to c. And since {ϱ c i (t i)} i I is consistent, we deduce that the rank of c is at least r. Then the proposed rary CSP has rank, at least, r. According to Theorem 4.7, solving a rary CSP instance can be done via its dual. If the latter problem is strongly (r + 1)consistent then it is globally consistent. A solution to the primal problem can, therefore, be deduced in polynomial time. 1 The form of the following tuple will slightly differ in the case where j is equal to 1, r 1 or r.
11 11 5 Directional rank of a CSP In order to weaken the requirements of Theorem 4.4, we derive a directional version of this theorem. But before we proceed, let us introduce some notations. Let P be a CSP whose variables are assumed to be totally ordered by, let (σ, ρ) be a rary constraint whose scope σ contains a variable x and let t be a tuple that instantiates the r 1 remaining variables of σ. We define the directional extension ρ x (t) of tuple t to variable x w.r.t. ρ and to be the subset of D x given by { ρ x ρ (t) = x (t) if y x, for all y σ otherwise D x (13) Definition 5.1 Let P be a CSP whose variables are totally ordered and let x be a variable of P. The directional rank of x is the size of the largest independent and consistent family of directional extensions to x. Definition 5.2 The directional rank of a CSP w.r.t a given ordering of its variables is the maximum directional rank over all its variables. Example 5.3 Consider again the binary CSP depicted in Figure 2 and assume the variable ordering (x 1, x 3, x 2, x 4 ). For this instance, the directional rank w.r.t. the chosen ordering is two or less since it cannot exceed the nondirectional rank. The directional rank of variable x 1 is zero since x 1 is the first variable in the ordering, and then by (13), all the directional extensions to x 1 are equal to {1, 2, 3}. Such extensions cannot form any independent family of extensions. The directional extensions to x 3 that differ from {1, 2, 3} are ρ x3 (x 1, v), v {1, 2, 3}. These three extensions cannot form an independent and consistent family of extensions containing more than one element. Thus, the directional rank of x 3 cannot exceed one. The directional extensions to x 2 are ρ x2 (x 1, u) and ϱ x2 (x 3, v) for all u, v {1, 2, 3}. It can be verified that any two of these extensions are either inconsistent or dependent. More precisely, if u = v = 1 or u, v {2, 3} then ρ x2 (x 1, u) and ϱ x2 (x 3, v) are inconsistent due to the constraint between x 1 and x 3, they are dependent otherwise. For instance, ρ x2 (x 1, 1) and ϱ x2 (x 3, 1) are inconsistent since the tuple (x 1, 1), (x 3, 1) is inconsistent due to the constraint between x 1 and x 3, while ρ x2 (x 1, 1) and ϱ x2 (x 3, 2) are dependent since they are respectively equal to {2, 3} and {2}. On the other hand, the directional rank of x 2 is at least one since { ρ x2 (x 1, 1)} is consistent and independent. Thus, the rank of x 2 is one. Because of the symmetry which exists between x 2 and x 4, we deduce that the directional rank of x 4 is also one. Thus, the problem considered in this example has directional rank one, which shows that the directional rank can be smaller than the nondirectional one. Computing the directional rank of a CSP can be done even faster than computing the nondirectional one. The reason is that when computing the directional rank of a variable x, only those extensions involving variables preceding x in the ordering require a computational effort, the other extensions are equal to D x, by definition. Assume that the arity r and the directional rank κ of the studied CSP are both bounded by a constant. To compute the directional rank of a variable x relative to a given ordering, one must consider the families of extensions to x involving at most κ + 1 distinct constraints. In addition to x, the scopes of these constraints must contain only variables preceding x in the ordering. Denote by e the maximum number of such constraints over all the variables of the CSP. The number of families of extentions having size κ + 1 or less is, therefore, bounded by O( e ( κ+1) d (r 1)( κ+1) ). Testing the independence of each of these families of extentions can be done in O(d) since their size is bounded by a constant. Thus, the time complexity of computing the directional rank of the whole CSP is O(n e ( κ+1) d (r 1)( κ+1)+1 ), and since e e and κ κ, computing the directional rank can be achieved more efficiently than computing the nondirectional one 1. 1 The number e is known as the width of the ordered hypergraph (Dechter, 2003) and we have e e in sparse hypergraphs. Another point worthmentioning is that κ e.
