Domain permutation reduction for Valued CSPs

Maher Helaoui1, Wady Naanaa2, and Bechir Ayeb1

1 Pole de Recherche en Informatique du Centre (PRINCE), Institut Superieurd’Informatique et des Techniques de Communication, 4011 Hammam Sousse

maher.helaoui@gmail.com, ayeb b@yahoo.com2Faculty of sciences, University of Monastir, Tunisia

wnaanaa@planet.tn

Abstract. Several combinatorial problems can be formulated as ValuedConstraint Satisfaction Problems (VCSPs) where constraints are definedthrough the use of valuation functions to reflect degrees of coherence. Thegoal is to find an assignment of values to variables with an overall finiteand optimal valuation. Despite the NP-hardness of this task, tractableversions can be obtained by forcing the allowable valuation functionsto have specific features. This is the case, for instance, of VCSPs withbinary and submodular valuation functions [17].In this paper, we are concerned with a problem generalizing submodularbinary VCSP, which we will call permuted submodular binary VCSP. Thelatter problem is obtained by independently applying permutations onthe domains of submodular binary VCSP. We show that VCSP instancesbuilt from permuted submodular binary functions satisfying an extracondition can be identified in O(n2d4) steps and solved, by means ofthe algorithm used for submodular binary VCSPs [2], in O(n3d3) steps,where n is the number of variables and d is the size of the largest domain.

1 Introduction

Constraint Satisfaction Problems (CSPs) provide a general and convenient frame-work to model and solve numerous combinatorial problems including planningand scheduling. In the standard CSP framework, the constraints are defined bycrisp relations, which specify the consistent combinations of values. However, inreal-world situations, one may need to express various degrees of consistency inorder to reflect the specificity of the problem at hand. The valued constraint sat-isfaction problems (VCSPs) approach [19] is intended to model such situations.Basically, a VCSP consists of a set of variables taking values in discrete setscalled domains. A valued constraint is defined through the use of a valuationfunction, which associates a degree of desirability to each combination of values.The problem is to find an assignment of values to variables from their respectivedomains with a finite and optimal global valuation. Finding such an assignmentor proving that none exists is known to be an NP-hard task [4].

The computational complexity of finding the optimal solution to a VCSP hasbeen studied in many works and several classes of tractable VCSPs have been

identified and solved [5, 12, 4]. Recall that a problem is said to be tractable ifand only if there exists a polynomial-time algorithm that solves it. Binary VCSPwith submodular binary valuation functions is one of these tractable class. Byexpressing VCSP instances with submodular binary funcitons as the problem offinding a minimum weighted cut of a weighted directed graph, it is possible tosolve them only in O(n3d3) steps, where n is the number of variables and d isthe size of the largest value domain [2].

Nonetheless, VCSPs resulting from modeling real situations are rarely lim-ited to submodular functions. In such cases, can we proceed in a more efficientmanner than an exhaustive search while keeping solution optimality? Is thereany mean to exploit submodularity in a less restrictive context? This paper isintended to contribute to providing positive answers to these questions. Ourapproach is inspired by the one described in [10] which, nonetheless, has beendedicated to crisp CSP. The latter approach can be summerized as follows: Givena crisp CSP instance, find and apply a permutation on the value domain of eachvariable in such a way that the constraint relations of the resultant CSP instanceare all max-closed. The latter property ensures the tractability of the solutionproblem [11]. Then solve, in polynomial-time, the permuted CSP instance. Fi-nally, having obtained a solution of the permuted instance, if any, apply thereverse permutation on each value domain to get a solution to the original CSPinstance. Of course, for the approach to be of any interest, the permutationsmust be found in polynomial time.

Our purpose is to extend the approach proposed in [10] to cope with VCSP.To this end, we chosed the submodular binary VCSP [2] as the tractable probleminto which, VCSP instances have to be transformed. The transformation processconsists in performing permutations on the value domains of the instances tobe solved so that the resulting instances fall into the submodular binary VCSPclass. We call the set of all the VCSP instances that can be transformed in thisway permuted submodular binary VCSP.

We show that, assuming an extra condition on the valuation functions em-ployed, permuted submodular binary VCSP instances can be identified inO(n2d4)steps and solved, by means of the algorithm used for submodular binary VCSPs,in O(n3d3) steps.

The paper is organized as follows: the next section introduces some definitionsand notations. Section 3, is devoted to specifying the particular VCSPs we areconcerned with. In section 4, we describe how the domain permutation theoryis extended in order to efficiently solve the VCSPs specified in Section 3. Wemention how this work can be extended in section 5 and conclude in section 6.Finally, in the appendix, we provide some proofs which were omitted in the maintext for clarity of exposition.

2 Definitions and notations

We begin by recalling and introducing some general notions. Then, we providethe notions that are related to the VCSP framework.

2.1 General notions

Let R be a binary relation.

– R is said to be functionnal on its first argument if and only if whenever 〈u, v〉and 〈u′, v〉 are in R, we have u = u′.

– R is said to be functionnal on its second argument if and only if whenever〈u, v〉 and 〈u, v′〉 are in R, we have v = v′.

– R is bijective if and only if it is functionnal on both of its arguments.

Let R be an equivalence relation over a finite and totally ordered set which isassumed to be D = {0, 1, . . . , |D| − 1}. We define the kernel of R, ker(R), to bethe subset of D obtained by selecting the minimal element from each equivalenceclass induced by R, that is

ker(R) = {min[v] | [v] ∈ D/R} (1)

where D/R denotes the set of the equivalence classes of R and [v] denotes theequivalence class of value v ∈ D. Since only one equivalence relation will be usedat a time throughout the paper, the kernel of the considered equivalence relationover D will simply be denoted by D. Also, for any v ∈ D, the minimal elementof [v], which is an element of D, will be denoted by v. Notice that [v] = [v] forall v ∈ D.

Let π be a permutation of D = ker(R). Recall that a permutation of a set Sis a bijection from S to S. We define the expantion of π to D to be the functionπ defined by

π(v) =∑

s∈D, π(s)<π(v)

| [s] | + | {s ∈ [v] | s < v} | (2)

Every permutation of D and its expantion to D, verify the following lemma3.

