On the relationship between disjunctive relaxations and minors in packing and covering problems

Leoni, V.; Nasini, G.

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Revista de la Unión Matemática Argentina

versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637

Rev. Unión Mat. Argent. v.46 n.1 Bahía Blanca ene./jun. 2005

On the relationship between disjunctive relaxations and minors in packing and covering problems

V. Leoni* and G. Nasini^†,

*Fundación Antorchas.
^†Universidad Nacional de Rosario, Argentina

Abstract
In 2002, Aguilera et al. analyzed the performance of the disjunctive lift-and-project operator defined by Balas, Ceria and Cornuéjols on covering and packing polyhedra, in the context of blocking and antiblocking duality. Their results generalize Lovász's Perfect Graph Theorem and a theorem of Lehman on ideal clutters. This study motivated many authors to work on the same ideas, providing alternative proofs and analyzing the behaviour of other lift-and-project operators in the same context.
In this paper, we give a survey of the results in the subject and add some new results, showing that the key of the behaviour of the disjunctive operator on these particular classes of polyhedra is the strong relationship between disjunctive relaxations and original relaxations associated to some minors.

1. INTRODUCTION

Many problems in Combinatorial Optimization can be formulated as 0 - 1 linear programs, where the set of feasible solutions may be seen as the set of integral solutions in a polyhedron , i.e. S = K ∩ ℤn .

In spite of optimizing a linear function over is equivalent to do it over

in the general case, the complete description of * K by linear inequalities is not known. Moreover, in most of the cases, even though a partial description is found, an exponential number of inequalities is involved.

A polyhedron K ⊂ ℝn will be called a relaxation of S ⊂ ℝn if S = K ∩ ℤn . Given two different relaxations and K ′ of such that S ⊂ K ′ ⊂ K , the bound obtained by optimizing over K ′ is tighter than the one obtained by optimizing over and we say that the relaxation is weaker than ′ K .

This fact motivates the definition of operators such that, applied on a relaxation of , in each step they obtain a new relaxation K ′ ⊂ K , arriving to conv(S) in a finite number of iterations.

This is the case of the and lift-and-project operators defined by Lovász and Schrijver [12] and the disjunctive operator ( BCC ) defined by Balas, Ceria and Cornuéjols [4].

In [1] and [2] the authors analyze the performance of the BCC operator on covering and packing polyhedra in the context of blocking and antiblocking duality, respectively. Their results generalize Lovász's Perfect Graph Theorem and its analogous on ideal clutters, due to Lehman. These results motivated Gerards et al. [8] and more recently, Lipták and Tunçel [11], to give alternative proofs in the case of the generalization of Lovász's Perfect Graph Theorem. In the same way, Leoni and Nasini [10] exposed alternative and simpler proofs for the case of the generalization of Lehman's theorem. Lipták and Tunçel [11] started the analysis of the behaviour of the and operators in the same context and, recently, Escalante et al. [6] completed this analysis, proving that similar generalizations do not exist for these operators.

The aim of this paper is to show that the key of the behaviour of the disjunctive operator observed in [1] and [2] is the strong relationship between disjunctive relaxations and original relaxations associated to some particular minors. This relationship becomes clear from the characterization of the extreme points of these relaxations.

This idea was initiated by Nasini [13], working in the particular case of the clique relaxation QST AB(G) of the stable set polytope in a graph . Independently, Lipták and Tunçel [11] obtained the same result.

In Section 2, we provide the fundamental definitions that will be treated in the rest of the paper.

In Section 3, we summarize the results on the extreme points of the disjunctive relaxations for set covering and set packing polyhedra. Besides, we show that the results on QST AB(G) can be extended to more general relaxations of .

In Section 4, we summarize the results on the extreme points of the blocker of the disjunctive relaxations for set covering polyhedra. Looking for similar relationships in the context of set packing polyhedra, we show that the antiblocker of the disjunctive relaxations of ST AB(G) obtained from QST AB(G) is also strongly related to the antiblocker of the clique relaxation associated to some particular subgraphs of the graph .

Finally, in Section 5 we summarize the results obtained in this work and present the conclusions.

