Constraint Satisfaction Problems : an attempt to transpose a dichotomy theorem

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Constraint Satisfaction Problems : an attempt to transpose a dichotomy theorem

Ir´en´ee Briquel

under the direction of Pascal Koiran

1 Introduction

Our work is motivated by two papers from Schaefer [10] and by Creignou and Hermann [3]

establishing dichotomy theorems for constraint satisfaction problems, together with definitions in algebraic complexity of similar problems ( [8], [6]).

A logical relation is a function that takes a boolean vector and returns a boolean. For a setS of logical relations, Schaefer defines in [10] aS-formula as a conjunction of logical relations from S, where the arguments of each relation are freely chosen among a set of variables. S-formulas are thus a subset of the boolean formulas. Schaefer defines the generalized satisfaction problem SAT(S) as the problem of deciding whether a given S-formula is satisfiable. For example, with as set S the set of all 3-clauses, SAT(S) is 3-SAT. Schaefer’s main result is, that depending on S, SAT(S) is either in P, or NP-complete.

This dichotomy theorem was transposed by Creignou and Hermann to the problems #SAT(S) of counting the number of satisfying assignments of a given S-formula. This time, #SAT(S) is either in FP, or #P-complete.

In fact, the sets S such that #SAT(S) is in FP are exactly the sets containing only affine constraints : constraints of the formx1⊕ · · · ⊕xm = 0 or x1⊕ · · · ⊕xn= 1.

Here we look for an algebraic equivalent to those problems, and try to extend further Schae- fer’s theorem. We use definitions introduced in [8] and [6] of what could be an algebraic extension of those problems : the polynomial associated to a boolean formulaφisP(φ)(X) =P

ǫφ(ǫ)X^ǫ, where X^ǫ is the product X₁^ǫ¹ × · · · × X_n^ǫⁿ. Our goal is also to study the complexity of the computation of families of polynomials associated toS-formulas, depending on S.

The following three sections are devoted to the search for a similar dichotomy theorem for our algebraic polynomials. We do not obtain a general dichotomy theorem, but we can conclude in several interesting cases : in fact, we show the VNP-completeness of families of polynomials associated to S-formulas when S is not reduced to affine constraints, but only we show that polynomials associated to affine formulas are computable in polynomial time for particular entries.

In the Appendices, we propose some proofs that could not be included in the main text, and further works. First, results obtained while accomplishing the previous study, such as new proofs of completeness in the boolean case thanks to the light of the algebraic case. But also independent results in relation with Koiran and Meer in [6], devoted to the expressive power of polynomials associated to formulas of bounded treewidth.

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2 From boolean to algebraic complexity

2.1 Constraint satisfaction problems

We define a logical relation to be a function from {0,1}^k to {0,1}, for some integer k, called the rank of the relation. Let us fix a finite set S = {φ1, . . . , φn} of logical relations. In [10]

, Schaefer defines an S-formula over r variables (x₁, . . . , x_r) to be any conjunction of boolean formulas, each of the formφ_i(x_i₁, . . . , x_i_m) whereφ_i ∈S, and m is the number of arguments of φi.

Schaefer defines also a generalized satisfaction problem SAT(S) as the problem of deciding whether a given S-formula is satisfiable. Schaefer obtains a remarkable result :

Theorem 1 [10] For any finite set of logical relationsS, the problemSAT(S)is either polynomial- time decidable or NP-complete.

This dichotomy theorem has known several extensions to various directions [2]. In particular, it was transposed by Creignou and Hermann to the problem of counting the number of satisfying assignments of a boolean formula. We remark, that the problem of counting the number of satisfying assignments of a formula is more difficult than to decide its satisfiability : if we know the number of satisfying assignments, we can check if this number is zero.

Theorem 2 [3] The problem#SAT(S), of counting the number of satisfying assignments of a given S-formula is either in FP(ie computable in polynomial time), or #P complete.

An amazing fact is, that the sets S for which #SAT(S) is #P-complete do not correspond to the ones for which SAT(S) is NP-complete. For example, #−2−SAT is #P-complete while 2−SAT is in P.

In fact, the sets S such that #SAT(S) are in FP are exactly the sets containing only affine constraints : constraints of the formx1⊕ · · · ⊕xm = 0 or x1⊕ · · · ⊕xn= 1.

Here we look for an algebraic equivalent to those problems, and try to extend further Schae- fer’s theorem.

2.2 Algebraic complexity theory : Valiant’s model

We work in the field of the algebraic complexity theory, and consider the model introduced by Valiant in [11]. We also give a small introduction to Valiant’s model. We fix a field of characteristic 0. All considered polynomials are over this field.

In Valiant’s model, we study the computation of multivariate polynomials. A p-family is a sequence f = (f_n) of multivariate polynomials such that the number of variables and the degree are p-bounded functions of n. An interesting example of p-family is the permanent family PER = (PER_n), where PER_n is the permanent of an n×n matrix with independent indeterminate entries.

We define the complexity off_nto be the minimum number of nodes of an arithmetic circuit (with nodes +,×,−) sufficient to computef_nwith as entry nodes the variablesX_i and constants ink.

Definition 1 VP A family isp-computable if the family of complexities of thef_nis ap-bounded family of integers. Those families constitute the complexity class VP.

The analogue of the class NP in Valiant’s model is the class VNP.

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Definition 2 VNP A p-family (f_n) is called p-definable iff there exists a p-computable family g= (gn) such that

f_n(X₁, . . . , X_p(n)) = X

ǫ∈{0,1}^q(n)

g_n(X₁, . . . , X_p(n), ǫ₁, . . . , ǫ_q(n))

The set of p-definable families form the class VNP.

We obviously note, that VP is included in VNP. We also introduce a notion of reduction : Definition 3 p-projection A polynomial f_n with v arguments is said to be a projection of a polynomial gm with uarguments, and we denote it fn≤gm, ifff(X1, . . . , Xv) =gm(a1, . . . , au) for some a_i chosen among the variables X_j and the constants in our field k.

A p-family f = (f_n) is a p-projection of g = (g_m) iff there exists a p-bounded function t:N→N such that

∃n₀∀n≥n₀, f_n≤g_t(n)

We denote it f ≤_pg.