12 12 THEOREM 5.4 Let P be a rary CSP instance whose variables are totally ordered. If P has directional rank κ and is directional strong ( κ(r 1) + 1)consistent then it is globally consistent. Proof We show that if P is directional strong ( κ(r 1) + 1)consistent w.r.t. a given ordering then a solution can be obtained, in a backtrackfree manner, by instantiating the variables following that ordering. Let t be a consistent instantiation of the κ(r 1) + k first variables in the ordering for some k, 1 k < n κ(r 1). Directional strong ( κ(r 1) + 1)consistency ensures that t exists for k = 1. We show that t can be extended to the next variable in the ordering, say x. Denote by I the subset of {1,..., C } such that x σ i for all i I and y x for all y σ i Notice that t instantiates all the variables in σ i \ {x} for all i I. As for the nondirectional version of this theorem, we distinguish two cases: (i) I κ We proceed exactly as we did in case (i) of the proof of Theorem 4.4 to show that t can be consistently extended to x. (ii) I > κ Since the directional rank of P is κ, the size of any independent and consistent family of extensions to x is bounded by κ. On the other hand, P is directional strong ( κ(r 1) + 1)consistent. Then we can proceed as in (ii) of the proof of Theorem 4.4 to deduce that t can be consistently extended to x. The advantage of the directional rank is that it cannot exceed the nondirectional one, and in some cases, it is much smaller. For instance, the rank of a binary CSP whose constraint graph is a tree can reach d, the size of the largest value domain in the CSP, while its directional rank is equal to one, if a variable ordering which places each parent variable before all its children is chosen. Then the level of local consistency required by Theorem 5.4 is lower than the one required by Theorem 4.4, which results in a superclass of tractable CSPs. The directional rank can be employed in identifying new classes of tractable CSPs. This can be achieved by searching for variable orderings which simultaneously ensure a given directional rank and the level of strong directional local consistency suggested by Theorem 5.4. The problem of searching for a variable ordering that satisfies the conditions of Theorem 5.4 for a given CSP instance can, in turn, be modeled as a CSP. We call such a CSP a variable ordering CSP. Moreover, we show that the variable ordering CSP is tractable, for CSPs having a bounded arity and a bounded directional rank. Cooper et al. (2010) proposed a similar approach for discovering variable orderings that render certain binary CSPs tractable. Let P = (X, D, C) be a CSP with bounded arity and assume that we want to find an ordering on the variables of P which satisfies the conditions of Theorem 5.4 for a fixed and bounded directional rank κ. Then, we can proceed as follows: First, enforce strong ( κ(r 1)+1)consistency on P. Then, construct the variable ordering CSP associated to P as follows: for each x X, create a variable O x which handles the position of x in the required ordering. The value domains of all the O x, x X is, therefore, {1,..., X }. The role of the constraints in the variable ordering CSP is to ensure a directional rank not exceeding κ. Proposition 3.3 suggests that dismissing independent families of extensions having size κ + 1, eliminates all the independent families of extensions having larger sizes. Accordingly, the constraints can be specified as follows: for every variable x X and for every independent and consistent family of extensions {ρ x i (t i)} i I such that I = κ + 1, add the constraint defined by O x < max y Y (O y) (14) where Y = i I σ i \{x}. All the independent families of extensions having size κ+1 must, therefore, be identified in order to know which constraints to add. In the worst case, there are O(ne κ+1 d (r 1)( κ+1) ) of such families. Testing the independence of a ( κ+1)membered family of extensions can be done in O(d) since κ is bounded by a constant.
