# Unifying and extending hybrid tractable classes of CSPs

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Journal of Experimental & Theoretical Artificial Intelligence Vol. 00, No. 00, Month-Month 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 NP-complete 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 NP-complete 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 max-closed 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 NP-complete, 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 k-consistent 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 k-consistent if and only if every consistent instantiation of any k 1 variables is k-consistent relative to every additional variable. If P is k-consistent for some k, 1 k X then it is said to be locally consistent and if it is i-consistent for all i : 1,..., k it is said to be strong k-consistent. Finally, if P is strong X -consistent then it is called globally consistent. The most used levels of local consistency are arc- and path-consistencies, which stand respectively for 2- and 3-consistency. Enforcing k-consistency, 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 k-consistency closure of P, may change. The time complexity of enforcing k-consistency 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 k-consistency can be lightened by considering a weak form of k-consistency called directional k-consistency (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 k-consistent 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 k-consistent if and only if it is directional i-consistent for all i : 1,..., k. As mentioned above, directional k-consistency is a weak form of (full) k-consistency in the sense that k- consistency entails directional k-consistency 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 r-ary 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 r-ary 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 n-ary 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 r-ary 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 r-ary 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 r-ary CSP is, at most, r. Next, we show that the dual of a r-ary CSP may have rank r. That is, it may contain a size-r 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 r-ary all-different constraints over value domain D = {1,..., r + 1}, where I = {1,..., r}. Suppose also that the primal problem contains a supplementary r-ary all-different 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 r-tuple 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 r-ary CSP has rank, at least, r. According to Theorem 4.7, solving a r-ary 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 r-ary 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 non-directional one. Computing the directional rank of a CSP can be done even faster than computing the non-directional 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 non-directional 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 r-ary 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 backtrack-free 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 non-directional 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 non-directional 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 max-closed. Thus, the variable ordering CSPs are defined over a max-closed constraint language. Such CSPs are known to be tractable (Jeavons and Cooper, 1995). A CSP instance with max-closed constraints can be solved in polynomial time by establishing generalized arc-consistency (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 path-consistency. The instance is already directional strong arc-consistent as it is. It remains to establish directional path-consistency, 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 path-consistency 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 path-consistency 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 path-consistent 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 k-consistency on the problem to be solved, for some k, then find a variable ordering following which the strong k-consistent closure of the problem has the appropriate directional rank.

14 m-tight constraints Dechter (1992) presented a theorem (Theorem 3.1) that states the following: If an r-ary 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 r-ary constraint relation ρ is called m-tight 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 m-tight for some m, 0 m d 1. Theorem 3.3 of (van Beek and Dechter, 1997) suggests that if the relations of a r-ary CSP instance are all m-tight 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 m-tightness 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 m-tightness. For this reason, many works have proposed further restrictions in order to obtain variations of m-tightness that are preserved while establishing local consistency. In (Zhang and Yap, 2006), the authors proposed the notion of weak m-tightness 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 m-tightness 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 two-fan relations. All these relations are 1-tight. Furthermore, the strong path-consistency closure of every implicational CSP instance is also implicational, and thus also 1-tight. Let as show that CSPs with m-tight 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 m-tight, 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 broken-triangle 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 arc-consistency and then selecting an appropriate value from each arc-consistent 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 two-membered 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 arc-consistency is needed to ensure global consistency. After establishing strong directional arc-consistency and, in accordance with Proposition 4.6, the directional rank of the resulting instance does not increase. We get, therefore, a strong directional arc-consistent CSP whose directional rank is one. According to Theorem 5.4, such an instance is globally consistent. This provides a rank-based proof of the tractability of the BTP class. 6.3 Row-convex constraints Another hybrid tractable class of binary CSP is the one involving row-convex constraints (van Beek and Dechter, 1995). This class is considered as a hybrid class because, in addition to the row-convexity of their constraints, instances in this class have to be strongly path-consistent in order to be globally consistent. Assuming a total order on the value domain of each variable, row-convex 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 row-convex 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 row-convex constraints have rank two or less. To this end, assume that there is a row-convex instance whose rank is greater that two. This means that there must exist a three-membered 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 row-convex, thus contradicting the hypothesis. Hence, CSPs with row-convex 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 path-consistent. The notion of row-convex constraints was strengthened in a way that yielded connected row-convex constraints (Deville et al., 1997). The advantange of connected row-convexity is that it is preserved by the relation operations performed when establishing path-consistency. As a result, CSPs involving exclusively connected row-convex constraints are tractable. Row-convex constraints was also generalized to tree-convex constraints in (Zhang and Yap, 2003), where the authors also used the notion of extension sets. In the context of tree-convex 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) + 1-consistency suffices to obtain the global consistency of r-ary CSPs. However, CSPs based on the latter two variations of row-convexity 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 path-consistency 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 k-consistency 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, Network-based heuristics for constraint-satisfaction 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 Backtrack-Free Search, Jour. ACM, 29(1), pp , Y. Deville, O. Barette and P. Van Hentenryck, Constraint satisfaction over connected row convex constraints, in: IJCAI-97, Nagoya, Japan, IJCAI Inc, 1997, volume 1, pp P. van Beek and R. Dechter, On the minimality and global consistency of row-convex 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 IJCAI-03, 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.