Definition 4 VNP-completeness A p-family g ∈ VNP is called VNP-complete iff any f in VNP is ap-projection of g.

The main result in Valiant’s theory is the VNP-completeness of the permanent [1].

2.3 The associated polynomial

Here we look for an algebraic equivalent to the problems SAT(S) and #SAT(S). We define the polynomial associated to a boolean formula in the following way :

Definition 5 To a boolean formula φ with m arguments, we can associate a polynomial P(φ) in a natural way :

P(φ)(X₁, . . . , X_m) = X

ǫ∈{0,1}^m

φ(ǫ)X₁^ǫ¹ × · · · ×X_m^ǫ^m

If we compute the polynomial associated toφon the entry (1, . . . ,1), we obtain the number of satisfying assignments of φ. So we can see the computation of this polynomial as a more general problem than the counting of the number of satisfying assignments of φ.

For eachp-family (φ_n)_n∈N ofS-formulas the family of polynomials (P(φ_n))_n∈N is ap-family.

We want to study the complexity of families of polynomials associated with S-formulas, depending on the choice of S.

We here remark, that for each setS, we had a unique problem #SAT(S), but we now have an infinite number of p-families of polynomials associated to p-families of S-formulas. We cannot expect to show, for a fixed setS, every family (P(φ_n)), where theφ_narep-boundedS-formulas, are VNP-complete : for every S, we can construct trivial families of associated polynomials that do not inherit all the complexity of S.

An equivalent to Schaeffer’s and Creignou and Hermann’s theorems would be to show, that for each setS of logical relations :

• either for allp-families of S-formulas (φ_n), (P(φ_n)) is in VP

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• or there exists a p-family ofS-formulas (φ_n) such that (P(φ_n)) is VNP-complete.

In the following section, we consider the case of the affine constraints, and fail to establish that the families of associated polynomials are computable in polynomial time. In the next section, we establish, that for sets S containing not affine constraints, there exists a VNP- complete family of polynomials associated to S-formulas.

3 The case of affine constraints

In this section, we study S-formulas, where S is reduced to affine constraints. We recall that our goal is to show either the belonging to FP of all families of associated polynomials, or the VNP-completeness of a certain family. Since it seems unlikely to find VNP-complete families of associated polynomials, we study is those families are easy to compute.

3.1 Computation of polynomial associated with affine formulas

We define affine formulas as the conjunction of constraints of the formsa₁⊕ · · · ⊕a_m = 0 and a1⊕ · · · ⊕an= 1, where (a1, . . . , an) are boolean variables.

A conjunction of affine formulas may be seen as a system of linear equations over the field Z/2Z. Since such system may be solved by Gaussian elimination, the number of satisfying assignments of the conjunction is computable in polynomial time.

Therefore, we may ask ourselves if the computation of polynomials associated to a p-family of affine formulas is in VP. We won’t establish this result, but we will prove that we can compute efficiently such polynomial on the values {1,−1}.

Let (φ_n)n∈N be a family of affine formulas of polynomial size. Let fix n an integer, and be m the number of variables of φ_n. As we remarked, we can solve the formula φ_n in polynomial time. The solution set S is an affine subspace of {0,1}ⁿ. Since φn accepts only the vectors in S, we have :

P(φn)(X1, . . . , Xm) = X

ǫ∈{0,1}^m

φ(ǫ)X^ǫ=X

η∈S

X^η

WhereX^ι is the product : X₁^ι¹× · · · ×X_m^ι^m ifι= (ι1, . . . , ιm). Letkbe the dimension ofS, and (e₁, . . . , e_k) a basis ofS. For all iin [0, k], let pute_i= (e_i,1, . . . , e_i,k). We can express :

P(φn)(X1, . . . , Xm) = X

η∈{0,1}^k

X₁^η¹ê^1,1^{⊕···⊕η}^kê^k,1× · · · ×Xm^η¹ê^1,m^{⊕···⊕η}^kê^k,m

Since this expression has an exponential number of monomials, it is not computable directly.

We do not know how to compute the polynomial in general, but in the case of entries in{1,−1}, the problem becomes easier.

In the case when everyX_ibelongs to{1,−1}, eachX_iverifies : X_i² = 1. Thus,X_i^η¹ê^1,i^{⊕···⊕η}^kê^k,i = X_i^η¹ê^1,1^+···+η^kê^k,1. The polynomial P(φ_n) may be factorized :

P(φ_n) = Y

i=1...k

X

ηi∈{0,1}

X₁^ηⁱêî,1×. . .×Xm^ηⁱêî,m

Each term of the product is now computable by a circuit of polynomial size. So this polynomial is computable by an arithmetic circuit of polynomial size. We have proved that the polynomial P(φ_n) is computable by an arithmetic circuit of polynomial size on entries in {−1,1}.

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We can also remark, that the polynomials associated with unique affine constraints (φ_n) are computed by circuits of polynomial size. Let fixφan affine constraintx1⊕ · · · ⊕xn=b. We can compute recursively the polynomials P_i¹ and P_i⁰ associated to the constraints x₁⊕ · · · ⊕x_i= 1 and x₁⊕ · · · ⊕x_i= 0 respectively. In fact, we have :

P_i¹(X₁, . . . , X_i) =X_i×P_i−1⁰ (X₁, . . . , X_i−1) +P_i−1¹ (X₁, . . . , X_i−1) and

P_i⁰(X₁, . . . , X_i) =X_i×P_i−1¹ (X₁, . . . , X_i−1) +P_i−1⁰ (X₁, . . . , X_i−1)

By dynamic programming, we can compute our polynomial in linear time. This method is still valid with a conjunction ofm affine constraints , but it asks to compute the intermediate polynomials for all combination of possible constants (b1, . . . , bm)∈ {0,1}^m in the right side of the equalities defining the constraints. So this method only ensures polynomial computations when the number of constraints grow logarithmically.

4 The not affine constraints

In this section, we establish the existence of VNP-complete families of polynomials associated withS-formulas, whenSis not reduced to affine constraints. This proof follows closely Creignou and Hermann’s proof of #P-completeness of the corresponding counting problems in [3].