Lemma 1. Let π be a permutation of D and let π be the expantion of π to D.Then, for all u, v ∈ D, π(u) < π(v) implies π(u) < π(v).

Furthermore,

Lemma 2. π is a permutation of D.

Converselly, given a permutation π′ of D, we define the function π′ given by

π′(v) = D(|{s ∈ D | π′(s) ≤ π′(v)}|) (3)

where D(k) is the kth smallest element of D.

Lemma 3. π′ is a permutation of D.

3 The proof of Lemma 1 to 3 are given in the appendix.

2.2 VCSP related notions

In the valued CSP framework (VCSP) [19], the set of possible valuations Eis assumed to be a totally ordered set with a minimum (⊥) and a maximum(⊤) element, equipped with a single monotonic binary operation ⊕ known asaggregation. These assumptions can be gathered in a valuation structure thatcan be specified as follows:

Definition 1. A valuation structure is defined as a tuple S = (E,⊕,�) suchthat:

– E is a set of valuations;– � is a total order on E;– ⊕ is a binary commutative, associative and monotonic operator.

Once the valuation structure is specified, we define the valued constraintsatisfaction problem (VCSP) we are concerned with as follows:

Definition 2. A valued constraint satisfaction problem (VCSP) instance is de-fined by a tuple (X,D,C, S) such that:

– X is a finite set of variables;– D is a finite set called the domain of the instance;– S = (E,⊕,�) is a fair valuation structure.– C is a set of valued constraints. Each valued constraint c is an ordered pair

(σ, φ) where σ ⊆ X is the scope of c and φ is a function from D|σ| to E.

The arity of a valued constraint is the size of its scope. The arity of a problem isthe maximum arity over its constraints. In this work, we are mainly concernedwith binary VCSPs, that is, VCSPs with unary and binary constraints only.Furthermore, the scopes of binary constraints are assumed to be ordered tuples.Hence, the constraint c = (〈xi, xj〉 , φ) differs from cT = (〈xj , xi〉 , φ

T ), whereφT , the transpose4 of φ, is such that φT (u, v) = φ(v, u), for all u, v ∈ D. Ofcourse, if c or cT is in C then both of them are in C.

As in [2], we shall assume throughout this paper that the set of valuationsE is the set of nonnegative integers (or reals) together with infinity (∞). Thelatter element will be used to designate a total incoherence whereas 0 will in-dicate a total coherence. Hence, the total order is ≤, the aggregation operatoris the sum (+) and its partial inverse is substraction (−). We obtain, therefore,the valuation structure (E,+,≤) which possesses all the properties outlinedin Definition 1. Moreover, for simplicity, we will assume in what follows thatD = {0, 1, . . . , |D| − 1}.

The valuation of an assignment t to a subset of variables V ⊆ X is obtainedby

4 We use the term transpose because, when binary functions are stored in matrices,the matrix associated to φT is the transpose of the matrix associated to φ.

ΦP (t) =∑

(σ,φ)∈~C,σ⊆V

φ(t ↓ σ) (4)

where t ↓ σ denotes the projection of t on the variables of σ and ~C is the sub-set {(〈xi, xj〉 , φ) ∈ C | i < j}. Hence, an overal optimal solution for a VCSPinstance P on n variables is an n-tuple t such that ΦP (t) is finite and minimalover all possible n-tuples.

In this paper, we are also led to cope with the (crisp) constraint satisfactionproblem (CSP), which is a VCSP with a valuation set E limited to {0,∞}. Inthis special case and since only solutions with finite costs are relevent, we canreplace every valuation function with a relation that exactly contains those tu-ples that have zero cost. The valuation structure S is, thereby, not necessary anymore. Consequently, a CSP can be defined by a triple (X,D,C), where every(crisp) constraint is defined by a scope and a relation specifying the tuples ofvalues allowed by the constraint.

Consider the set of all functions φ : D2 −→ E. Since only one valuation set(E) is used throughout the paper, we will simply write FD to designate thisset of functions. The set of all binary relations over domain D, in turn, will bedenoted by RD.

The valuation functions employed in defining the VCSPs considered in thispaper are elements of FD. These elements can be limited to specific subsets ofFD in order to obtain tractable VCSPs. Similarly, the binary relations used ina binary CSP can be limited to specific subsets of RD in order to get tractableCSPs. Restricting the functions (resp. relations) employed in defining a VCSP(resp. a CSP) is captured in the notion language. Hence, a valued binary con-straint language over domain D is any subset of FD, and a binary constraintlanguage over domain D is any subset of RD. For any valued constraint languageGD ⊆ FD, we will refer to the set of all VCSP instances with valuation functionsin GD by VCSP(GD). Similarly, for any constraint language SD ⊆ RD, we willrefer to the set of all CSP instances with relations in SD by CSP(SD).

We also use the following known tractable binary CSP class. Denote byfunct1V (resp. funct2V ) the (crisp) constraint language of all binary rela-tions over domain V that are functional on their first (resp. second) argumentand by bijectV the valued constraint language of all bijective binary relationsover domain V . Hence, CSP(bijectV ) for instance will refer to the set of allbinary CSP instances with relation in bijectV .

3 Modularity related VCSPs

To specify the VCSPs we are conserned with, we need to introduce the followingpredicates. Let φ be any function of FD, we associate to φ three quaternarypredicates modφ, submodφ and ssubmodφ that are defined as follows

modφ(u, u′, v, v′) ⇔def φ(u, u′) + φ(v, v′) = φ(u, v′) + φ(v, u′) (5)

submodφ(u, u′, v, v′) ⇔def φ(u, u′) + φ(v, v′) ≤ φ(u, v′) + φ(v, u′) (6)

(7)

where u, v, u′, v′ are any elements of D.We can easily see that these predicates verify the following equivalences.

modφ(u, u′, v, v′) ⇔ modφ(v, v

′, u, u′)

modφ(u, u′, v, v′) ⇔ modφT (u′, u, v′, v)

for all u, v, u′, v′ in D. Furthermore, these equivalences hold also if we replacethe modφ predicate by the submodφ one.

Definition 3. A binary function φ ∈ FD is submodular if and only if submodφ(u, u′, v, v′)

holds for all u, v, u′, v′ ∈ D such that u < v and u′ < v′.