### 3 No-Wait Job Shops with Variable Processing Times

3 No-Wait Job Shops with Variable Processing Times In this chapter we assume that, on top of the classical no-wait job shop setting, we are given a set of processing times for each operation. We may select

### Some Applications of Graph Bandwidth to Constraint Satisfaction Problems

Some Applications of Graph Bandwidth to Constraint Satisfaction Problems Ramin Zabih Computer Science Department Stanford University Stanford, California 94305 Abstract Bandwidth is a fundamental concept

### The Encoding Complexity of Network Coding

The Encoding Complexity of Network Coding Michael Langberg Alexander Sprintson Jehoshua Bruck California Institute of Technology Email: mikel,spalex,bruck @caltech.edu Abstract In the multicast network

### A Fast Arc Consistency Algorithm for n-ary Constraints

A Fast Arc Consistency Algorithm for n-ary Constraints Olivier Lhomme 1 and Jean-Charles Régin 2 1 ILOG, 1681, route des Dolines, 06560 Valbonne, FRANCE 2 Computing and Information Science, Cornell University,

### Rigidity, connectivity and graph decompositions

First Prev Next Last Rigidity, connectivity and graph decompositions Brigitte Servatius Herman Servatius Worcester Polytechnic Institute Page 1 of 100 First Prev Next Last Page 2 of 100 We say that a framework

### Path Consistency on Triangulated Constraint Graphs*

Path Consistency on Triangulated Constraint Graphs* Christian Bliek ILOG 1681 Route des Dolines 06560 Valbonne, France bliekqilog.fr Djamila Sam-Haroud Artificial Intelligence Laboratory Swiss Federal

This article was originally published in a journal published by Elsevier, and the attached copy is provided by Elsevier for the author s benefit and for the benefit of the author s institution, for non-commercial

### On the Space-Time Trade-off in Solving Constraint Satisfaction Problems*

On the Space-Time Trade-off in Solving Constraint Satisfaction Problems* Roberto J. Bayardo Jr. and Daniel P. Miranker Department of Computer Sciences and Applied Research Laboratories The University of

### Bound Consistency for Binary Length-Lex Set Constraints

Bound Consistency for Binary Length-Lex Set Constraints Pascal Van Hentenryck and Justin Yip Brown University, Box 1910 Carmen Gervet Boston University, Providence, RI 02912 808 Commonwealth Av. Boston,

### 2-C3OP: An Improved Version of 2-Consistency

-COP: An Improved Version of -Consistency Marlene Arangú, Miguel A. Salido, Federico Barber Instituto de Automática e Informática Industrial Universidad Politécnica de Valencia. Valencia, Spain Abstract

### 6. Lecture notes on matroid intersection

Massachusetts Institute of Technology 18.453: Combinatorial Optimization Michel X. Goemans May 2, 2017 6. Lecture notes on matroid intersection One nice feature about matroids is that a simple greedy algorithm

### Binary vs. Non-Binary Constraints

Binary vs. Non-Binary Constraints Fahiem Bacchus Department of Computer Science University of Toronto Toronto, Canada fbacchus@cs.toronto.edu Xinguang Chen Department of Computing Science University of

### Constraint Networks. Constraint networks. Definition. Normalized. Constraint Networks. Deduction. Constraint. Networks and Graphs. Solving.