13 13 x ρ x 1 2 o o x4 ρ x3 Figure 3. The constraint graph of a binary CSP on four variables. Hence, identifying all the independent families of extensions can be achieved in O(ne κ+1 d (r 1)( κ+1)+1 ) steps. As can be deduced from (14), the arity of the resulting constraints is bounded by ( κ + 1)(r 1) + 1, which is a constant since we assumed that r and κ are constants. In addition, the number of these constraints is polynomially bounded by O(ne κ+1 ). Thus, constructing a variable ordering CSP associated with a CSP instance having a bounded arity and a bounded rank requires polynomial space. On the other hand, the relations defining the constraints of the variable ordering CSP, see (14), are maxclosed. Thus, the variable ordering CSPs are defined over a maxclosed constraint language. Such CSPs are known to be tractable (Jeavons and Cooper, 1995). A CSP instance with maxclosed constraints can be solved in polynomial time by establishing generalized arcconsistency (Bessière and Régin, 1997) provided the constraints have arities bounded by a constant. Hence, the problem of identifying the hybrid class of tractable CSPs based on Theorem 5.4 is tractable provided the directional rank and the constraint arities are both bounded by a constant. Example 5.5 Consider again the binary CSP instance used in the previous examples, which is, nonetheless, slightly modified by discarding the constraint between x 1 and x 3, (see Figure 3). The directional rank of the resulting instance is also bounded by two since the same relations, (ρ and ϱ), are used. In fact, the directional rank of the modified instance is exactly two whatever the variable ordering with regard to which it is determined. For instance, if the chosen ordering places x 1 in the last position then the extensions ρ x1 (x 2, 1) and ϱ x1 (x 4, 1) are independent and consistent, and then, the instance has directional rank two. If another variable is placed in the last position, a similar situation occurs, due to the symmetry between the variables. By Theorem 5.4, the level of directional local consistency needed to obtain global consistency is directional strong pathconsistency. The instance is already directional strong arcconsistent as it is. It remains to establish directional pathconsistency, say, following the natural order, i.e., (x 1, x 2, x 3, x 4 ). Establishing directional pathconsistency following this ordering involves a single step which consists in establishing pathconsistency along the path (x 1, x 4, x 3 ). This action consists in introducing a new constraint between x 1 and x 3, which allows solely the tuples of D 1 D 3 that can be consistently extended to x 4. These tuples are exactly those contained in ρ. Thus, the resulting instance is exactly the one considered in the previous examples. In summary, we see that only relations ρ and ϱ are used in the strong directional pathconsistency closure of the considered CSP instance. We have already noticed that these two relations cannot yield any independent family of subsets that contains more that two members. Hence, the strong directional pathconsistent instance has directional rank two or less, and so, by Theorem 5.4, is globally consistent. 6 Related work The proposed theory can be used to prove the tractability of many existing structural, relational or hybrid classes of CSPs. All the tractable classes that will be discussed below, and probably many others, can be identified by the approach described in Section 5, which can be summarized as follows: enforce strong kconsistency on the problem to be solved, for some k, then find a variable ordering following which the strong kconsistent closure of the problem has the appropriate directional rank.
14 mtight constraints Dechter (1992) presented a theorem (Theorem 3.1) that states the following: If an rary CSP instance, with largest value domain having size d is strongly (d(r 1) + 1)consistent then it is globally consistent. In (van Beek and Dechter, 1997), the authors proved a stronger theorem which relates the level of local consistency required to guarantee the global consistency to the tightness of the constraint relations. A rary constraint relation ρ is called mtight if, for any variable x constrained by ρ and any instantiation t of the remaining r 1 variables, either ρ x (t) m or ρ x (t) = D x. Clearly, any CSP instance is mtight for some m, 0 m d 1. Theorem 3.3 of (van Beek and Dechter, 1997) suggests that if the relations of a rary CSP instance are all mtight then strong ((m + 1)(r 1) + 1)consistency ensures global consistency. The class of tractable problems specified by the latter theorem can be viewed as a hybrid class because the theorem does not exclusively rely on a structural or a relational property. Indeed, although the mtightness is a relational property, the level of local consistency of a problem often depends on the problem structure. Unfortunately, establishing the required level of local consistency may destroy mtightness. For this reason, many works have proposed further restrictions in order to obtain variations of mtightness that are preserved while establishing local consistency. In (Zhang and Yap, 2006), the authors proposed the notion of weak mtightness at level k, a variation of m tightness which is preserved by establishing local consistency. We notice here that the latter work used the notion of intersecting extensions to define the proposed