Submodular functions of any arity are widely studied because they involvetractable discrete optimization problems [15, 16, 20, 21, 23].

Denote by submodD the language of all submodular functions of FD. Hence,VCSP(submodD) will designate the class of VCSPs with valuation functionsin submodD. This VCSP class will be called the class of submodular VCSPinstances.

Definition 4. Let φ be in FD. Two values u, v ∈ D are said to be modularequivalent with regard to φ, (notation u ∼φ v), if and only if modφ(u, u

′, v, v′)holds for all u′, v′ ∈ D.

Property 1. For any φ ∈ FD, ∼φ is an equivalence relation over D.

Proof. Reflexivity and symmetry hold trivially. Transitivity also holds if we as-sume that for every u ∈ D there exists v ∈ D such that φ(u, v) is finite.

⊓⊔

Since ∼ φ is an equivalence relation for all φ ∈ FD, we can consider thekernel of ∼φ which is defined according to (1). Hence, among the functions ofFD, we can distinguish those which induce an equivalence relation whose kernelis equal to D.

Definition 5. Let φ be in FD. φ is said to be prime if and only if for all u, v ∈D, u ∼φ v implies u = v.

In other words, φ is prime if and only if there is no two distinct elementsof D that are modular equivalent with regard to φ. It is easy to see that primefunctions can be equivalently specified by the condition ker(∼φ) = D. Denoteby primeD the language of all prime functions of FD. VCSP(primeD) will des-ignate the class of VCSPs with valuation functions in primeD. This VCSP classwill be called the class of prime VCSP instances.

The notion of modular equivalence can be extended to cope with functions.

Definition 6. Two functions φ, φ′ ∈ FD are modular equivalent, φ ∼ φ′, if andonly if ∼φ = ∼φ′ .

Observe that modular equivalent functions induce equivalence relations thathave the same kernel.

Property 2. ∼ is an equivalence relation over FD.

Proof. This follows immediatly from Definition 6.

⊓⊔

One of the equivalence class of ∼ is primeD. Indeed, all prime functions arepairwise modular equivalent since they induce the same equivalence relation overD, namely, the equality relation. Denote by equivD any equivalence class of ∼,the instances in VCSP(equivD) will be referred to as modular equivalent VCSPinstances.

4 Domain permutation for VCSP

The VCSP(submodD) class highlighted in the previous section is essential to ourpresent work because it is known to be tractable [2]. In other respects, in [10],the authors proposed a theory relying on domain permutations whose aim is totransform (crisp) CSP instances into CSP instances over a tractable constraintlanguage, namely the max-closed constraint language [11].

In this section, we study the problem of discovering domain permutationsthat transform a given binary VCSP instance into an instance in the VCSP(submodD)class. We call the set of all binary VCSP instances for which such transforma-tion is possible permuted submodular binary VCSP. The tool that will be used todiscover the required permutations is the domain pemutation reduction theory[10], which is adapted and extended to cope with valued constaints.

4.1 Definitions

We begin by presenting the definitions needed for applying the domain pemuta-tion reduction theory to the VCSP framework.

Denote by ΠD the set of all permutations of D. Let 〈π, π′〉 be an orderedpair of permutations of D, that is, 〈π, π′〉 ∈ Π2

D and let (v, v′) be an orderedpair of D2. Applying 〈π, π′〉 to (v, v′) results in the ordered pair 〈π, π′〉 (v, v′) =(π(v), π′(v′)) and applying 〈π, π′〉 to a binary function φ ∈ FD, yields the binaryfunction 〈π, π′〉φ defined by

〈π, π′〉φ(v, v′) = φ(π−1(v), π′−1(v′)), for all (v, v′) ∈ D2 (8)

Definition 7. Let P = (X,D,C, S) be a VCSP instance. A domain permuta-tion for P is a mapping Π assigning to each variable x ∈ X a permutation fromΠD.

– For any scope σ = 〈xi, xj〉, we define Π(σ) = 〈Π(xi), Π(xj)〉 ∈ Π2D.

– For any valued constraint (σ, φ) ∈ C, we define Π(σ, φ) = (σ,Π(σ)φ), whereΠ(σ)φ is defined by (8).

– The pemuted constraint set is given by Π(C) = {Π(σ, φ) | (σ, ρ) ∈ C}.– The permuted instance is defined by Π(P ) = (X,D,Π(C), S)

Let P be a VCSP instance. If there exists a domain permutation Π forP such that Π(P ) is in VCSP(submodD) then we say that P is reductible tosubmodD. Our aim is to determine whether a given VCSP instance is reductibleto submodD. This problem is called the reduction problem into submodD andresolving it efficiently is the key to extending the VCSP(submodD) tractableclass. To express this problem in the constraint satisfaction framework, we needto adapt the notion of a lifted relation, (see Definition 20 of [10]), in order tohandle valued constraints.

Definition 8. We define the relation lifting any function φ ∈ FD into submodD,ρ(φ, submodD), to be the following binary relation over ΠD:

ρ(φ, submodD) = {〈π, π′〉 ∈ Π2D | 〈π, π′〉φ ∈ submodD}

4.2 Reducing prime VCSPs

To begin with, we show that the reduction problem into submodD is tractablein the case where the original VCSP instance has its valuation function in theprimeD valued constraint language. In a subsequent step, we will consider thereduction problem into submodD for original VCSPs constructed from a largervalued constraint language.

Definition 9. The lifted (crisp) constraint language of primeD into submodD

is L(primeD, submodD) = {ρ(φ, submodD) | φ ∈ primeD}.

Definition 10. Let P = (X,D,C, S) be a VCSP instance. We define the liftedinstance for P into submodD, L(P, submodD), to be the (crisp) CSP instance(X,ΠD, {(σ, ρ(φ, submodD)) | (σ, φ) ∈ C}).

Lemma 4. L(primeD, submodD) ⊆ Funct1ΠD.