1 Satisfaction Problems Albert-Ludwigs-Universität Freiburg networks networks and Stefan Wölfl, Christian Becker-Asano, and Bernhard Nebel October 27, 2014 October 27, 2014 Wölfl, Nebel and Becker-Asano

### The Structure of Bull-Free Perfect Graphs

The Structure of Bull-Free Perfect Graphs Maria Chudnovsky and Irena Penev Columbia University, New York, NY 10027 USA May 18, 2012 Abstract The bull is a graph consisting of a triangle and two vertex-disjoint

### Constraint Satisfaction Problems

Constraint Satisfaction Problems Search and Lookahead Bernhard Nebel, Julien Hué, and Stefan Wölfl Albert-Ludwigs-Universität Freiburg June 4/6, 2012 Nebel, Hué and Wölfl (Universität Freiburg) Constraint

### Relational Database: The Relational Data Model; Operations on Database Relations

Relational Database: The Relational Data Model; Operations on Database Relations Greg Plaxton Theory in Programming Practice, Spring 2005 Department of Computer Science University of Texas at Austin Overview

### Edge Colorings of Complete Multipartite Graphs Forbidding Rainbow Cycles

Theory and Applications of Graphs Volume 4 Issue 2 Article 2 November 2017 Edge Colorings of Complete Multipartite Graphs Forbidding Rainbow Cycles Peter Johnson johnspd@auburn.edu Andrew Owens Auburn

### 9.5 Equivalence Relations

9.5 Equivalence Relations You know from your early study of fractions that each fraction has many equivalent forms. For example, 2, 2 4, 3 6, 2, 3 6, 5 30,... are all different ways to represent the same

### A Uniform View of Backtracking

A Uniform View of Backtracking Fahiem Bacchus 1 Department. of Computer Science, 6 Kings College Road, University Of Toronto, Toronto, Ontario, Canada, M5S 1A4, fbacchus@cs.toronto.edu? Abstract. Backtracking

### A New Algorithm for Singleton Arc Consistency

A New Algorithm for Singleton Arc Consistency Roman Barták, Radek Erben Charles University, Institute for Theoretical Computer Science Malostranské nám. 2/25, 118 Praha 1, Czech Republic bartak@kti.mff.cuni.cz,

### The External Network Problem

The External Network Problem Jan van den Heuvel and Matthew Johnson CDAM Research Report LSE-CDAM-2004-15 December 2004 Abstract The connectivity of a communications network can often be enhanced if the

### These notes present some properties of chordal graphs, a set of undirected graphs that are important for undirected graphical models.

Undirected Graphical Models: Chordal Graphs, Decomposable Graphs, Junction Trees, and Factorizations Peter Bartlett. October 2003. These notes present some properties of chordal graphs, a set of undirected

### Paths, Flowers and Vertex Cover

Paths, Flowers and Vertex Cover Venkatesh Raman M. S. Ramanujan Saket Saurabh Abstract It is well known that in a bipartite (and more generally in a König) graph, the size of the minimum vertex cover is

### Small Survey on Perfect Graphs

Small Survey on Perfect Graphs Michele Alberti ENS Lyon December 8, 2010 Abstract This is a small survey on the exciting world of Perfect Graphs. We will see when a graph is perfect and which are families

### Math 5593 Linear Programming Lecture Notes

Math 5593 Linear Programming Lecture Notes Unit II: Theory & Foundations (Convex Analysis) University of Colorado Denver, Fall 2013 Topics 1 Convex Sets 1 1.1 Basic Properties (Luenberger-Ye Appendix B.1).........................