2. DEFINITIONS AND PRELIMINARIES

The disjunctive operator defined by Balas, Ceria and Cornuéjols [4] can be applied on polyhedra n K ⊂ [0, 1] . After a lift-and-project iteration of the operator on j ∈ {1,...,n} , they obtain

Applying iteratively the operator on a subset of indices J = {i1,..,ij} ⊂ {1,...,n} , they proved that

( ) Pi1(Pi2(...(Pij(K)))) = conv {x ∈ K : xj ∈ {0,1}for all j ∈ J } := PJ(K).

For any subset J ⊂ {1,...,n} , PJ(K) will be called a disjunctive relaxation of ( ) K * := conv K ∩ ℤn .

Clearly, P{1,...,n}(K) = K * . This last property allows us to talk about the minimum number of iterations needed to find K * , which is called the disjunctive index of . It is also clear that K * = K if and only if the disjunctive index of is equal to zero.

Given a graph with N = {1,...,n} as set of nodes, let us recall that Q ⊂ N is a clique in if in every pair in , its nodes are adjacent. Moreover, S ⊂ N is a stable set in if in no pair of , its nodes are adjacent. Clearly, a clique in a graph is a stable set in its complementary graph .

The stable set polytope ST AB(G) of is the convex hull of the incidence vector of its stable sets. The clique relaxation QST AB(G) of is defined as

∑ QST AB(G) = {x ∈ ℝn : x ≤ 1 for all Q clique in G}. + i i∈Q

In [1], the authors study the behaviour of the disjunctive operator on polyhedra K = QST AB(G) .

Following Lovász's Perfect Graph Theorem in [12], if a graph is perfect then its complementary graph -- G is also perfect. From the polyhedral characterization of perfect graphs given by Chvátal [5], i.e is perfect if and only if QST AB(G) = ST AB(G) , we have that the disjunctive index of is zero if and only if the disjunctive index of -- QST AB( G) is zero. Moreover, it is known that if is minimally non perfect (mnp), then -- G is also mnp and has only one fractional extreme point. It holds that if a graph is mnp , the disjunctive indices of QST AB(G) and are both equal to one.

These well-known results have been generalized in [1] proving that

Theorem 1.
For any graph , the disjunctive indices of and coincide.

From this result, the disjunctive index of QST AB(G) may be considered as a measure of the "imperfection" of . In this sense, Theorem 1 shows that any graph is as "imperfect" as its complementary graph, generalizing Lovász's Perfect Graph Theorem. This result is proved in [1] as a consequence of the behaviour of the disjunctive relaxation on the context of antiblocker duality.

If C K denotes the antiblocker of a polyhedron defined by

it is well known that

The proof of Theorem 1 in [1] is a direct corollary of the following stronger result.

Theorem 2.
For any J ⊂ {1,...,n} , ( ) P [P (QST AB(G))]C = [QST AB(G)]C = ST AB(G)- J J .

Similar results have been obtained in [2], working on set covering polyhedra.

A clutter is a set of non-comparable subsets —called edges— of a set N = {1,...,n} , called the vertex set.

Given a clutter over , the blocker of , b(C) , is the clutter of minimal vertex covers of , i.e. minimal subsets of satisfying

We will denote by M (C) the 0 - 1 matrix (with entries 0 or 1) whose rows are the characteristic vectors of the edges of . Clearly, M (C) has no dominating rows.

Given a clutter , x ∈ {0,1}n is the incidence vector of a vertex cover of if and only if M (C)x ≥ 1 . The set covering polyhedron associated to is defined as

where Q(C) = {x ∈ ℝn+ : M (C)x ≥ 1} is called the original relaxation of Q *(C) .

In this case, considering the blocker of a polyhedron defined as

and the blocker b(C) of a clutter , it is well-known [7] that

In order to analyze the "disjunctive behaviour" over blocking polyhedra, the authors in [2] had to define an extension ¯Pj of the disjunctive operator, as

where n Q0(C) := Q(C) ∩ [0,1] . They showed that ¯Pj preserves the main properties of . In particular, for any J ⊂ {1,...,n} ,

The disjunctive index is also well defined, since

¯P{1,...,n}(Q(C)) = P{1,...,n}(Q0(C)) + ℝn+ = Q *0(C) + ℝn+ = Q *(C).

Ideal clutters, defined by Lehman, are those for which Q(C) is integral, or equivalently, those for which the disjunctive index is zero.