There exists exactly tree logical relations with two arguments that are not affine : the positive two-clauses OR₀ : (x,y) →(x∨y), the implicative two-clauses OR₁ : (x,y)→ (x∨y) and the negative two-clauses OR₂ : (x,y)→(x∨y).

We successively consider the sets S reduced to {OR₂},.{OR₀} and {OR₁} and search for VNP-complete families of polynomials associated to S-formulas. Then we will show, that we can reduce one of those tree cases to each set S not reduced to affine constraints.

4.1 Negative 2-clauses

Here we consider the setS ={OR₂}={(x,y)→(x∨y)}. We prove that there exists a p-family of polynomials (P(φ_n)) associated to a p-family of S-formulas (φ_n), that is VNP-complete.

The partial permanent PER^∗(A) of a matrixA= (A_i,j) is defined as follows : PER^∗(M) =X

π

Y

i∈defπ

A_iπ(i)

where the sum is over all injective partial maps from [1, n] to [1, n]. It is shown in [1], that the partial permanent defines a VNP-complete family of polynomials. The partial permanent may be seen as the family of polynomials (P(φ_n)) associated with a family of formulas (φ_n), such thatφ_nrecognizes the matrices of partial maps from [1, n] to [1, n]. What is interesting, is that such formulaφn may be simply defined as follows :

φ_n(ǫ) = ^

i,j,k:j6=k

ǫ_ij ∨ǫ_ik∧ ^

i,j,k:i6=k

ǫ_ij∨ǫ_kj

The first conjunction ensures that the matrixǫhas no more than one 1 on each row ; the second ensures that ǫhas no more than one 1 on each column. Soφn accepts exactly the partial maps from [1, n] to [1, n]. Since (φ_n) is a family of {OR₂}-formulas, we have :

Theorem 3 (φ_n) is a p-family of {OR₂}-formulas such that the p-family of associated polynomials (P(φ_n)) isVNP-complete.

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4.2 Positive 2-clauses

Here we consider the setS ={OR₀}={(x,y)→(x∨y}. We search a problem that would be naturally reduced to a conjunction of such formulas.

Actually, we can consider a problem of vertex covering in a graph G = (V, E). A vertex cover ofG is a setS ∈V such that for each edge e= (u, v)∈E, we have : (u ∈S)∨(v∈S).

We associate to each vertex vi ∈V a weight Xi : we will say that G is vertex weighted. The vertex covering polynomial of Gwill be the polynomial

V CP(G) =X

S

Y

vi∈S

X_i

where the sum is taken on all the covering sets of G. Now let us associate to each v_i ∈ V a boolean variableǫi with the meaning, thatvi is chosen in our covering set whenǫi is true. For each edge between two vertices v_i and v_j, the covering of the edge may be expressed byǫ_i∨ǫ_j. So the polynomial V CP(G) may be expressed by

V CP(G) = X

ǫ∈{0,1}^|V^|

^

(vi,vj)∈E

ǫ_i∨ǫ_j X^ǫ

So this polynomial may be expressed as a polynomial associated to a{OR₀}-formula. More- over, the size of theS-formula isp-bounded by the size of the graph. Therefore, each p-family of covering polynomials may be identified to a p-family of polynomials associated withS-formulas.

So we search for a VNP-complete family of covering polynomials.

For an integer k, we note V CP_k(G) the polynomial obtained by adding the weights of all vertex coverings of G cardinality k. So V CP_k(G) is defined like V CP(G), but with a sum over only the coverings S such that |S|=k. The problem of counting the vertex coverings of a certain cardinality in a graph is known to be #P-complete. Actually, we have a reduction from #3−SAT to this problem (see [4]). We adapt this reduction to obtain a projection from polynomials associated to boolean functions to polynomials of the typeV CP_k(G).

From the previous section, we have a VNP-complete family of polynomials associated to a conjunction of 2-clauses. Thus, we can simplify our reduction by projecting only the polynomials associated with conjunctions of 2-clauses. We could even consider conjunctions of 2-clauses of the type OR₂, but it would not simplify our reduction. Our projection follows the one given in [4].

Let φ be a conjunction of 2-clauses. We construct the (weighted) graph G_φ this way : for every variable ǫ_i inφ, we add two vertices ǫ_i and ǫ_i, linked by an edge. The edge between the two vertices will ensure that at least one is chosen in the covering set. The allowed cardinality k for a covering set will ensure that exactly one is chosen . This choice will correspond to the assignment of the variable ǫ_i. For every clausec_i inφ, we add also two linked vertices c_i,1 and c_i,2, each of them linked with the vertex of the corresponding literal in the clause. Again, the choice ofk will ensure, that exactly one of the two vertices c_i,1 and c_i,2 is chosen.

If we admit this fact, that the choice of the cardinality kwill ensure, that exactly one of the vertices of each pairǫ_i andǫ_i, orc_i,1 andc_i,2 is chosen, we have :

Lemma 1 To each satisfying assignment ofφcorrespond covering sets of cardinalityk, each of them containing the vertices corresponding to the assigned values of the variables. Furthermore, no covering set of cardinality k correspond to unsatisfying assignments.

Proof. Since each covering set of cardinality k contain exactly one of the two verticesǫi and ǫi

for all variableǫ_i, a unique assignment of the variables correspond to it.

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c_1,1 c_1,2 c_2,1 c_2,2 c_3,1 c_3,2 d_2,1

e_2,1 e_3,1

d_2,2 d_3,1

e_2,2

d_3,2 e_3,2 e_1,2

d_1,2

e_1,1

d_1,1

ǫ₃ ǫ₃

ǫ₂

ǫ₁ ǫ₁ ǫ₂

Figure 1: Graph G_φ obtained for φ= (ǫ₂∨ǫ₁)∧(ǫ₁∨ǫ₃)∧(ǫ₂∨ǫ₃)

An assignment is satisfying when at least one literal of each clause is satisfied. We construct a corresponding set of cardinalitykby adding vertices to the setS of the vertices corresponding to literals set to true in the assignment. For a given clausec_i, suppose without loss of generality that the first literal is satisfied. Then this literal belongs toS, and the edge between this literal and c_i,1 is covered. By adding c_i,2 to S, we ensure the covering of all the edges implied in this clause. By choosing similarly one vertexc_j,1 orc_j,2 for all clausec_j, we construct a covering set of cardinalityk of G_φ corresponding to our assignment.