Proof. We have to prove that every relation in L(primeD, submodD) is func-tionnal on its first argument. To this end, we assume that there exists ρ inL(primeD, submodD) such that ρ is not functionnal on its first argument andproceed to get a contradiction. If ρ is not functionnal on its first argument thenthere must exist 〈π1, π〉 , 〈π2, π〉 ∈ ρ such that π1 6= π2. This implies that thereexists u, v ∈ D,u < v such that

π1(u) < π1(v) ∧ π2(u) > π2(v) ∨ π1(u) > π1(v) ∧ π2(u) < π2(v)

Assume, without loss of generality, that the lift-hand side of the disjunctionholds. On the other hand, ρ ∈ L(primeD, submodD) implies that there existsφ ∈ primeD such that ρ = ρ(φ, submodD). Furthermore, since φ ∈ primeD,there must exist u′ < v′ such that ¬modφ(u, u

′, v, v′), otherwise, u and v wouldbe modular equivalent and therefore φ would not be prime. Here, we distinguishtwo cases:

– π(u′) < π(v′): let φ1 = 〈π1, π〉φ. Since 〈π1, π〉 ∈ ρ, we must have φ1 ∈submodD, and then submodφ1

(π1(u), π(u′), π1(v), π(v

′)) must be true. By(8), we obtain submodφ(u, u

′, v, v′). Similarly, let φ2 = 〈π2, π〉φ. Since 〈π2, π〉 ∈ρ, we must have φ2 ∈ submodD, and then submodφ2

(π2(v), π(u′), π2(u), π(v

′)).By (8), we obtain submodφ(v, u

′, u, v′). But submodφ(u, u′, v, v′) and submodφ(v, u

′, u, v′)yield modφ(u, u

′, v, v′), thus a contradiction.

– π(u′) > π(v′): we can proceed in the same manner as in the first case todeduce the same contradiction.

Hence, every relation in L(primeD, submodD) is functionnal on its first argu-ment.

⊓⊔

Lemma 4 is the first stage for proving that the lifted instance of a VCSP(primeD)instance into submodD is built from a particular constraint language, namely,the bijectD constraint language. To derive this result, we need to establish thefollowing two identities.

Lemma 5. 5 Let φ ∈ FD and let π, π′ ∈ ΠD, then (〈π, π′〉φ)T = 〈π′, π〉φT .

5 We assume that the transpose operation has more priority than applying permuta-tions to functions.

Proof. For all u, v ∈ D, we have

(〈π, π′〉φ)T (u, v) = 〈π, π′〉φ(v, u)

= φ(π−1(v), π′−1(u))

= φT (π′−1(u), π−1(v))

= 〈π′, π〉φT (u, v)

⊓⊔

Lemma 6. For all φ ∈ FD, we have ρ(φ, submodD) = ρ(φT , submodD)T .

Proof. Exploiting the fact that the transpose of a submodular binary functionis also submodular and using the identity given in Lemma 5, we obtain

ρ(φ, submodD) = {〈π, π′〉 ∈ Π2D | 〈π, π′〉φ ∈ submodD}

= {〈π, π′〉 ∈ Π2D | (〈π, π′〉φ)T ∈ submodD}

= {〈π, π′〉 ∈ Π2D | 〈π′, π〉φT ∈ submodD}

= {〈π′, π〉 ∈ Π2D | 〈π′, π〉φT ∈ submodD}T

= ρ(φT , submodD)T

⊓⊔

As a first result, we prove that the lifted instance of a prime VCSP intosubmodD is a bijective binary CSP.

Theorem 1. If P is in VCSP(primeD) then L(P, submodD) is in CSP(bijectΠD).

Proof. Let P = (X,D,C, S) be in VCSP(primeD). We have to prove that everyconstraint ρ(φ, submodD) of L(P, submodD) is in bijectΠD

.Let (σ, φ) ∈ C. Since P is in VCSP(primeD), φ must be in primeD. By

Lemma 4, this guarantees that ρ(φ, submodD) is in funct1ΠD. On the other

hand, (σ, φ) ∈ C implies (σT , φT ) ∈ C, where σT is equal to σ in the reverseorder. It follows that φT is also in primeD. Again by Lemma 4, this guaranteesthat ρ(φT , submodD) ∈ funct1ΠD

, and then ρ(φT , submodD)T ∈ funct2ΠD.

According to Lemma 6, we obtain ρ(φ, submodD) ∈ funct2ΠD, and since we

have already proved that ρ(φ, submodD) ∈ funct1ΠD, we get ρ(φ, submodD) ∈

bijectΠD. Hence, the result.

⊓⊔

In [22], the authors proposed a O(|X|2|D|) algorithm for solving functionnalbinary CSP, a class that includes the bijective binary CSP one. Hence, the liftedCSP instance resulting from the reduction into submodD of a VCSP(primeD)instance, can be solved in O(n2|ΠD|) steps. Unfortunatelly, this time complexityis not polynomial on the size of the VCSP instance to be reduced, because |ΠD| =|D|! is factorial, unless |D| is O(1). Theorem 1 provides, however, a preliminaryresult indicating that the reduction problem into submodD is polynomial-timesolvable for VCSP(primeD) instances with a bounded domain.

4.3 Reducing modular equivalent VCSPs

As it was outlined above, FD is partitioned into equivalence classes by the mod-ular equivalence relation on functions, ∼, and primeD is just one of these equiv-alence classes. Let equivD be any equivalence class of FD, VCSP(equivD) willbe called a modular equivalent VCSP.

Consider the problem of reducing instances in VCSP(equivD) for any equivD

in FD/ ∼. Clearly, substituting equivD to primeD yields a more general reduc-tion problem. However, in what follow, we show that the reduction problem ofany instance in VCSP(equivD) into submodD is tractable.

Definition 11. Let P = (X,D,C, S) be in VCSP(equivD). The kernel of P isthe VCSP P = (X, D, C, S), where

– D = ker(∼φ), for any φ ∈ equivD.– C = {(σ, φ) | (σ, φ) ∈ C and φ is the restriction of φ to D2}.

We show that the valuation functions involved in the kernel of any instancein VCSP(equivD) are prime.

Lemma 7. Let φ ∈ FD and let (u, u′, v, v′), (x, x′, y, y′) ∈ D4 be such that u ∼φ

x, u′ ∼φTx′, v ∼φ y and v′ ∼φT y′. Then we have

(i) modφ(u, u′, v, v′) ⇔ modφ(x, u

′, v, v′)(ii) modφ(u, u

′, v, v′) ⇔ modφ(x, x′, y, y′)

Moreover, these equivalences also hold if we replace the mod predicate by submodor by ssubmod.