### Repairing Preference-Based Argumentation Frameworks

Repairing Preference-Based Argumentation Frameworks Leila Amgoud IRIT CNRS 118, route de Narbonne 31062, Toulouse Cedex 09 amgoud@irit.fr Srdjan Vesic IRIT CNRS 118, route de Narbonne 31062, Toulouse Cedex

### Conflict based Backjumping for Constraints Optimization Problems

Conflict based Backjumping for Constraints Optimization Problems Roie Zivan and Amnon Meisels {zivanr,am}@cs.bgu.ac.il Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, 84-105,

### MA651 Topology. Lecture 4. Topological spaces 2

MA651 Topology. Lecture 4. Topological spaces 2 This text is based on the following books: Linear Algebra and Analysis by Marc Zamansky Topology by James Dugundgji Fundamental concepts of topology by Peter

### Topic: Local Search: Max-Cut, Facility Location Date: 2/13/2007

CS880: Approximations Algorithms Scribe: Chi Man Liu Lecturer: Shuchi Chawla Topic: Local Search: Max-Cut, Facility Location Date: 2/3/2007 In previous lectures we saw how dynamic programming could be

### Disjoint Support Decompositions

Chapter 4 Disjoint Support Decompositions We introduce now a new property of logic functions which will be useful to further improve the quality of parameterizations in symbolic simulation. In informal

### On 2-Subcolourings of Chordal Graphs

On 2-Subcolourings of Chordal Graphs Juraj Stacho School of Computing Science, Simon Fraser University 8888 University Drive, Burnaby, B.C., Canada V5A 1S6 jstacho@cs.sfu.ca Abstract. A 2-subcolouring

### Open and Closed Sets

Open and Closed Sets Definition: A subset S of a metric space (X, d) is open if it contains an open ball about each of its points i.e., if x S : ɛ > 0 : B(x, ɛ) S. (1) Theorem: (O1) and X are open sets.

### Complexity Results on Graphs with Few Cliques

Discrete Mathematics and Theoretical Computer Science DMTCS vol. 9, 2007, 127 136 Complexity Results on Graphs with Few Cliques Bill Rosgen 1 and Lorna Stewart 2 1 Institute for Quantum Computing and School

### NP-Hardness. We start by defining types of problem, and then move on to defining the polynomial-time reductions.

CS 787: Advanced Algorithms NP-Hardness Instructor: Dieter van Melkebeek We review the concept of polynomial-time reductions, define various classes of problems including NP-complete, and show that 3-SAT

### Maximal Monochromatic Geodesics in an Antipodal Coloring of Hypercube

Maximal Monochromatic Geodesics in an Antipodal Coloring of Hypercube Kavish Gandhi April 4, 2015 Abstract A geodesic in the hypercube is the shortest possible path between two vertices. Leader and Long

### Decreasing the Diameter of Bounded Degree Graphs

Decreasing the Diameter of Bounded Degree Graphs Noga Alon András Gyárfás Miklós Ruszinkó February, 00 To the memory of Paul Erdős Abstract Let f d (G) denote the minimum number of edges that have to be

### Constraint Propagation: The Heart of Constraint Programming

Constraint Propagation: The Heart of Constraint Programming Zeynep KIZILTAN Department of Computer Science University of Bologna Email: zeynep@cs.unibo.it URL: http://zeynep.web.cs.unibo.it/ What is it

### Time and Space Lower Bounds for Implementations Using k-cas

Time and Space Lower Bounds for Implementations Using k-cas Hagit Attiya Danny Hendler September 12, 2006 Abstract This paper presents lower bounds on the time- and space-complexity of implementations

### Optimality certificates for convex minimization and Helly numbers

Optimality certificates for convex minimization and Helly numbers Amitabh Basu Michele Conforti Gérard Cornuéjols Robert Weismantel Stefan Weltge October 20, 2016 Abstract We consider the problem of minimizing

### v V Question: How many edges are there in a graph with 10 vertices each of degree 6?

ECS20 Handout Graphs and Trees March 4, 2015 (updated 3/9) Notion of a graph 1. A graph G = (V,E) consists of V, a nonempty set of vertices (or nodes) and E, a set of pairs of elements of V called edges.