Conversely, a covering set of cardinalitykcorresponding to a non satisfying assignment must contain the set S of vertices corresponding to the literals set to true in the assignment. Since this assignment is not satisfying, there exists a clause ci such that neither of the literals of the clause is verified. So neither of the edges betweenc_i,1 and c_i,2 and the corresponding literals is covered byS. Since only one of those two vertices may be chosen if we limit our covering set to the cardinality k, it is impossible to cover the two edges. No covering set of cardinality k correspond to this assignment. 2

However, we don’t have a bijection between the satisfying assignments and the covering sets : in the case when the two literals of a clause are satisfied, a corresponding covering set may containc_i,1 as well asc_i,2. To cope with this, for each literal linked to a vertex c_i,j, we add two vertices d_i,j and e_i,j, and two edges between the negation of the literal and d_i,j, and between d_i,j and e_i,j; we will also choose k to allow exactly one of those two vertices in a covering set.

This vertex will be freely chosen when the negation of the literal is satisfied, and also when at most one literal of the clause is satisfied. This way, for a satisfying assignment, exactly one free choice per clause is possible to have a corresponding covering set.

Figure 1 shows an example of such construction.

Let v denote the number of vertices of G_φ. We can choose the value ofk:

Lemma 2 Let k=v/2 andS be a covering set ofG_φof cardinality k. For each pair of vertices

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(ǫ_i, ǫ_i), (c_i,1, c_i,2) or (d_i,j, e_i,j), exactly one of the two vertices belongs to S.

Proof. Every vertex of G_φ belongs to one pair of the form (ǫi, ǫi), (ci,1, ci,2) or (di,j, ei,j). Since each of those pairs are linked by an edge in G_φ, at least one of the two vertices of each pair belongs to S. So |S| ≥ v/2. Since |S|=k =v/2, exactly one of the two vertices of each pair belongs to S. 2

This fact ensures us a correspondence between satisfying assignments ofφand covering sets of cardinalityk of G_φ :

Lemma 3 Let us denote by c the number of clauses of φ. To a satisfying assignment of φ correspond exactly 2^c covering sets of cardinality k of G_φ.

Proof. Let C be a covering set ofG_φ of cardinalityk, corresponding to a satisfying assignment S of φ. Among the vertices corresponding to literals, C must contain exactly the vertices corresponding to the literals set to true inS. Let consider a clause ci=x∨y, and see which of the vertices ofG_φ added to express this clause may be chosen inC.

When exactly one literal is satisfied, let say the first one x,C must containc_i,2 to cover the edge between ci,2 and y. Since x is not in C, to cover the edges (ei,1, di,1) and (di,1, x) with only one of the two vertices e_i,1 and d_i,1, C must contain d_i,1. On the other hand, y is in C;

C may containe_i,2 as well as d_i,2. Two different restrictions from C to the vertices expressing that clause are possible.

When the two literals are satisfied, their negations are not in C, and C must contain d_i,1 and d_i,2, and e_i,1 and e_i,2. But c_i,1 as well as c_i,2 may be chosen. Once more, two different restrictions fromC to the vertices expressing that clause are possible.

Since the choices of the restrictions of S to the vertices expressing each clause are independent, 2^c choices of C are possible. To S correspond 2^c different covering sets C of cardinality k.

2

We may suppose that an order among the vertices of G_φ is fixed, such that we can denote G_φ(w₁, . . . , w_c) the weighted graph obtained by adding the weightw_i to theith vertex ofG_φ. Theorem 4 We can give to the w_i values among the X_i and the constants of the field such that V CP_k(G_φ(w)) =P(φ)(X). In other words, P(φ) is a projection of V CP_k(G_φ).

Proof. Let us denote by c the number of clauses of φ. We saw that to a satisfying assignment of φcorrespond exactly 2^c covering sets of cardinality k ofGφ.

Let us consider a satisfying assignment of φand denote by I the set of indices i such that ǫ_i is true in this assignment. This assignment contributes to P(φ) with a monomial Q

i∈IX_i. But the corresponding covering sets contain the vertices ǫi fori∈I and the vertices ǫi for i /∈I. So if we add weights X_i to all the verticesǫ_i, the weight of a corresponding covering set will contain the monomialQ

i∈IX_i.

We also remark that each covering set of cardinality k of G_φ contain exactly c vertices of the typec_i,j. If we add weights 1/2 to the vertices c_i,j and weights 1 to all remaining vertices, the weight of each corresponding covering set will be ₂¹c

Q

i∈IX_i. So the sum of the weights of the 2^c covering sets associated to our assignment equals Q

i∈IX_i. Finally, with those weights w_i,V CP_k(G_φ(w)) equals P(φ)(X). 2

We now define a family of graphs (G_n) with the previous construction.

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Definition 6 Let us denote (G_n) the family of vertex weighted graphs defined by G_n = G_φ_n for every integer n, where the family (φn) is the family of conjunctions of 2-clauses such that (P(φ_n))is the family of partial permanents. Let(k_n) be the family of integers associated to the graphs (G_n), such that P(φ_n) is a projection of V CP_k_n(G_n) for alln.

We immediately have :

Theorem 5 (V CP_k_n(G_n)) is ap-family of VNP-complete polynomials.

Proof. The VNP-complete family (P(φn)) is a projection of the family (V CP_k_n(Gn)). The family of graphs (G_n) is also polynomially growing, and thus (V CP_k_n(G_n)) is p-definable. It is so a VNP-complete family. 2

To complete our sequence of reductions, we need to show that our p-family (V CP_k_n(G_n)) is a reduction of a p-family of the form (V CP(G^′_n)).

We do not know if it is possible to find a projection to fit to the definition of the classical VNP-completeness. The projection seems to be a rather strong notion of reduction : it allows one unique call to the hard problem, and thus corresponds to the notion of polynomial many-one counting reduction for counting problems. In the case of counting problems, a weaker notion of reduction is used, allowing many calls to the hard problem : the polynomial time transduction with oracles. Some problems are known to be #P-complete only with this notion of reduction, while polynomial many-one reductions are unknown (see [5]). In [1], B¨urgisser proposes an analogue of the polynomial transduction with oracles for Valiant’s setting.