Proof.(i) modφ(u, u

′, v, v′) is equivalent to

φ(u, u′) + φ(v, v′) = φ(u, v′) + φ(v, u′) (9)

Adding φ(x, v′) to both members of (9), we get the following equivalent inequality

φ(x, v′) + φ(u, u′) + φ(v, v′) = φ(x, v′) + φ(u, v′) + φ(v, u′) (10)

On the other hand, u ∼φ x implies modφ(u, x, u′, v′), which is equivalent to

φ(u, u′) + φ(x, v′) = φ(u, v′) + φ(x, u′) (11)

Substituting the right-hand side of (11) to the left-hand side of (11) in (10), weobtain

φ(x, u′) + φ(v, v′) = φ(x, v′) + φ(v, u′)

which is equivalent to modφ(x, u′, v, v′).

(ii) We have

modφ(u, u′, v, v′) ⇔ modφ(x, u

′, v, v′)

⇔ modφ(y, v′, x, u′)

⇔ modφT (y′, y, u′, x)

⇔ modφT (x′, x, y′, y)

⇔ modφ(x, x′, y, y′)

⊓⊔

Lemma 8. Let φ be in FD and let D = ker(∼φ). If φ is modular equivalent toits transpose then φ, the restriction of φ to D2, is in primeD.

Proof. of Lemma 8. We must prove φ is prime. That is, for all u, v ∈ D, u ∼φ

v implies u = v. Assume that u ∼φ v and u 6= v for some u, v ∈ D andproceed to get a contradiction. The latter inequality implies that u ≁φ v sinceboth u and v are minimal in their respective equivalence classes. According toDefinition 4, this implies that there exist u′, v′ ∈ D such that ¬modφ(u, u

′, v, v′),and then ¬modφT (u′, u, v′, v). Moreover, since φ is modular equivalent to φT ,we must have ker(∼φT ) = D. It follows that u ∼φT u, v ∼φT v, u′ ∼φT u′

and v′ ∼φT v′. Then, by Lemma 7, we obtain ¬modφT (u′, u, v′, v), and then¬modφ(u, u

′, v, v′). Exploiting the fact that φ is the restriction of φ to D2, weobtain ¬modφ(u, u

′, v, v′), which means that u ≁φ v. Hence, a contradiction.

⊓⊔

Theorem 2. If P is in VCSP(equivD) then the kernel of P is in VCSP(primeD).

Proof. Let P = (X,D,C, S) be any instance of VCSP(equivD) and let (σ, φ)be any constraint of P , the kernel of P . By Definition 11, there must exist aconstraint (σ, φ) ∈ C such that φ is the restriction of φ to D2, where D is thekernel of ∼φ. Moreover, (σ, φ) ∈ C implies that φ ∈ equivD. Also, (σ, φ) ∈ Cimplies that (σT , φT ) ∈ C, where σT is equal to σ in the reverse order. It followsthat φT ∈ equivD, and then φ is modular equivalent to its transpose. Accordingto Lemma 8, φ must be in primeD. We deduce that P is in VCSP(primeD).

⊓⊔

Next, we show that the reduction problem of a VCSP(equivD) instance intosubmodD is closely related to the reduction of the instance kernel.

Lemma 9. Let φ ∈ FD be a function which is modular equivalent to its trans-pose, let π, π′ be two permutations of D = ker(∼φ) and π, π′ their respectiveexpantion to D. Then we have the following

〈π, π′〉φ ∈ submodD ⇔ 〈π, π′〉 φ ∈ submodD

Proof.(⇒) suppose that 〈π, π′〉φ ∈ submodD and 〈π, π′〉 φ /∈ submodD. The latterstatement implies that there exist u, v, u′, v′ ∈ D such that u < v, u′ < v′ and¬submod〈π,π′〉φ(u, u

′, v, v′). By (8), we obtain ¬submodφ(π−1(u), π′−1(u′), π−1(v), π′−1(v′)).

Since φ is the restriction of φ to D2 and both π and π′ are permutations of D,we obtain

¬submodφ(s, s′, t, t′) (12)

where u = π(s), u′ = π(s′), v = π(t) and v′ = π(t′). Moreover, since s, s′, t, t′ ∈D ⊆ D and π is a permutation of D, there must exist s, s′, t, t′ ∈ D such thats = π(s), s′ = π(s′), t = π(t) and t′ = π(t′). By making substitutions in (12), weobtain

¬submodφ(π−1(s), π′−1(s′), π−1(t), π′−1(t′)) (13)

Applying (8) to (13), we get

¬submod〈π,π′〉φ(s, s′, t, t′) (14)

Finally, since u < v and u′ < v′, we also have π(s) < π(t) and π(s′) < π(t′).Applying Lemma 1, we get π(s) < π(t) and π(s′) < π(t′), and then s < t ands′ < t′. In this case, (14) implies that 〈π, π′〉φ is not in submodD, which con-tradicts the hypothesis.

(⇐) Suppose that 〈π, π′〉φ /∈ submodD and 〈π, π′〉 φ ∈ submodD. The formerstatement implies that there exist u, v, u′, v′ ∈ D such that u < v, u′ < v′ and¬submod〈π,π′〉φ(u, u

′, v, v′). By (8), we obtain

¬submodφ(π−1(u), π′−1(u′), π−1(v), π′−1(v′)) (15)

Since π is a permutation of D, and u, v, u′, v′ ∈ D, there must exist s, t, s′, t′ ∈D such that u = π(s), v = π(t), u′ = π(s′) and v′ = π(t′). Thus, (15) isequivalent to ¬submodφ(s, s

′, t, t′). By Lemma 7 and the fact that φ is modularequivalent to its transpose, we obtain ¬submodφ(s, t, s

′, t′). In addition, since φis the restriction of φ to D2, we obtain

¬submodφ(s, s′, t, t′) (16)

Moreover, since π is a permutation of D, there must exist w, x, w′, x′ ∈ Dsuch that π(s) = w, π(t) = x, π(s′) = w′ and π(t′) = x′. By making substitutionsin (16), we obtain

¬submodφ(π−1(w), π′−1(w′), π−1(x), π′−1(x′))

and, by (8), we get

¬submod〈π,π′〉φ(w, w′, x, x′) (17)

Moreover, since π(s) = u < v = π(t) and π(s′) = u′ < v′ = π(t′), byLemma 1, we must have π(s) ≤ π(t) and π(s′) ≤ π(t′). In addition, we musthave s 6= t and s′ 6= t′ otherwise (16) will not hold. Since π is bijective, wededuce that π(s) < π(t) and π(s′) < π(t′) and then w < x and w′ < x′. Thetwo latter inequalities with (17) imply that 〈π, π′〉 φ is not in submodD, thus acontradiction.