### A note on the pairwise Markov condition in directed Markov fields

TECHNICAL REPORT R-392 April 2012 A note on the pairwise Markov condition in directed Markov fields Judea Pearl University of California, Los Angeles Computer Science Department Los Angeles, CA, 90095-1596,

### Lecture 15: The subspace topology, Closed sets

Lecture 15: The subspace topology, Closed sets 1 The Subspace Topology Definition 1.1. Let (X, T) be a topological space with topology T. subset of X, the collection If Y is a T Y = {Y U U T} is a topology

### STANLEY S SIMPLICIAL POSET CONJECTURE, AFTER M. MASUDA

Communications in Algebra, 34: 1049 1053, 2006 Copyright Taylor & Francis Group, LLC ISSN: 0092-7872 print/1532-4125 online DOI: 10.1080/00927870500442005 STANLEY S SIMPLICIAL POSET CONJECTURE, AFTER M.

### On the Max Coloring Problem

On the Max Coloring Problem Leah Epstein Asaf Levin May 22, 2010 Abstract We consider max coloring on hereditary graph classes. The problem is defined as follows. Given a graph G = (V, E) and positive

### Topology Homework 3. Section Section 3.3. Samuel Otten

Topology Homework 3 Section 3.1 - Section 3.3 Samuel Otten 3.1 (1) Proposition. The intersection of finitely many open sets is open and the union of finitely many closed sets is closed. Proof. Note that

### Optimal and Suboptimal Singleton Arc Consistency Algorithms

Optimal and Suboptimal Singleton Arc Consistency Algorithms Christian Bessiere LIRMM (CNRS / University of Montpellier) 161 rue Ada, Montpellier, France bessiere@lirmm.fr Romuald Debruyne École des Mines

### Applied Lagrange Duality for Constrained Optimization

Applied Lagrange Duality for Constrained Optimization Robert M. Freund February 10, 2004 c 2004 Massachusetts Institute of Technology. 1 1 Overview The Practical Importance of Duality Review of Convexity

### Semantic Forcing in Disjunctive Logic Programs

Semantic Forcing in Disjunctive Logic Programs Marina De Vos and Dirk Vermeir Dept of Computer Science Free University of Brussels, VUB Pleinlaan 2, Brussels 1050, Belgium Abstract We propose a semantics

### Term Algebras with Length Function and Bounded Quantifier Elimination

with Length Function and Bounded Ting Zhang, Henny B Sipma, Zohar Manna Stanford University tingz,sipma,zm@csstanfordedu STeP Group, September 3, 2004 TPHOLs 2004 - p 1/37 Motivation: Program Verification

### On k-dimensional Balanced Binary Trees*

journal of computer and system sciences 52, 328348 (1996) article no. 0025 On k-dimensional Balanced Binary Trees* Vijay K. Vaishnavi Department of Computer Information Systems, Georgia State University,

### Connectivity, Graph Minors, and Subgraph Multiplicity

Connectivity, Graph Minors, and Subgraph Multiplicity David Eppstein Department of Information and Computer Science University of California, Irvine, CA 92717 Tech. Report 92-06 January 10, 1992 Abstract

### arxiv: v2 [cs.ds] 18 May 2015

Optimal Shuffle Code with Permutation Instructions Sebastian Buchwald, Manuel Mohr, and Ignaz Rutter Karlsruhe Institute of Technology {sebastian.buchwald, manuel.mohr, rutter}@kit.edu arxiv:1504.07073v2

### An Eternal Domination Problem in Grids

Theory and Applications of Graphs Volume Issue 1 Article 2 2017 An Eternal Domination Problem in Grids William Klostermeyer University of North Florida, klostermeyer@hotmail.com Margaret-Ellen Messinger

### Topological properties of convex sets

Division of the Humanities and Social Sciences Ec 181 KC Border Convex Analysis and Economic Theory Winter 2018 Topic 5: Topological properties of convex sets 5.1 Interior and closure of convex sets Let

### CS 188: Artificial Intelligence Fall 2011

CS 188: Artificial Intelligence Fall 2011 Lecture 5: CSPs II 9/8/2011 Dan Klein UC Berkeley Multiple slides over the course adapted from either Stuart Russell or Andrew Moore 1 Today Efficient Solution