First we introduce the oracle computation :

Definition 7 The oracle complexity L^g(f) of a polynomial f with respect to the oracle polynomial g is the minimum number of arithmetic operations +,−,∗ and evaluations of g over previously computed values that are sufficient to compute f from the indeterminates Xi and constants in k.

Definition 8 Let us consider two p-families f = (f_n) and g = (g_n). We have a polynomial oracle reduction, or c-reduction denoted f ≤_c g, if there exist a p-bounded function t:N→ N such that the map n7→L^g^t(n)(f_n) isp-bounded.

We remark that the relation≤_c is transitive, and that the projectionf ≤_p gimplies f ≤_c g : iff is a projection of q,f(X) may be computed by a single call to the oracle g on entries in the variables and the constants of the field. We remark also that :

g∈VP∧f ≤_cg⇒f ∈VP

We may define a larger notion of VNP-completeness, based on the polynomial oracle reduction. A p-family f will be VNP-hard if every p-family in VNP is constructively reductible to P. It is VNP-complete if in addition,f ∈VNP. Since the projection ≤_p implies the reduction

≤_c, the new class of VNP-complete families contains all the classic VNP-complete class.

We remark, that for every vertex weighted graph G and any integer k, the polynomial V CP_k(G) is the homogeneous component of degree k of V CP(G). But we have the following result from [1] :

Lemma 4 Let f be a polynomial in the variables X₁, . . . , X_n. For any δ such that δ ≤deg f, let denote f^(δ) the homogeneous component of degree δ of f. Then, L^f(f^(δ)) is polynomially bounded in the degree off.

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Proof. We first remark, that for all λ∈k, we have :

deg f

X

i=0

λ^kf^(k)(X) =f(λX)

where λX denotes (λX₁, . . . , λX_n). We can thus compute the homogenous components of f by an interpolation algorithm on the polynomial inλfor deg f + 1 different values ofλ.

2

We remark, that given a circuit computing the polynomialf, a simple transformation returns a circuit of size polynomial in the size of the previous circuit and the degree off computing any homogeneous component of f. It suffices to replace each gate of the circuit by deg f + 1 gates, each carrying an homogeneous component of the previous gate. The construction is explained in [8].

Theorem 6 1. (V CP_k_n(Gn))≤c (V CP(Gn)).

2. The family (V CP(G_n)) isVNP-complete, with respect to c-reduction.

3. There exist a VNP-complete family, with respect to c-reduction, of polynomials associated to a {OR0}-family of boolean formulas.

4.3 Implicative 2-clauses

Here we consider the set S ={OR₁} ={(x,y) → (x∨y)}. Those logical relations are called implicative, because x∨y is equivalent to y ⇒ x. To find a VNP-complete (with respect to c-reduction) family associated toS-formulas, we follow the chain of reductions from [9] and [7]

that permitted to establish, that #SAT(S) is #P-complete.

Those articles show consecutively, that the problems of counting the independent sets, the independent sets in a bipartite graph, the antichains in partial ordered sets (posets), the ideals in posets, and finally satisfaction of implicative 2-clauses are #P-complete. We will consider polynomial equivalents to those problems.

First, an independent set in a graph G= (V, E) is a setI ⊆V such that no pair of vertices of I are linked by an edge in E. In a weighted graph G, the weight of the vertex v being given by w(v), we can define the independent set polynomial like the vertex cover polynomial :

IP(G) =X

I

Y

v∈I

w(v)

where the sum is taken over all independent setsI ⊆V, and the polynomial of the independent sets of cardinalityk :

IP_k(G) = X

I:|I|=k

Y

v∈I

w(v)

We remark that I is an independent set ofGif and only if V\I is a covering set of G.

Lemma 5 For every conjunction of 2-clausesφ, P(φ) is a projection ofIP_k(G_φ), whereG_φ is defined like previously, andk is the half of the number of vertices of G_φ.

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u v

s₁:−1

s3:−1

s₂:−1 Figure 2: The gadget subgraph

Proof. According to lemma 3, for every satisfying assignment of φ correspond 2^c covering sets of cardinality k of G_φ, such that each of them contain the vertices associated to satisfied variables of the assignment, the vertices associated to the negation of the others, and half the vertices of the type c_i,j,d_i,j and e_i,j.

We can also associate to each satisfying assignment the 2^c complementaries of the associated covering sets. This gives a map from the satisfying assignments of φto the independent sets of cardinality kof G_φ. If we define the weight functionwby giving a weight X_i to the vertices of the type ǫ_i, 1/2 to each vertex of the type c_i,j and 1 to the other vertices, the weight of each independent set of cardinality k will be the same weight, than the one of its complementary with the weight functionw^′ considered in the previous section. So, the polynomialIP_k(G_φ(w)) equals the polynomialV CP_k(G_φ(w^′)), and thus : IP_k(G_φ(w)) =P(φ)(X). 2

Corollary 1 The family IP_k(G_n) is VNP-complete, where the family (G_n) is introduced in definition 6.

Like previously, since for all weighted graph G and integer k, IP_k(G) is the homogeneous component of degreek ofIP(G), we have : (IP_k_n(G_n))≤_c (IP(G_n)). Thus, we have :

Theorem 7 The family (IP(G_n)) isVNP-complete with respect to c-reduction.

The next step is to transform a graph G= (V, E) into a bipartite graph, without changing the polynomial IP(G). We construct the graph G^′ by replacing each edge (u, v) in G by the gadget subgraph shown in figure 2.

We claim the following :

Lemma 6 Given a graphGand an edge(u, v)ofG, the graphG₁ obtained fromGby replacing (u, v) by the gadget subgraph verifies : IP(G₁) =−IP(G).

Proof. Let I be the independent sets of the graph G_\(u,v) obtained by suppressing the edge (u, v) in G. This set I may be partitioned into four sets I_∅,I_u,I_v and I_u,v of the independent sets of G_\(u,v) containing respectively none of the vertices u and v, just the vertex u, just the vertexv, and the two vertices u and v. We remark that the sets I_u,I_v and I_∅ form a partition of the independent sets of G. In fact, a set S is an independent set of Gif and only if it is an independent set of G_\(u,v) and it does not containu andv, so if and only if it belongs toI\I_u,v.