⊓⊔

Theorem 3. Let P be in VCSP(equivD). Then P is reductible to submodD

if and only if its kernel is reductible to submodD.

Proof. Let P = (X,D,C, S). According to Theorem 2, P ∈ VCSP(equivD)implies that the kernel of P , P = (X, D, C, S) is in VCSP(primeD). Moreover,for any domain permutation Π of P , Definition 7 imposes that

(σ, φ) ∈ C ⇔ (σ,Π(σ)φ) ∈ Π(C) (18)

Definition 11, in turn, imposes that

(σ, φ) ∈ C ⇔ (σ, φ) ∈ C (19)

Again from Definition 7 and for any domain permutation Π of P , we obtain

(σ, φ) ∈ C ⇔ (σ, Π(σ)φ) ∈ Π(C) (20)

From (18), (19) and (20), we deduce that

(σ,Π(σ)φ) ∈ Π(C) ⇔ (σ, Π(σ)φ) ∈ Π(C) (21)

(⇒) Suppose that P is reductible to VCSP(submodD), which means that thereexists a domain permutation Π = 〈π1, . . . , πn〉, where each πi, 1 ≤ i ≤ n is apermutation of D and such that Π(P ) ∈ VCSP(submodD). Consider the tupleΠ = 〈π1, . . . , πn〉, where each πi, 1 ≤ i ≤ n is derived from πi, 1 ≤ i ≤ naccording to (3), that is

πi(v) = D(card{s ∈ D | πi(s) ≤ πi(v)})

According to Lemma 3, each πi, 1 ≤ i ≤ n is a permutation of D. Hence, Π isa domain permutation of P .

Now, we have to show that Π(P ) ∈ VCSP(submodD). Let (σ, Π(σ)φ) be anyconstraint of Π(C). By (21), (σ,Π(σ)φ) must be in Π(C), and since Π(P ) ∈VCSP(submodD), we must have Π(σ)φ ∈ submodD. By Lemma 9, we inferthat Π(σ)φ ∈ submodD. Π(P ) is, therefore, in VCSP(submodD), which meansthat Π lifts P into VCSP(submodD) and then P is reductible to submodD.

(⇐) Suppose that P is reductible to VCSP(submodD), which means that thereexists a domain permutation Π = 〈π1, . . . , πn〉, where each πi, 1 ≤ i ≤ n is

a permutation of D such that Π(P ) ∈ VCSP(submodD). Consider the tupleΠ = 〈π1, . . . , πn〉 where each πi, 1 ≤ i ≤ n is defined, in accordance withLemma 2, as follows:

πi(v) =∑

s∈D,πi(s)<πi(v)

card[s] + card{u ∈ [v] | u < v}

According to Lemma 2, each πi, 1 ≤ i ≤ n is a permutation of D. Hence, Πis a domain permutation of P .

Now, we show that Π(P ) ∈ VCSP(submodD). Let (σ,Π(σ)φ) be any con-straint of Π(C). By (21), (σ, Π(σ)φ) must be in Π(C), and since Π(P ) ∈VCSP(submodD), we must have Π(σ)φ ∈ submodD. By Lemma 9, we inferthat Π(σ)φ ∈ submodD. Π(P ) is, therefore, in VCSP(submodD), which meansthat Π lifts P into VCSP(submodD) and then P is reductible to submodD.

⊓⊔

Finally, we show that it is tractable to determine whether a given binaryVCSP instance is modular equivalent. This can be done by computing the mod-ular equivalence relation associated with every valuation function used in theinstance and cheking whether the obtained relations are all the same. Com-puting the modular equivalence relation for a simple binary funciton can beachieved in O(|D|3) steps using the algoritihm proposed in [14] (Algorithm 2),which outputs the equivalence classes of the relation. Hence, computing the mod-ular equivalence relations for all the constraints takes O(|X|2|D|3) steps. Theequality test between pairs of equivalence relations can be done in O(|D|) steps.Thus, for the pairwise comparison of all the relations, O(|X|4|D|) supplementarysteps are needed.

Proposition 1. The modular equivalent binary VCSP class is tractably identi-fiable.

Example 1. Consider the V CSP P1 over the valuation structure Q+, P1 is com-posed of four variables X1, X2, X3 and X4 whose cost constraints are composedsimultaneously of strictly supermodular functions φ (for example φ(a, b) = a2 +b2 − ab) and strictly submodular functions φ’ for example φ(a, b)’= ab as shownin Figure 1. So P1 is prime. We can easily verify that P1 /∈ V CSP (SubmodD).Now if we permute the values order of odd variables or we permute the valuesorder of even variables (see Figure 2). We can easily verify that after domainpermutation P1 become composed of both strictly submodular relations ϕ andϕ’, so now P1 ∈ V CSP (SubmodD) over the valuation structure Q+. P1 can besolved in polynomial time by OSAC [?] or by VAC [9] over Q+.

We suppose now that P1 is over the valuation structure Qm. We can also verifythat after domain permutation P1 become composed of both strictly submodularrelations ϕ and ϕ’, so now P1 ∈ V CSP (SubmodD) over the valuation structureQm ∀m ∈ IN. According to [9] P1 can be solved in polynomial time by VAC over

X1 X2

X4 X3

Fig. 1. A binary VCSP with strictly supermodular functions φ and strictly submodularfunctions φ’.

Qm only if P1 is VAC.