### The Resolution Width Problem is EXPTIME-Complete

The Resolution Width Problem is EXPTIME-Complete Alexander Hertel & Alasdair Urquhart November 24, 2007 Abstract The importance of width as a resource in resolution theorem proving has been emphasized

### The problem of minimizing the elimination tree height for general graphs is N P-hard. However, there exist classes of graphs for which the problem can

A Simple Cubic Algorithm for Computing Minimum Height Elimination Trees for Interval Graphs Bengt Aspvall, Pinar Heggernes, Jan Arne Telle Department of Informatics, University of Bergen N{5020 Bergen,

### Division of the Humanities and Social Sciences. Convex Analysis and Economic Theory Winter Separation theorems

Division of the Humanities and Social Sciences Ec 181 KC Border Convex Analysis and Economic Theory Winter 2018 Topic 8: Separation theorems 8.1 Hyperplanes and half spaces Recall that a hyperplane in

### 2. CONNECTIVITY Connectivity

2. CONNECTIVITY 70 2. Connectivity 2.1. Connectivity. Definition 2.1.1. (1) A path in a graph G = (V, E) is a sequence of vertices v 0, v 1, v 2,..., v n such that {v i 1, v i } is an edge of G for i =

### A Greedy Approach to Establish Singleton Arc Consistency

A Greedy Approach to Establish Singleton Arc Consistency Christophe Lecoutre and Stéphane Cardon CRIL-CNRS FRE 2499, Université d Artois Lens, France {lecoutre, cardon}@cril.univ-artois.fr Abstract In

### Checking for k-anonymity Violation by Views

Checking for k-anonymity Violation by Views Chao Yao Ctr. for Secure Info. Sys. George Mason University cyao@gmu.edu X. Sean Wang Dept. of Computer Sci. University of Vermont Sean.Wang@uvm.edu Sushil Jajodia

### 1 Variations of the Traveling Salesman Problem

Stanford University CS26: Optimization Handout 3 Luca Trevisan January, 20 Lecture 3 In which we prove the equivalence of three versions of the Traveling Salesman Problem, we provide a 2-approximate algorithm,

### MARCHING CUBES AND VARIANTS

CHAPTER MARCHING CUBES AND VARIANTS In the introduction, we mentioned four different approaches to isosurface construction. In this chapter, we describe one of those approaches to isosurface construction,

### Two Characterizations of Hypercubes

Two Characterizations of Hypercubes Juhani Nieminen, Matti Peltola and Pasi Ruotsalainen Department of Mathematics, University of Oulu University of Oulu, Faculty of Technology, Mathematics Division, P.O.

### A geometric non-existence proof of an extremal additive code

A geometric non-existence proof of an extremal additive code Jürgen Bierbrauer Department of Mathematical Sciences Michigan Technological University Stefano Marcugini and Fernanda Pambianco Dipartimento

### / Approximation Algorithms Lecturer: Michael Dinitz Topic: Linear Programming Date: 2/24/15 Scribe: Runze Tang

600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Linear Programming Date: 2/24/15 Scribe: Runze Tang 9.1 Linear Programming Suppose we are trying to approximate a minimization

### A generalization of Mader s theorem

A generalization of Mader s theorem Ajit A. Diwan Department of Computer Science and Engineering Indian Institute of Technology, Bombay Mumbai, 4000076, India. email: aad@cse.iitb.ac.in 18 June 2007 Abstract

### Chapter 1. Preliminaries

Chapter 1 Preliminaries 1.1 Topological spaces 1.1.1 The notion of topological space The topology on a set X is usually defined by specifying its open subsets of X. However, in dealing with topological

### Backtracking algorithms for disjunctions of temporal constraints

Artificial Intelligence 120 (2000) 81 117 Backtracking algorithms for disjunctions of temporal constraints Kostas Stergiou a,, Manolis Koubarakis b,1 a APES Research Group, Department of Computer Science,