Lehman's theorem on ideal clutters [9] establishes that if a clutter is ideal then so is its blocker. In other words, Lehman's theorem says that the disjunctive index of Q(C) is zero if and only if the disjunctive index of Q(b(C)) is zero. Moreover, it is known that if is minimally non ideal ( mni ), then b(C) is also mni and Q(C) has only one fractional extreme point. It holds that if a clutter is mni , the disjunctive indices of Q(C) and Q(b(C)) are both equal to .

Lehman's theorem is generalized in [2] in the following sense:

Theorem 3. For any clutter , the disjunctive indices of and coincide.

Once again, the disjunctive index of Q(C) may be considered as a measure of the "non-idealness" of , and Theorem 3 shows that any clutter is as "non-ideal" as its blocker, generalizing Lehman's theorem.

The proof of Theorem 3 in [2] is a direct corollary of the following stronger result (the analogous of Theorem 2) in the context of blocker duality.

Theorem 4. For any J ⊂ {1,...,n} , [ ]B B ¯PJ( P¯J (Q(C)) ) = [Q(C)] = Q *(b(C)) .

Proofs of Theorem 2 and Theorem 4 given respectively in [1] and [2] are based on the characterization of valid inequalities of the disjunctive relaxations. In those proofs and also in the alternative ones given in [8], the relationship between the disjunctives relaxations and some particular minors is not noticed.

3. DISJUNCTIVE RELAXATIONS, EXTREME POINTS AND MINORS

Let n K ⊂ [0,1] be a polyhedron and consider for S ⊂ N ,

and

If is a fixed subset of , S ⊂ J and $-- S = J \S$ , let us define

It is not hard to see [1] that

and, if V (K) is the set of extreme points of ,

Given a graph G = (N, E) with N = {1, ...,n} and considering K = QST AB(G) , it is clear that K S is not empty if and only if is a stable set in . Moreover, denoting by Γ (S) = {i ∈ N : i is adjacent to some j ∈ S} , if x ∈ KS , xi = 0 for all i ∈ Γ (S) . Therefore, calling -- ^S = S ∪ Γ (S) , we have

Given S ⊂ J , we write any n x ∈ ℝ as -- x = (x, ^x) , with -- x ∈ ∣J∪^S∣ ℝ . Therefore, denoting by $G \S$ (respectively G ⊖ S) , the subgraph obtained from by the deletion (destruction) of nodes in , the following lemma holds.

Lemma 5. Given , and , is an extreme point of if and only if and is an extreme point of $-- QST AB((G \S) ⊖ S)$ .

This lemma is the key of the proof in [13] of Theorem 6 below.

Theorem 6. [13] For any , if and only if $G \J$ is perfect.

The generalization of Lovász's Perfect Graph Theorem given by Theorem 1 is a direct consequence of the previous theorem.

The "symmetry" of Theorem 1 and Theorem 3 can be also expressed by similar relationships between the extreme points of the disjunctive relaxations of the set covering polyhedron associated to a clutter and those associated to some minors of .

Given a clutter over N = {1,...,n} and j ∈ N , C ∕j denotes the clutter obtained from by the contraction of , i.e. the clutter defined over N - {j} , whose edges are the minimal elements of ${S \ {j}: S edge of C}$ . Also, $C \j$ denotes the clutter obtained from by the deletion of , i.e. the clutter defined over N - {j} , whose edges are the edges of not containing . Given two disjoints subsets , of , $C ∕R\S$ denotes the minor of obtained by the contraction of nodes in and deletion of nodes in . It is known that $b(C∕R \S) = b(C) ∕S\R$ .

In this way, in [10] it is proved that

Theorem 7. [10] For any , if and only if $-- C∕R \R$ is ideal for every R ⊂ J .

Let us notice that, meanwhile the integrality of PJ (QST AB(G)) involves the perfection of only the subgraph induced by $N \ J$ (see Theorem 6), the "symmetric" result for set covering polyhedra involves the idealness of many minors. In other words, in this case there is not a particular minor of which guarantees the integrality of P¯J (Q(C)) . This difference should be found in the characterization of the extreme points of the disjunctive relaxations of Q(C) . We present here an scheme of the proof of Theorem 7 in [10].