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Since the vertices s₁,s₂ ands₃are only connected to G₁ through the vertices uandv, in an independent setSofG1, the possible choices forS∩ {s1, s2, s3}when the other vertices ofS are fixed does only depend on S∩ {u, v}. So it only depends whether S\{s₁, s₂, s₃} belongs toI_∅, I_u,I_v orI_u,v. Let us call I_∅^∗,I_u^∗,I_v^∗ and I_u,v^∗ those possible subsets of {s₁, s₂, s₃}completing an independent set of G_\(u,v) into an independent set of G1 when this set contains none of u and v, only u, onlyv, or the two vertices respectively. With those notations, we can identify the set I₁ of the independent sets ofG₁ and the union (I_∅×I_∅^∗)∪(I_u×I_u^∗)∪(I_v×I_v^∗)∪(I_u,v×I_u,v^∗ ).

For a set of vertices S, let denoteW(S) the weight Q

a∈Sw(a) of this set. We have :

IP(G₁) = X

S∈I1

W(S) = X

S∈(I_∅×I_∅^∗)

W(S) + X

S∈(Iu×I_u^∗)

W(S)

+ X

S∈(Iv×I_v^∗)

W(S) + X

S∈(Iu,v×I_u,v^∗ )

W(S)

= X

S∈I∅

W(S)· X

S∈I_∅^∗

W(S) +· · ·+ X

S∈Iu,v

W(S)X_i· X

S∈I_u,v^∗

W(S) But we have :

• P

S∈I_∅^∗W(S) =w(s₁)w(s₂) +w(s₁) +w(s₂) +w(s₃) + 1 =−1

• P

S∈I_u^∗W(S) =w(s₂) +w(s₃) + 1 =−1

• P

S∈Iv^∗W(S) =w(s₁) +w(s₃) + 1 =−1

• P

S∈I_u,v^∗ W(S) =w(s₃) + 1 = 0 Thus,

IP(G₁) =−X

S∈I_∅

W(S)− X

S∈Iu

W(S)− X

S∈Iv

W(S) =−IP(G) sinceIu,Iv and I_∅ form a partition of the independent sets of G. 2

Since G^′ is obtained by replacing all edges of G by the gadget subgraph, we prove easily thatIP(G^′) = (−1)^eIP(G), whereeis the number of edges ofG. We see easily that the graph G^′ is a bipartite graph : if we consider the setsU of all vertices ofGand all vertices of typee_s, and V of all vertices of types₁ ands₂, every edge ofG^′ links a vertex of U and a vertex of V. By applying this transformation to the graphs (G_n) such that (IP(G_n)) is a VNP-complete family, we prove the following :

Lemma 7 There exists a family of bipartite graphs(G^′_n)such that(IP(Gn))is aVNP-complete family of polynomials, with respect to the c-reduction.

In the following, we won’t use only this lemma, but we will use the structure of the family (G^′_n) provided by our transformation of the family (Gn) into bipartite graphs. For example, we will use the fact that if we denote by V₁ and V₂ the partite sets of G^′_n, in one of those two sets, say V₁, all vertices have weight −1 : they are all vertices of the type s₁ or s₁2, with the previous notations.

In [9], the authors remark, that given a bipartite graph, we can construct naturally a partial ordered set. Given bipartite graph G = (V₁, V₂, E), we define the partial order (X,≤) with X =V1∪V2, and givenx and y inX, x≤y if and only if x ∈V1, y ∈V2 and (x, y)∈E. We see easily that ≤is transitive and antisymmetric. We recall the definition of an antichain :

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Definition 9 (Antichain) An antichain A in a poset(X,≤)is a subset ofX such that for all pair (x, y) of elements of A, x andy are incomparable.

We define the antichain polynomial of a (weighted) poset (X,≤) as the polynomial : AP(X) =X

A

Y

x∈A

w(x) where the sum is over all antichains A of (X,≤).

Let us consider a bipartite graph Gand its corresponding poset (X,≤). We see easily, that a set S ⊆ X is an antichain in (X,≤) if and only if it is independent in G. We thus have : AP(X) = IP(G). Let us denote by (X_n) the family of posets associated to our family of bipartite graphs (G^′_n). We can identify the families (AP(X_n)) and (IP(G^′_n)).

Let us define the notion of ideal in a poset.

Definition 10 (Ideal) An ideal I in a poset (X,≤) is a subset of X such that for all x ∈I, ally such that y≤x belong to I.

We can also define the ideal polynomial IP P(X) of the ideals in a poset (X,≤) : IP P(X) =X

I

Y

x∈I

w(x) where the sum is over all ideals I of (X,≤).

Given an ideal I in a poset (X,≤), the maximal elements of I form an antichain A : since they are maximal in I, they cannot be compared. Conversely, given an antichain A, the set of elements x that are less than an element of A form an ideal. We verify easily that those transformations are bijective and reciprocals. We thus have a bijection between the ideals and the antichains of a given poset. This fact suffices to the authors of [9], since the bijection proves, that a poset has the same number of antichains and ideals; the counting problems are thus equivalent.

But in our case, our goal is to express the family (AP(X_n)) as a family (IP P(X_n^′)), for a family of posets (X_n^′). Since the ideals and the antichains do not correspond to the same sets of elements, their weights are not equal; we cannot identify simply AP(X) andIP P(X) for any posetX.

We do not know how to reduce a family of antichain polynomials into ideal polynomials in general, but in the case of the family (AP(X_n)), since the structure of the family (G^′_n) is particular, the problem is easier. We claim the following :

Theorem 8 For all integer n, we have : AP(X_n) =IP P(X_n)

The proof is placed in Appendix 1, to ease the reading of this section.

Corollary 2 There exists a VNP-complete p-family of polynomials of the form (IP P(X_n)), with respect to the c-reduction.