### NON-CENTRALIZED DISTINCT L-DIVERSITY

NON-CENTRALIZED DISTINCT L-DIVERSITY Chi Hong Cheong 1, Dan Wu 2, and Man Hon Wong 3 1,3 Department of Computer Science and Engineering, The Chinese University of Hong Kong, Hong Kong {chcheong, mhwong}@cse.cuhk.edu.hk

### Hardness of Subgraph and Supergraph Problems in c-tournaments

Hardness of Subgraph and Supergraph Problems in c-tournaments Kanthi K Sarpatwar 1 and N.S. Narayanaswamy 1 Department of Computer Science and Engineering, IIT madras, Chennai 600036, India kanthik@gmail.com,swamy@cse.iitm.ac.in

### ROUGH MEMBERSHIP FUNCTIONS: A TOOL FOR REASONING WITH UNCERTAINTY

ALGEBRAIC METHODS IN LOGIC AND IN COMPUTER SCIENCE BANACH CENTER PUBLICATIONS, VOLUME 28 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1993 ROUGH MEMBERSHIP FUNCTIONS: A TOOL FOR REASONING

### 5 The Theory of the Simplex Method

5 The Theory of the Simplex Method Chapter 4 introduced the basic mechanics of the simplex method. Now we shall delve a little more deeply into this algorithm by examining some of its underlying theory.

### Preferred directions for resolving the non-uniqueness of Delaunay triangulations

Preferred directions for resolving the non-uniqueness of Delaunay triangulations Christopher Dyken and Michael S. Floater Abstract: This note proposes a simple rule to determine a unique triangulation

### Partitioning Orthogonal Polygons by Extension of All Edges Incident to Reflex Vertices: lower and upper bounds on the number of pieces

Partitioning Orthogonal Polygons by Extension of All Edges Incident to Reflex Vertices: lower and upper bounds on the number of pieces António Leslie Bajuelos 1, Ana Paula Tomás and Fábio Marques 3 1 Dept.

### Chapter 3. Set Theory. 3.1 What is a Set?

Chapter 3 Set Theory 3.1 What is a Set? A set is a well-defined collection of objects called elements or members of the set. Here, well-defined means accurately and unambiguously stated or described. Any

### Advanced Algorithms Class Notes for Monday, October 23, 2012 Min Ye, Mingfu Shao, and Bernard Moret

Advanced Algorithms Class Notes for Monday, October 23, 2012 Min Ye, Mingfu Shao, and Bernard Moret Greedy Algorithms (continued) The best known application where the greedy algorithm is optimal is surely

### Graph Connectivity G G G

Graph Connectivity 1 Introduction We have seen that trees are minimally connected graphs, i.e., deleting any edge of the tree gives us a disconnected graph. What makes trees so susceptible to edge deletions?

### arxiv: v2 [math.co] 13 Aug 2013

Orthogonality and minimality in the homology of locally finite graphs Reinhard Diestel Julian Pott arxiv:1307.0728v2 [math.co] 13 Aug 2013 August 14, 2013 Abstract Given a finite set E, a subset D E (viewed

### Constraint Satisfaction Problems. slides from: Padhraic Smyth, Bryan Low, S. Russell and P. Norvig, Jean-Claude Latombe

Constraint Satisfaction Problems slides from: Padhraic Smyth, Bryan Low, S. Russell and P. Norvig, Jean-Claude Latombe Standard search problems: State is a black box : arbitrary data structure Goal test

### Structural Characterizations of Schema-Mapping Languages

Structural Characterizations of Schema-Mapping Languages Balder ten Cate University of Amsterdam and UC Santa Cruz balder.tencate@uva.nl Phokion G. Kolaitis UC Santa Cruz and IBM Almaden kolaitis@cs.ucsc.edu

### Separators in High-Genus Near-Planar Graphs

Rochester Institute of Technology RIT Scholar Works Theses Thesis/Dissertation Collections 12-2016 Separators in High-Genus Near-Planar Graphs Juraj Culak jc1789@rit.edu Follow this and additional works

### Discrete Optimization. Lecture Notes 2

Discrete Optimization. Lecture Notes 2 Disjunctive Constraints Defining variables and formulating linear constraints can be straightforward or more sophisticated, depending on the problem structure. The

### Formal Model. Figure 1: The target concept T is a subset of the concept S = [0, 1]. The search agent needs to search S for a point in T.