In the following, will be a fixed subset of . For R ⊂ J, with $-- R = J \R$ , let us consider the polyhedron

where

-- { } GR := x ∈ ℝn : xi ≥ 1 for i ∈ R . +

It can be easily seen that

Then, every extreme point of P¯J (Q(C)) is an extreme point of , for some R ⊂ J . Moreover, writing any x ∈ ℝn as -- x = (x,^x), where -- x ∈ ℝ ∣J∣ and denoting by xR ∈ {0,1}∣J∣ the characteristic vector of R ⊂ J , we have

Lemma 8. Given and , is an extreme point of if and only if and is an extreme point of $-- Q(C ∕R \R)$ .

The difference with the "stable set case" appears noting that it is not true that for any R ⊂ J , every extreme point of is an extreme point of P¯J (Q(C)) . However, it can be proved that

Lemma 9. [10] Let be an extreme point of , for some and a minimal subset of such that . Then, is an extreme point of and also, an extreme point of P¯J (Q(C)) .

Now, we derive Theorem 7, the "symmetric" of Theorem 6:

Proof. (of Theorem 7)

Suppose that for some R ⊂ J , $-- C ∕R \R$ is not ideal, i.e. that $-- Q(C ∕R \R)$ has some fractional extreme point . Then, R (x ,^x) is a fractional extreme point of . From Lemma 9, there exists S ⊂ R such that (xS, ^x) is a fractional extreme point of P¯J (Q(C)) , and then ¯PJ(Q(C)) is not integral. Therefore, ¯PJ(Q(C)) ⁄= Q *(C) .

To see the converse, if $C∕R-\R$ is ideal for every R ⊂ J , from Lemma 8 follows that is an integral polyhedron for every R ⊂ J . Therefore, ¯ PJ (Q(C)) is also integral.

Now we come back to Theorem 6. Let us observe that, in other words, this theorem states that for any graph , the disjunctive index of QST AB(G) is the minimum number of nodes that it is necessary to delete in in order to obtain a perfect graph.

A similar result have been found in [3], working on the matching polytope in a graph. Let us recall that a matching in a graph G = (N, E) is a subset of edges where any two of them are incident to a same node of . The matching polytope M AT CH(G) is defined as the convex hull of the incidence vector of the matchings in . The natural original relaxation of is

( ) { ∑ } K ≤(G) = x ∈ ℝ ∣E ∣: xij ≤ 1 for all i ∈ N . ( + ) j:(i,j)∈E

It is known that K (G) = M AT CH(G) ≤ if and only if is bipartite. In [3], it is proved that the disjunctive index of K ≤(G) is the minimum number of edges that must be taken off from in order to obtain a bipartite graph.

This fact does not seem to be surprising if we recall that the matching polytope of a graph is exactly the stable set polytope of the line graph L(G) of . However, it cannot be seen as a particular case of Theorem 6 because the relaxation K ≤(G) of ST AB(L(G)) is, in general, weaker than QST AB(L(G)) .

In a more general context, in a set packing problem we are given a clutter over a set and optimize over * K (C) , the convex hull of 0 - 1 vectors in { } K(C) = x ∈ ℝ ∣N ∣: M (C)x ≤ 1 + .

If we consider the associated graph G(C) of defined by a set of nodes equal to , the vertex set of , and where two nodes in G(C) are adjacent if there exists an edge of that contains both of them, it is not hard to see that K *(C) = ST AB(G(C)) and K(C) is a relaxation, weaker than QST AB(G(C)) .

When K(C) = QST AB(G(C)) we say that the clutter is conformal. For this family of clutters we have the results on the disjunctive relaxations presented in Section 2.

The rest of this section is devoted to generalize Theorem 6 for other families of clutters. We are looking for families of clutters C where, for any in C, the disjunctive index of the relaxation K(C) coincides with the minimum number of nodes we must delete from G(C) in order to obtain a graph G ′ , such that G′ = G(C ′) for some C′ ∈ C and K(C ′) = ST AB(G ′) .

Given a family C of clutters and a graph , let us say that is a C-graph if G = G(C) for some C ∈ C .

We need a consistent definition for the family we are looking for. For this purpose, we have to impose some conditions.

Definition 10. A family of clutters C is hereditary if it satisfies the following conditions:

if and are in C and , then ;

if and is a vertex of , then $C \ i$ is in C.

Then, we have

Definition 11. Given an hereditary family of clutters C and a C-graph such that with , is C-perfect if and only K(C) = ST AB(G(C)) .