concepts. Another work related to mtightness is the one concerning the implicational (or 0/1/all) CSPs (Cooper et al., 1994). This is a tractable class of binary problems. It groups together particular binary problems in that they involve solely three types of binary relations: complete, permutation and twofan relations. All these relations are 1tight. Furthermore, the strong pathconsistency closure of every implicational CSP instance is also implicational, and thus also 1tight. Let as show that CSPs with mtight relations have rank m + 1 or less. To this end, assume the converse and proceed to get a contradiction. According to Definition 4.2, there must exist a variable x such that κ x > m + 1. According to Definition 4.1, there must exist an independent and consistent family of extensions, {ρ x i (t i)} i I, whose size I is greater that m + 1. From Proposition 3.4, we obtain min i I ρ x i (t i) > m, and then ρ x i (t i) > m for all i I. Since all the ρ i, i I are mtight, we must have ρ x i (t i) = D x for all i I. This implies that {ρ x i (t i)} i I is dependent, thus contradicting the hypothesis. We deduce that Theorem 4.4 generalizes also Theorem 3.3 of (van Beek and Dechter, 1997). 6.2 The broken triangle property In a recent article, Cooper et al. (2010) proposed the broken triangle property (BTP) as a means to ensure the tractability of certain binary CSPs. A binary CSP instance verifies the brokentriangle property w.r.t. a variable ordering if and only if for every three variables x y z such that ({x, z}, ρ) and ({y, z}, ϱ) are constraints of the problem and for all u D x and v D y such that the tuple (x, u), (y, v) is consistent, we have 1 ρ z (x, u) ϱ z (y, v) ρ z (x, u) ρ z (y, v) (15) A binary CSP verifying the broken triangle property can be solved in polynomial time by first establishing strong arcconsistency and then selecting an appropriate value from each arcconsistent value domain. As mentioned by the authors themselves, the set of all CSPs possessing the broken triangle property form a hybrid class of tractable binary CSPs. 1 See Lemma 3 of (Cooper et al., 2010).
15 15 To show the similarity between the BTP and the notion of directional rank proposed in this paper, we equivalently write statement (15) as follows ρ z (x, u) = ρ z (x, u) ϱ z (y, v) ϱ z (y, v) = ρ z (x, u) ρ z (y, v) By (2), this suggests that any twomembered family of directional extensions to any variable of a binary CSP instance verifying the BTP is dependent. Thus, all instances in the BTP class have directional rank one or less, and then, by Theorem 5.4, only strong arcconsistency is needed to ensure global consistency. After establishing strong directional arcconsistency and, in accordance with Proposition 4.6, the directional rank of the resulting instance does not increase. We get, therefore, a strong directional arcconsistent CSP whose directional rank is one. According to Theorem 5.4, such an instance is globally consistent. This provides a rankbased proof of the tractability of the BTP class. 6.3 Rowconvex constraints Another hybrid tractable class of binary CSP is the one involving rowconvex constraints (van Beek and Dechter, 1995). This class is considered as a hybrid class because, in addition to the rowconvexity of their constraints, instances in this class have to be strongly pathconsistent in order to be globally consistent. Assuming a total order on the value domain of each variable, rowconvex constraints can be defined as follows: Let x, y be two variables forming the scope of a binary constraint ({x, y}, ρ) and let v D x and a, b, c D y such that a y b y c, where y denotes the total order on D y. The constraint ({x, y}, ρ) is rowconvex if and only if for all such a, b, c whenever (v, a) and (v, c) are in ρ, (v, b) must also be in ρ. Let us show that binary CSPs with rowconvex constraints have rank two or less. To this end, assume that there is a rowconvex instance whose rank is greater that two. This means that there must exist a threemembered independent family of extensions {ρ y 1 (x 1, a), ρ y 2 (x 2, b), ρ y 3 (x 3, c)} to some variable y of the instance. This implies that there must exist ā, b, c D y such that 1 and and b, c ρ y 1 (x 1, a) ā / ρ y 1 (x 1, a) ā, c ρ y 2 (x 2, b) b / ρ y 2 (x 2, b) ā, b ρ y 3 (x 3, c) c / ρ y 3 (x 3, c) Since ā, b and c play a symmetric role in the last three statements, we can assume, without loss of generality, that ā y b y c. The second statement yields, therefore, (b, ā), (b, c) ρ 2 and (b, b) / ρ 2, which means that ρ 2 is not rowconvex, thus contradicting the hypothesis. Hence, CSPs with rowconvex constraints have rank two or less, and then Theorem 4.4 can be used to show that such instances are globally consistent provided that they are strong pathconsistent. The notion of rowconvex constraints was strengthened in a way that yielded connected rowconvex constraints (Deville et al., 1997). The advantange of connected rowconvexity is that it is preserved by the relation operations performed when establishing pathconsistency. As a result, CSPs involving exclusively connected rowconvex constraints are tractable. Rowconvex constraints was also generalized to treeconvex constraints in (Zhang and Yap, 2003), where the authors also used the notion of extension sets. In the context of treeconvex constraints, the domain values are 1 The following statements correspond to the configuration depicted in Figure 1(c).