Finally, if we suppose that P1 is a fuzzy CSP, and ⊕ is the MAX operator.We can also verify that after domain permutation P1 become composed of bothstrictly submodular relations ϕ and ϕ’, so now P1 ∈ V CSP (SubmodD) whoseoperator is not strictly monotonic.

Example 2. Consider the V CSP P1 over the valuation structure Q+, P1 is com-posed of four variables whose cost constraints are composed of both strictlysubmodular and strictly supermodular functions as shown in Figure 3. So P1 isprime. We can easily verify that P1 /∈ V CSP (SubmodD). Now if we permutethe values order of odd variables (see Figure 4) or we permute the values orderof even variables (see Figure 5), We can easily verify now that after permutationP1 ∈ V CSP (SubmodD) over the valuation structure Q+. According to [9] P1

can be solved in polynomial time by VAC over Q+.

We suppose now that P1 is over the valuation structure Qm, and m = 7. We caneasily verify that after permutation P1 ∈ V CSP (SubmodD) over the valuationstructure Q7. According to [9] P1 can be solved in polynomial time by VAC overQm only if P1 is VAC.

Finally if we suppose that P1 is a fuzzy CSP, and ⊕ is the MAX operator.We can also verify that after permutation P1 ∈ V CSP (SubmodD) where theoperator is not strictly monotonic.

X1 X2

X4 X3

Fig. 2. A binary VCSP with strictly submodular relations ϕ and ϕ’ after domainpermutation of the values of odd or even variables.

Example 3. Consider the V CSP P2, over the valuation structure Q, composedof sixteen variables whose cost constraints are composed of both strictly sub-modular and strictly supermodular functions, so P2 is prime, as shown in Figure6. We can easily verify that P2 /∈ V CSP (SubmodD).Now if we reverse the values order of even variable (see Figure 7), we can easilyverify, after this permutation reduction, that P2 ∈ V CSP (SubmodD). We haveproved that finding this permutation can be done in polynomial time over thevaluation structure Q, so we get a polynomial super-class of submodular VCSP.

Example 4. Consider the VCSP P composed of n variables whose constraintgraph is defined as follows:Two variables of the same parity are connected by a set of strictly submodu-lar functions. While, two variables of different parity are connected by a set ofstrictly supermodular functions. We can easily verify that P /∈ V CSP (SubmodD).By reversing the values order of even (or odd) variables, we obtain a constraintgraph where all constraints are defined by submodular functions. We have provedthat finding this permutation can be done in polynomial time, over the valuationstructure Q, so we get a polynomial super-class of submodular VCSP.

1

3

2 4

Fig. 3. A binary VCSP with strictly submodular and strictly supermodular functions.

5 Related work

In [10], the authors proposed a theory whose aim is to extend the max-closedCSP tractable class. Their approach consists in discovering permutations which,when applied to every value domain, result in a max-closed CSP instance. Ifsuch permutations exist, the permuted instance is solved in polynomial timeand then a solution to the original instance is deduced by simply inverting thepermutations.

The primary advantage of the approach is that the seach for domain permu-tations itselt is expressed as a CSP referred to as the lifted constraint instance.In some cases, the latter CSP has the advantage of beying tractable. This is thecase, for instance, for bounded arity CSP with Boolean domain. Nonetheless,have proved that finding is an NP-complet even for binary CSPs thee-valueddomains.

1

3

2 4

Fig. 4. A permutation of the values of odd variables.

renamable Horn class

A set of clauses is renamable Horn if there exists a replacement of some literalswith their negated versions which transforms all clauses into Horn clauses.

Recal that a Horn clause is a disjunction of literals in which there at mostone positive literal.

The idea of domain permutation was also used by E. Chen et al in view oftransforming binary CSPs into the connected row convex tractable class [1].

but it appeared that their algorithm recognizes only a subset of the binaryCSP instances that can be transformed into the CRC class. In fact, it has beenproved that the connected row convex identification problem is intractable fordomains of size four or more [10].

Schlesinger [18], in the context of image processing, suggested an equivalentversion of this work that does not carry the hard constraints.

1

3

2 4

Fig. 5. A permutation of the values of even variables.

When a problem P is Virtual Arc Consistency (VAC) [8, 9], it is known thatthe problem Bool(P ) has a nonempty closed arc consistency. This allows VACto inherit different classes of tractable problems that are resolved by arc consis-tency in CSPs. Indeed, VAC can solve the problems of minimizing submodularfunctions, which is a non-trivial polynomial language of VCSP, on the valua-tion structure Q+ [3]. Where Q+ = 〈Q+ ∪ {∞},+,≥〉 and Q+ represents allnon-negative rational numbers. It is already known that OSAC solves the VCSPwith submodular cost functions [7]. In [9] authors have shown that VAC canobtaine optimal solution of a V CSP whose cost functions are all submodularfor domains order are known or unknown. On a valuation structure Q+.

X1 X2

X4 X3

Fig. 6. A binary VCSP with stricly submodular and strictly supermodular functions.

6 Conclusion

The domain permutation reduction problem into the submodular valued con-straint language is tractable for all VCSP instances built from modular equiva-lent binary functions.

References

1. Chen E., Zhang Z., Wang X., Aihara K. An algorithm for fast recognition of con-nected row-convex constraint networks. In JFPLC 2001, The Tenth International

French Speaking Conference on Logic and Constraint Programming, 43–58, April2001.

2. Cohen, D. A., Cooper, M. C., Jeavons, P. G., Krokhin, A. A.: A maximal tractableclass of soft constraint satisfaction. Journal of Artificial Intelligence Research 221–22, 2004.

3. Cohen, D. A., Cooper, M. C., Jeavons, P. G., Krokhin, A. A.: The complexity ofsoft constraint satisfaction. Artificial Intelligence 170, 983–1016, 2006.

4. Cohen, D. A., Cooper, M. C., Jeavons, P. G.: Generalising submodularity andhorn clauses: Tractable optimization problems defined by tournament pair multi-morphisms. Theoretical Computer Science 401, 36–51, 2008.