Condition 1 in Definition 10 implies that C-perfection in Definition 11 is well defined. Besides, it is not hard to see that $G(C \i) = G(C) \{i}$ . Then, Condition 2 implies that any node induced subgraph of a C-graph is a C-graph, and moreover, any node induce subgraph of a C-perfect graph is C-perfect.

According to Definition 11, when C is the family of conformal clutters, a C-perfect graph is a perfect graph. Also, when C is the family of clutters with at most two vertices in each edge, for any clutter in C we have,

∣N∣ K(C) = FRAC(G(C)) := {x ∈ {0,1} : xi + xj ≤ 1, for all (i,j) edge in G(C)},

and it is known that C-perfect graphs are bipartite graphs.

This definition also includes K ≤(G) as a relaxation of M AT CH(G) = ST AB(L(G)) . In this case, the elements of C have vertex set equal to the set of edges of a graph G = (N, E) , and the edges are the subsets of incident in to a node in . Besides, K(C) = K (G) ≤ and G(C) = L(G) . It is known that L(G) is C-perfect if and only if is bipartite.

Now, the generalization of Theorem 6 will be formulated as

Theorem 12. If belongs to an hereditary family of clutters C and , for any , if and only if $G \J$ is C-perfect.

To prove Theorem 12, it only remains to recall that

where -- KS = K(C) ∩ GS ∩ H S and notice that again, is not empty if and only if is a stable set in G(C) . Moreover, if x ∈ KS , xi = 0 for all i ∈ Γ (S) in G(C) . Therefore, calling -- S^= S ∪ Γ (S) , we have

It is not difficult to see that -- n x = (x, ^x) ∈ ℝ with -- x ∈ J∪^S ℝ ∣ ∣ is an extreme point of if x ∈ GS ∩ H ^S and is an extreme point of $K(C \(S ∪ ^S))$ . Moreover, $^ ^ -- G(C \(S ∪ S)) = G(C)\(S ∪ S) = (G(C) \S) ⊖ S$ .

Proof. (of Theorem 12)

Clearly, PJ (K(C)) = ST AB(G(C)) if and only PJ(K(C)) is an integral polyhedron. Equivalently, from the previous observations, $^ K(C \(S ∪ S)$ has to be integral, for any S ⊂ J . Equivalently, $K(C \(S ∪ S^)) = ST AB(G(C) \(S ∪ ^S))$ . Finally, PJ (QST AB(G(C))) = ST AB(G(C)) if and only if $-- (G(C) \S) ⊖ S$ is C-perfect for all S ⊂ J . Since $-- (G(C) \S) ⊖ S$ is a node induced subgraph of $G(C) \J$ , the result follows.

Now, for any graph , the results for QST AB(G) and K ≤(G) may be seen as particular cases of the previous theorem. We also have that the disjunctive index of F RAC(G) is the minimum number of nodes we must delete from in order to obtain a bipartite graph.

4. DISJUNCTIVE RELAXATIONS IN THE CONTEXT OF POLYHEDRAL DUALITY

The relationship between extreme points and minors has been also found in the context of "polyhedral duality" for set covering polyhedra [10]. The extreme points of the blocker of a disjunctive relaxation are characterized in the following theorem:

Theorem 13. [10] Let and . Then, if and only if $α^∈ [Q(C ∕S\S)]B$ . Moreover, is an extreme point of if and only if is an extreme point of $[ --]B Q(C ∕S\ S)$ .

Theorem 4 easily follows from the previous result.

In this section, we will find similar relationships between the antiblocker of a disjunctive relaxation of ST AB(G) and some particular subgraphs of a graph .

In the following, will be denote a fixed subset of . Let -- n α = (α, ^α) ∈ ℝ with -- α ∈ ℝ ∣J∣ and α ∈ [ST AB(G)]C ∩ GS ∩ HS-- for some S ⊂ J .

From the observations made by Gerards, Maróti and Schrijver in [8], we know that if αi > 0 and j ∈ S ( j ⁄= i ), then (i,j) is an edge of In particular, since αi = 1 for all i ∈ S ; is a clique in . Moreover, if i ∕∈ J and α > 0 i , then is a node of ------- G ▽ S := G ⊖ S .