16 16 REFERENCES viewed as the nodes of a tree and the extension sets must correspond to connected subtrees. If this holds, strong 2(r 1) + 1consistency suffices to obtain the global consistency of rary CSPs. However, CSPs based on the latter two variations of rowconvexity remain of rank two or less, and so the theoretical results presented in (Deville et al., 1997) and (Zhang and Yap, 2003) can also be derived in the framework of ranked CSPs. 7 Conclusion In this paper, we proposed a theoretical tool devoted to identifying hybrid classes of tractable CSPs. The identification schema is polynomial provided that the problems under study involve solely constraints of bounded arities. We also showed that the tractability of many known CSP classes can be proven via the proposed theory. This work can be developed further in an attempt to characterize hybrid tractable classes in a more precise way. For instance, we can address the following question: what are the binary CSPs that have rank two or less and whose pathconsistency closure has also rank two or less? According to Theorem 4.4, such problems are tractable. The answer to this question is of interest because any CSP of bounded arity can be polynomially transformed into a binary CSP whose rank is two or less. This CSP is simply the dual of the dual problem. References C. Bessiere and J. Régin, Arc consistency for general constraint networks: preliminary results, in: Proceedings of IJCAI 97, M.C. Cooper, An optimal kconsistency algorithm, Artif. Intell., 41(1), pp , M.C. Cooper, D.A. Cohen and P.G. Jeavons, Characterizing Tractable Constraints, Artif. Intell., 65(2), pp , M.C. Cooper, P.G. Jeavons and A.Z. Salamon, Generalizing constraint satisfaction on trees: Hybrid tractability and variable elimination, Artif. Intell., 174(9 10), pp , R. Dechter and J. Pearl, Networkbased heuristics for constraintsatisfaction problems, Artif. Intell., 34(1) pp. 1 38, R. Dechter, From local to global consistency, Artif. Intell., 55, pp , R. Dechter, Constraint Processing, Morgan Kaufmann Ed E.C. Freuder, A Sufficient Condition for BacktrackFree Search, Jour. ACM, 29(1), pp , Y. Deville, O. Barette and P. Van Hentenryck, Constraint satisfaction over connected row convex constraints, in: IJCAI97, Nagoya, Japan, IJCAI Inc, 1997, volume 1, pp P. van Beek and R. Dechter, On the minimality and global consistency of rowconvex constraint networks, Jour. ACM, 42(3), pp , P. van Beek, R. Dechter, Constraint tightness and looseness versus local and global consistency Jour. ACM, 44(4), pp , P.G. Jeavons and M.C. Cooper, Tractable constraints on ordered domains, Artif. Intell., 79(2), pp , Y. Zhang and R.H.C. Yap, Consistency and set intersection, in: Proceedings of IJCAI03, Acapulco, Mexico, IJCAI Inc, 2003, pp Y. Zhang and R.H.C. Yap, Set intersection and Consistency in Constraint Networks, Jour. Artif. Intell. Resear., 27, pp , 2006.
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