To conclude, we remark, that the ideal polynomial in a poset (X,≤) may be expressed simply as a polynomial associated to a S-formula. We associate to each x_i ∈X a boolean variable ǫ_i with the meaning, that x_i belongs to an ideal whenǫ_i is true. For every pair (x_i, x_j)∈X such that x_i ≤x_j, the condition x_j ∈I ⇒x_i ∈I may be expressed by (ǫ_j ⇒ǫ_i), or (ǫ_i∨ǫ_j). Thus, we have :

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IP P(X) = X

ǫ∈{0,1}^|X|

^

(i,j):xi≤xj

ǫi∨ǫj X^ǫ

We can easily verify, that the family of S-formulas (ψn) coding the relations in the posets (X_n) is p-bounded. Thus, we have :

Theorem 9 There exists aVNP-complete family, with respect to c-reductions, of polynomials associated to a p-family of {OR1}-formulas

4.4 The general case

We show that for every set of logical relations not reduced to affine ones, we can find a VNP- complete family of associated polynomials :

Theorem 10 Let S be a finite set of logical relations, such that some are not affine. Then there exists a VNP-complete family of polynomials associated to a p-family of S-formulas.

The proof of this result is an analogue of the proof of the #P-completeness of the corresponding counting problems given in [2]. We will discuss the different steps of the proof, without giving the proof of all intermediate lemmas, and show how it is adaptable to our context.

The authors use the notion of perfect and faithful implementation (definition 5.1) :

Definition 11 A conjunction of α boolean constraints {f₁, . . . , f_α} over a set of variables x= {x₁, . . . , x_n}andy ={y₁, . . . , y_n}is a perfect and faithful implementation of a boolean formula f(x), if and only if

1. for any assignment of values to xsuch that f(x) is true, there exists a unique assignment of values to y such that all the constraints f_i(x, y) are satisfied.

2. for any assignment of values to x such that f(x) is false, no assignment of values to y can satisfy more than (α−1) constraints.

The fact, that the implementation is perfect refers to the existence of an assignment of values to y such that all the constraints are satisfied, when f(x) is true. The adjective faithful refers to the fact, that there is a unique y to do so.

We refer to the set x as the function variables and the set y as the auxiliary variables.

We say, that a setSof logical relationsimplements perfectly and faithfullya boolean formula f(x) if there is a S-formula that implementsf(x) perfectly and faithfully. We also extend the definition to logical relations : a set S of logical relations implements perfectly and faithfully a logical relation f if S implements perfectly and faithfully every application of f to a set of variables x.

Let us denote F the logical relation x→(x= 0). From [2], lemma 5.30, we have :

Lemma 8 If a logical relation f is not affine, then {f,F} implements at least one of the three logical relations OR₀, OR₁ or OR₂ perfectly and faithfully.

The following lemma, analogue to lemma 5.15 from [2], shows that perfect and faithful implementation provide a mechanism to do projections from the polynomials associated to sets of logical relations.

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Lemma 9 Let S and S^′ be two sets of logical relations such that every relation of S can be perfectly and faithfully implemented by S^′. Then every p-family of polynomials associated to a p-family of S-formulas is a projection of a p-family of polynomials associated to a p-family of S^′-formulas.

Proof. Let (φn) be a p-family of S-formulas, and let us fix an integer n.

Let x = {x₁, . . . , x_p} be the set of variables of the formula φ_n. This formula φ_n is a conjunction of logical relationsf_i ∈S applied on variables from{x₁, . . . , x_p}. If we replace each of those relations f_i by a perfect and faithful implementation using constraints inS^′, using for each f_i a new set of auxiliary variables, we obtain a conjunctionψ_n of logical relations fromS^′ applied on variable setx∪y, where y={y₁, . . . , y_q} is the union of the auxiliary variables sets added for each logical relation f_i.

Since all implementations are perfect and faithful, every assignment to x that satisfies all constraints ofφ_ncan be extended by a unique assignment tox∪ythat satisfies all constraints of ψ_n. Conversely, for an assignment toxthat does not satisfy all constraints ofφ_n, no assignment to x∪y can extend the previous one and satisfy every constraint ofψ_n.

Since ψ_n is a conjunction of logical relations fromS^′ applied on a set of variables x∪y,ψ_n is a S^′-formula. Furthermore, the number of constraints of ψ_n is bounded by the product of the number of constants of ψn and the maximum number of logical relations from S^′ needed to implement a logical relation from S - which does not depend on n. The size of ψ_n is also polynomially bounded in the size of φ_n. We have :

P(φ_n)(X₁, . . . , X_p) = X

ǫ∈{0,1}^p

φ_n(ǫ)X^ǫ

= X

ǫ∈{0,1}^p,y∈{0,1}^q

ψ_n(ǫ, y)X^ǫ

= X

ǫ∈{0,1}^p,y∈{0,1}^q

ψ_n(ǫ, y)X^ǫ1^y¹. . .1^y^q

= P(ψ_n)(X₁, . . . , X_p,1, . . . ,1)

Finally, the family (P(φn)) is a projection of the family (P(ψn)), which is a p-family of polynomials associated to S^′-formulas. 2

From the two previous lemmas, and from the VNP-completeness of families of polynomials associated to{OR₀}- ,{OR₁}- and{OR₃}-formulas, we get, that for every set of logical relations S such that S contains non affine relations, there exists a VNP-complete family of polynomials associated toS∪ {F}-formulas. To get rid of the logical relation {F}, the authors of [2] need to re-investigate the expressiveness of a non affine relation, and distinguish various cases. For our polynomial problems, we can easily force a boolean variable to be set to false by giving to the associated polynomial variable the value 0. We can now give the proof of theorem 10 : Proof. Let (φ_n) be a p-family of S∪ {F}-formulas such that (P(φ_n)) is VNP-complete. The existence of such a family is ensured by lemmas 8 and 9.

Let us consider an integer n. φ_n(x₁, . . . , x_n) is a conjunction of logical relations from S applied to variables from x and and constraints of the form (x_i = 0). We remark, that if φn(x) contains the constraint (xi = 0), then the variableXi does not appear in the polynomial P(φ_n)(X₁, . . . , X_n) : all the monomials containing the variable X_i have null coefficients. If we suppress from the conjunction the constraint (x_i = 0), and instead replace the corresponding

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variableX_i by 0, we obtain exactly the same polynomial : the monomials such thatX_i appears in it have null coefficients; the others correspond to assignments such thatxi= 0. Let us denote ψ_nthe formula obtained by suppressing from φ_n all the constraints of the form (x_i = 0).