Now, let be an extreme point of [PJ (QST AB(G))]C ∩ GS ∩ HS-- , for some S ⊂ J and J ⊂ N . Since [P (QST AB(G))]C ⊂ [ST AB(G)]C J , α ∈ [ST AB(G)]C ∩ G ∩ H -- S S . Then, we have $C ^α ∈ [QST AB(G \J)]$ and ^αi = 0, when is not a node of G ▽ S. In the following theorem, we prove that the converse also holds, i.e.

Theorem 14. Let . If $[QST AB(G \J )]C$
and when is not a node of then . Moreover, if is an extreme point of then is an extreme point of $C [QST AB(G \J)]$ .

Proof. Let -- x = (x, ^x) be an extreme point of PJ(QST AB(G)) with -- ∣J∣ x ∈ {0,1} and $^x ∈ QST AB(G \J)$ . Let us define -- R = {i ∈ J : xi = 1} . We need to show that αx ≤ 1 . Since αx = ∣R ∩ S ∣ + ^α ^x , clearly if ∣R ∩ S∣ = 0 . If ∣R ∩ S ∣ ⁄= 0 , then ∣R ∩ S∣ = 1 and it only remains to prove that ^α^x = 0.

Let R ∩ S = {k} and $i ∈ N \J$ with ^αi > 0 . Since (i, k) is an edge of , -- ^xi + xk ≤ 1, obtaining that ^xi = 0 .

For the second part, let us assume that is a convex combination of two points ^α1 and ^α2 in $[QST AB(G \J)]C$ . Defining α1 = (α, ^α1) and α2 = (α, ^α2), from the previous results we have αi ∈ [PJ (QST AB(G))]C for i = 1,2 and is a convex combination of and .

From this characterization follows an easy new alternative proof for Theorem 2.

Proof. (of Theorem 2)

Since -- C ST AB( G) ⊂ PJ ([PJ (QST AB(G))] ) , it suffices to prove that the polyhedron PJ ([PJ (QST AB(G))]C ) is integral. If -- α = (α,α^) is any extreme point of it, then is an extreme point of [PJ (QST AB(G)]C . Moreover, α ∈ [ST AB(G)]C ∩ GS ∩ HS,- for some S ⊂ J. From Theorem 14, we have that is an extreme point of $C ----- [QST AB(G \J )] = ST AB( G \J )$ and then, is integral.

5. SUMMARY AND CONCLUSIONS

In order to make the symmetry of the results between the "packing" and "covering" cases clearer and to completely understand the relationship between the disjunctive relaxations and the original problem associated to some minors, let us summarize the results on the extreme points of the disjunctive relaxations and those of their "dual polyhedra".

Given a clutter over N = {1,...,n} and J ⊂ N , let

{ ∣N ∣ } K(C) = x ∈ ℝ + : M (C)x ≤ 1 and

First, for the disjunctive relaxations, we have:

with is an extreme point of if and only if there exists such that and is an extreme point of with $-- G(C ′) = (G(C) \S) ⊖ S$ ;
with is an extreme point of if and only if there exists such that and is an extreme point of $-- Q(C ∕S\S)$ .

For the "dual polyhedra" of the disjunctive relaxations, let α = (α, ^α) ∈ ℝn

with

. We have:

When is the clutter of maximal cliques in a graph : is an extreme point of if and only if there exists such that and is an extreme point of $C [QST AB(G \J)]$ ;
is an extreme point of if and only if there exists such that and is an extreme point of $[ --]B Q(C ∕S \S)$ .

We believe that the aim of the paper is achieved: the results obtained reveal that the way the disjunctive operator behaves is due to the fact that the extreme points of a disjunctive relaxation and those of its "dual polyhedron" are strongly related to the same combinatorial problem on a minor of the original clutter. From these results we can say that the BCC

operator "preserves" the original combinatorial structure of the problem. The "negative" results obtained analyzing the behaviour of other lift-and-project operators in the same context ([11] and [6]) are due to the fact that the

and

operators do not share this "combinatorial" property of the BCC

operator.

References

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V. Leoni
Departamento de Matemática
Facultad de Ciencias Exactas, Ingeniería y Agrimensura
Pellegrini 250
2000- Rosario, Argentina.
valeoni@fceia.unr.edu.ar

G. Nasini
Departamento de Matemática
Facultad de Ciencias Exactas, Ingeniería y Agrimensura
Pellegrini 250
2000- Rosario, Argentina.
nasini@fceia.unr.edu.ar

Recibido: 28 de junio de 2003
Aceptado: 15 de noviembre de 2005