Although this paper analyzes shaping with respect to its benefits on search problems, the reader should recognize that shaping is often intimately related to reinforcement learning. The objective in reinforcement

### The Branch & Move algorithm: Improving Global Constraints Support by Local Search

Branch and Move 1 The Branch & Move algorithm: Improving Global Constraints Support by Local Search Thierry Benoist Bouygues e-lab, 1 av. Eugène Freyssinet, 78061 St Quentin en Yvelines Cedex, France tbenoist@bouygues.com

### We assume uniform hashing (UH):

We assume uniform hashing (UH): the probe sequence of each key is equally likely to be any of the! permutations of 0,1,, 1 UH generalizes the notion of SUH that produces not just a single number, but a

### A Model of Machine Learning Based on User Preference of Attributes

1 A Model of Machine Learning Based on User Preference of Attributes Yiyu Yao 1, Yan Zhao 1, Jue Wang 2 and Suqing Han 2 1 Department of Computer Science, University of Regina, Regina, Saskatchewan, Canada

### Designing Views to Answer Queries under Set, Bag,and BagSet Semantics

Designing Views to Answer Queries under Set, Bag,and BagSet Semantics Rada Chirkova Department of Computer Science, North Carolina State University Raleigh, NC 27695-7535 chirkova@csc.ncsu.edu Foto Afrati

### A Fast Algorithm for Optimal Alignment between Similar Ordered Trees

Fundamenta Informaticae 56 (2003) 105 120 105 IOS Press A Fast Algorithm for Optimal Alignment between Similar Ordered Trees Jesper Jansson Department of Computer Science Lund University, Box 118 SE-221

### Product constructions for transitive decompositions of graphs

116 Product constructions for transitive decompositions of graphs Geoffrey Pearce Abstract A decomposition of a graph is a partition of the edge set, giving a set of subgraphs. A transitive decomposition

### Revisiting the bijection between planar maps and well labeled trees

Revisiting the bijection between planar maps and well labeled trees Daniel Cosmin Porumbel September 1, 2014 Abstract The bijection between planar graphs and well labeled trees was published by Cori and

### A Connection between Network Coding and. Convolutional Codes

A Connection between Network Coding and 1 Convolutional Codes Christina Fragouli, Emina Soljanin christina.fragouli@epfl.ch, emina@lucent.com Abstract The min-cut, max-flow theorem states that a source

### Basic Graph Theory with Applications to Economics

Basic Graph Theory with Applications to Economics Debasis Mishra February, 0 What is a Graph? Let N = {,..., n} be a finite set. Let E be a collection of ordered or unordered pairs of distinct elements

### Dynamic heuristics for branch and bound search on tree-decomposition of Weighted CSPs

Dynamic heuristics for branch and bound search on tree-decomposition of Weighted CSPs Philippe Jégou, Samba Ndojh Ndiaye, and Cyril Terrioux LSIS - UMR CNRS 6168 Université Paul Cézanne (Aix-Marseille

### Structure and Complexity in Planning with Unary Operators

Structure and Complexity in Planning with Unary Operators Carmel Domshlak and Ronen I Brafman ½ Abstract In this paper we study the complexity of STRIPS planning when operators have a single effect In

### 9.1 Cook-Levin Theorem

CS787: Advanced Algorithms Scribe: Shijin Kong and David Malec Lecturer: Shuchi Chawla Topic: NP-Completeness, Approximation Algorithms Date: 10/1/2007 As we ve already seen in the preceding lecture, two

### arxiv: v1 [cs.ds] 19 Feb 2014

Turing Kernelization for Finding Long Paths and Cycles in Restricted Graph Classes Bart M. P. Jansen 1 University of Bergen, Norway. Bart.Jansen@ii.uib.no arxiv:1402.4718v1 [cs.ds] 19 Feb 2014 Abstract.

### Relational Databases

Relational Databases Jan Chomicki University at Buffalo Jan Chomicki () Relational databases 1 / 49 Plan of the course 1 Relational databases 2 Relational database design 3 Conceptual database design 4