Since P(φ_n)(X₁, . . . , X_n) =P(ψ_n)(y₁, . . . , y_n), where y_i is 0 if the constraint (x_i = 0) was inserted inφn, andXi otherwise, we have, that (P(φn)) is a projection from (P(ψn)). Thus, the family (P(ψ_n)) is VNP-complete. We see easily that the (ψ_n) are S-formulas. We constructed a p-family of S-formulas such that the associated family of polynomials is VNP-complete.

2

5 Conclusion

In this study, we have a general approach of the polynomials associated to boolean formulas.

This way to write polynomials is quite expressive : all polynomials with coefficients in{0,1}may be expressed this way, and this is the case of many usual families, such as the permanent. We could not establish our wished dichotomy theorem, but we established the existence of (weakly) VNP-complete families associated toS-formulas for all setsS not reduced to affine constraints.

During this study, we considered many uncommon families of polynomials : polynomials on vertex weighted graphs where less studied than on edge weighted graphs in Valiant’s theory [1];

polynomials on weighted posets are also new. We thus extend the collection of known VNP- complete problems, which is much smaller than the NP-complete one. The use of the notion of oracle reduction to show the weak VNP-completeness of families of polynomials is also new.

We finally have a better comprehension of a new tool, the polynomial associated to a boolean formula, but also give a small contribution to Valiant’s theory.

5.1 Further results

The Appendices, besides proof of theorem 8 placed there to ease the reading of the proof, present further results, obtained while accomplished the previous study for Appendices 2 and 3, and one independent result, Appendix 4. In the second Appendix, we establish, that the formula recognizing permutation matrices cannot be expressed as a conjunction of affine constraints.

The goal is to show, that affine constraints have a low expression power : the permanent cannot be expressed as a polynomial associated to affine constraints.

In the section 4, polynomials on vertex weighted graphs are considered. Their study was motivated by results in [9] on the corresponding counting problems on vertex sets. But in [11], Valiant uses polynomials on edge weighted graphs as an intermediate to show the

#P-completeness of the permanent on (0,1)-matrices - a major result in counting complexity. We can follow this method to adapt our proofs of VNP-completeness into new proofs of

#P-completeness of the original counting problems. The reductions are given in Appendix 3.

This method to consider simple graphs as particular cases of weighted graphs seems also an interesting way to find reductions, and to determinate which reductions may be simplified.

In Appendix 4, we establish, that a boolean formula that recognizes the permutation matrices cannot have a bounded treewidth. The goal is still to find links between the complexities of a boolean formula and its associated polynomial. This was first a direct proof of a result from Koiran an Meer in [6], but leaded to a more general result : no restriction on the size of the boolean formula is needed.

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References

[1] Peter B¨urgisser. Completeness and Reduction in Algebraic Complexity Theory. Number 7 in Algorithms and Computation in Mathematics. Springer, 2000.

[2] N. Creignou, S. Khanna, and M. Sudan. Complexity classification of boolean constraint satisfaction problems. SIAM monographs on discrete mathematics. 2001.

[3] Nadia Creignou and Miki Hermann. Complexity of generalized satisfiability counting problems. Information and Computation, 125:1–12, 1996.

[4] Ding-Zhu Du and Ker-I Ko. Theory of Computational Complexity. Wiley-Interscience Series in Discrete Mathematics and Optimization. J. Wiley and Sons, 2000.

[5] Mark Jerrum. Counting, Sampling and Integrating : Algorithms and Complexity. Lectures in Mathematics - ETH Z¨urich. Birkh¨auser, Basel.

[6] Pascal Koiran and Klaus Meer. On the expressive power of CNF formulas of bounded Tree- and Clique- Width.To appear in Proc WG’08.

[7] Nathan Linial. Hard enumeration problems in geometry and combinatorics. SIAM Journal of Algebraic and Discrete Methods, 7(2):331–335, 1986.

[8] Guillaume Malod. Polynˆomes et coefficients. PhD thesis, Universit´e Claude Bernard - Lyon I, 2003.

[9] J. Scott Provan and Michael O. Ball. The complexity of counting cuts and of computing the probability that a graph is connected. SIAM Journal of Computing, 12(4):777–788, 1983.

[10] T. J. Schaefer. The complexity of satisfiability problems. InConference Record of the 10th Symposium on Theory of Computing, pages 216–226, 1978.

[11] L. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8:189–201, 1979.

[12] I. Weneger.Branching Programs and Binary Decision Diagrams : Theory and Applications.

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u v

s₁:−1

s₃:−1

s₂:−1 Figure 3: A part of the poset

A Appendices

A.1 Proof of lemma 8

Theorem 11 For all integer n, we have : AP(X_n) =IP P(X_n) Proof.

Let fix an integer n. We recall that the bipartite graph G^′_n = (V₁, V₂, E) is constituted of subgraphs of the type of figure 2, for all edge (u, v) of the initial graph G_n. So (if we identify X_n and the oriented graph obtained by representing the relation ≤ by an arrow from x to y for all x, y, such thatx≤y) X_n is constituted of subgraphs of the type of figure 3, for all edge (u, v) of G_n.

Let consider an antichain A of X_n and the corresponding ideal I. We remark that the set I\A of elements added from the antichain to the ideal correspond to vertices ofV1, since they are not maximal elements of the antichain. Thus, they have weight −1. So to an antichain A correspond an ideal I composed of the elements of A and a certain number of elements of weight−1. I may also have a different weight thanA. For example, ifAis reduced to a vertex of the type u, such that in the initial graphG_n, u is only connected to 1 other vertex v, the weight of A will be w(u), whereas the weight of I will be −w(u) : I also contains the element of the type e1 introduced in the gadget subgraph replacing the edge (u, v).

Fortunately, by changing the correspondence between antichains and ideals, we can preserve the weights : we will construct a bijection from the antichains to the ideals ofX_nthat preserves the weights in lemma 10, and thus we have :

AP(X_n) =IP P(X_n) 2

Lemma 10 There exists a bijection (different from the natural one considered previously) from the antichains to the ideals of X_n, this one keeping the weights unchanged.

Proof. Let consider an antichain A and its corresponding ideal I in the natural bijection. We will construct a new corresponding idealI^′ such that the weights of A andI^′ are equal.