5. Creignou, N., Khanna, S., Sudan, M.: Complexity Classification of Boolean Con-straint Satisfaction Problems. In SIAM Monographs on Discrete Mathematics and

Applications, 7 SIAM, 2001.6. Cooper, M. C.: Arc consistency for soft constraints. Artificial Intelligence 154

199–227, 2004.7. Cooper, M. C. Minimization of locally-defined submodular functions by Optimal

Soft Arc Consistency. Constraints 13 (4), 2008.8. Cooper M. C., de Givry S., Sanchez M., Schiex T., Zytnicki M. Virtual arc con-

sistency for weighted csp. In Procceding of AAAI-08 Chicago, IL, 2008a.

X1 X2

X4 X3

Fig. 7. A permutation reduction of a binary VCSP with strictly submodular andstrictly supermodular functions to a VCSP ∈ VCSP(Submod).

9. Cooper, M. C., de Givryb S., Sanchezb M., Schiex T., Zytnicki M., Werner T.,Soft arc consistency revisited. Artificial Intelligence 174, 449–478, 2010.

10. Green, M., Cohen, D. Domain permutation reduction for constraint satisfactionproblems. Artificial Intelligence, 172, 1094–1118, 2008.

11. Jeavons, P.G., Cooper, M.C. Tractable constraints on Ordered domains. ArtificialIntelligence, 79 (2) : 327339, 1995.

12. Khatib, L., Morris, P., Morris, R., Rossi, F.: Temporal constraint reasoning withpreferences. In Proceedings of the 17th International Joint Conference on Artificial

Intelligence IJCAI-01 Seattle, USA (2001) 322–327, 2001.13. Mohr, R., Henderson, T.C. Arc and path consistency revisited, Artificial Intelli-

gence, 28, 225233, 1986.14. Naanaa, W., Helaoui, M., Ayeb, B. A decomposition method for valued CSPs In

SofT’10 - 10th Workshop on Preferences and Soft Constraints, 2010.15. Narayanan, H. Submodular Functions and Electrical Networks, North-Holland,

Amsterdam, 1997.16. Nemhauser, G., Wolsey, L. Integer and Combinatorial Optimization, John Wiley

and Sons, 1988.17. Orlin J.B. A faster strongly polynomial time algorithm for submodular function

minimization. Mathematical Programming, 118 (2) 237–251, 2009.18. Schlesinger, D. Exact Solution of Permuted Submodular MinSum Problems, In

Energy Minimization Methods in Computer Vision and Pattern Recognition, num-ber 4679–2007 in LNCS, 28–38, 2007.

19. Schiex, T., Fargier, H., Verfaillie, G.: Valued constraint satisfaction problems: hardand easy problems. In Proceedings of the 14th IJCAI Montreal, Canada, 631–637,1995.

20. Schrijver, A. A combinatorial algorithm minimizing submodular functions instrongly polynomial time, Journal of Combinatorial Theory, Series B 80 (2) 346–355, 2000.

21. Topkis, D. Supermodularity and Complementarity, Princeton University Press,1998.

22. Zhang, Y., Yap, R. H. C., Jaffar,.J : Functional Elimination and 0/1/All Con-straints. In AAAI IAAI, 175–180, 1999.

23. Zivn, S., Jeavons, P.G. Classes of submodular constraints expressible by graphcuts. In Proceedings of the 14th International Conference on Principles and Prac-

tice of Constraint Programming, CP’08, in Lecture Notes in Computer Science,5202, Springer, 2008.

A Proof of Lemma 1 to 3

Proof. of Lemma 1Suppose the left-hand side of the inequality is true. We have, therefore

π(u) =∑

s∈D, π(s)<π(u)

| [s] | + |{s ∈ [u] | s < u}|

<∑

s∈D, π(s)<π(u)

| [s] | + | [u] |

<∑

s∈D, π(s)≤π(u)

| [s] |

<∑

s∈D, π(s)<π(v)

| [s] |

<∑

s∈D, π(s)<π(v)

| [s] | + | {s ∈ [v] | s < v} |

< π(v)

⊓⊔

Proof. of Lemma 2First, we check that π is a mapping of D. Indeed, π(v) exists for all v ∈ D andthen, by (2), π(v) exists for all v ∈ D. Furthermore, for all v ∈ D, π(v) is anonnegative integer that verifies

π(v) =∑

s∈D,π(s)<π(v)

| [s] | + | {s ∈ [v] | s < v} |

≤∑

s∈D,π(s)<π(v)

| [s] | + | [v] | − 1

≤∑

s∈D,π(s)≤π(v)

| [s] | − 1

≤ |D| − 1

π(v) is therefore a mapping from D to D.

Next, we prove π is injective. Assume the converse, that is, π(u) = π(v) forsome u, v ∈ D such that u 6= v. We distinguish two cases:

– [u] = [v], which implies that u = v and, since π is bijective, π(u) = π(v).Thus,

∑

s∈D, π(s)<π(u)

| [s] | =∑

s∈D, π(s)<π(v)

| [s] |

By (2), the cardinality of {s ∈ [u] | s < u} and {s ∈ [v] | s < v} must be thesame, which implies that u = v and contradicts the hypothesis.

– [u] 6= [v], which implies that u 6= v, and then, that π(u) 6= π(v) since πis bijective. Assume, without loss of generality, that π(u) < π(v). ApplyingLemma 1, we get π(u) < π(v), which is a contradiction.

In both cases, we derived a contradiction. Thus, π is an injective mapping fromD to D. It is, therefore, a permutation of D.

⊓⊔

Proof. of Lemma 3We can see, from (3), that π′(v) exists for all v ∈ D and is in D. Indeed,v ∈ D ⊆ D then π′(v) exsits for all v ∈ D. Moreover, the arguments of functionD(.) are integers of {1, . . . , |D|}. Thus, π′(v) is in D.

Now, we prove that π′ is injective. So, we assume that there exist u, v ∈ Dsuch that u 6= v and π′(u) = π′(v) and proceed to get a contradiction. u 6= vimplies that π′(u) 6= π′(v) since π′ is bijective. Assume, without loss of generality,that π′(u) < π′(v). This implies that {s ∈ D | π′(s) < π′(u)} must containless elements than {s ∈ D | π′(s) ≤ π′(v)}, and since D(.) is clearly strictlyincreasing, we obtain π′(u) < π′(v). Hence, a contradiction. π′ is, therefore, aninjective mapping from D to D, which implies that π′ is a permutation of D.

⊓⊔