1 Introduction
Abstract

We study the set covering polyhedron related to circulant matrices. In particular, our goal is to characterize the first Chvátal closure of the usual fractional relaxation. We present a family of valid inequalities that generalizes the family of minor inequalities previously reported in the literature and includes new facet-defining inequalities. Furthermore, we propose a polynomial time separation algorithm for a particular subfamily of these inequalities.

Keywords: set covering   circulant matrices   Chvátal closure

Generalized minor inequalities for the set covering polyhedron related to circulant matrices

Paola B. Tolomei111ptolomei@fceia.unr.edu.ar

Departamento de Matemática, Facultad de Ciencias Exactas, Ingeniería y Agrimensura, Universidad Nacional de Rosario, Santa Fe, Argentina

and CONICET, Argentina

Luis M. Torres222luis.torres@epn.edu.ec

Centro de Modelización Matemática ModeMat - Escuela Politécnica Nacional, Quito, Ecuador

1 Introduction

The weighted set covering problem can be stated as

where is an matrix with entries, , and is the vector having all entries equal to one. The SCP is a classic problem in combinatorial optimization with important practical applications (crew scheduling, facility location, vehicle routing, to cite a few prominent examples), but hard to solve in general. One established approach to tackle this problem is to study the polyhedral properties of the set of its feasible solutions. [5, 11, 14, 15].

The set covering polyhedron is defined as the convex hull of all feasible solutions of SCP. Its fractional relaxation is the feasible region of the linear programming relaxation of SCP, i.e.,

It is known that SCP can be solved in polynomial time if belongs to the particular class of circulant matrices defined in the next section. Hence, it is natural to ask whether an explicit description in terms of linear inequalities can be provided for in this case, an issue that has been addressed in several recent studies by researchers in the field (see [2, 7, 8, 12] among others). For the related set packing polytope of circulant matrices,

such a description follows from the results published in [13].

Bianchi et al. introduced in [7] a family of facet-defining inequalities for which are associated with certain structures called circulant minors. Moreover, the authors presented in [8] two families of circulant matrices for which is completely described by this class of minor inequalities, together with the full-rank inequality and the inequalities defining , usually denoted as boolean facets. The existence of a third family of circulant matrices having this property follows from previous results obtained by Bouchakour et al. [9] in the context of the dominating set polytope of some graph classes.

If an inequality is valid for a polytope and , then is valid for the integer polytope . This procedure is called Chvátal-Gomory rounding, and it is known that the system of all linear inequalities which can be obtained in this way defines a new polytope , the first Chvátal closure of . Moreover, iterating this procedure yields in a finite number of steps. An inequality is said to have Chvátal rank of if it is valid for the -th Chvátal closure of a polytope. All inequalities mentioned above have Chvátal rank less than or equal to one.

With the aim of investigating if the results in [8, 9] can be generalized to all circulant matrices, we have tried to characterize the first Chvátal closure of for any circulant matrix . In particular, we addressed the question whether the system consisting of minor inequalities, boolean facets, and the full-rank inequality is sufficient for describing . We have obtained a negative answer to this question in the form of a new class of valid inequalities for which contains minor inequalities as a proper subclass. All inequalities from this class have Chvátal rank equal to one, and besides, some of them define new facets of , as we show by an example.

This paper is organized as follows. In the next section, we introduce some notation and preliminary results required for our work. In Section 3 we describe our approach for computing the first Chvátal closure of and define the new class of generalized minor inequalities. A separation algorithm for a particular subclass of these is provided in Section 4. Finally, some conclusions and possible directions for future work are discussed in Section 5. A preliminary version of this article appeared without proofs in [16].

2 Notations, definitions and preliminary results

For , let denote the additive group defined on the set , with integer addition modulo . Throughout this article, if is a matrix of order , then we consider the columns (resp. rows) of to be indexed by (resp. by ). In particular, addition of column (resp. row) indices is always considered to be taken modulo (resp. modulo ). Two matrices and are isomorphic, denoted by , if can be obtained from by permutation of rows and columns. Moreover, we say that a row of is a dominating row if for some other row of , .

Given , the minor of obtained by contraction of , denoted by , is the submatrix of that results after removing all columns with indices in and all dominating rows. In this work, when we refer to a minor of we always consider a minor obtained by contraction.

Let with , and for every . With a little abuse of notation we will also use to denote the incidence vector of this set. A circulant matrix is the square matrix of order whose -th row vector is . Observe that , where is the -th canonical vector in .

A minor of is called a circulant minor if it is isomorphic to a circulant matrix . As far as we are aware, circulant minors were introduced for the first time in [12], where the authors used them as a tool for establishing a complete description of ideal and minimally nonideal circulant matrices. More recently, Aguilera [1] completely characterized the subsets of for which is a circulant minor. We review at next his main result, as some terms will be needed for the separation algorithm presented in Section 4.

Given , the digraph has vertex set and is an arc of if . We call arcs of the form short arcs and arcs of the form long arcs. Associated with any directed cycle in , three parameters can be defined: its number of short arcs, its number of long arcs, and the number of turns around the set of nodes it makes. Hence, the relationship must hold for the integers and . In our current notation, Theorem 3.10 of [1] states the following.

Theorem 2.1.

[1] Let be integers verifying and let such that . Then, the following are equivalent:

  1. is isomorphic to .

  2. induces in disjoint simple dicycles , each of them having the same parameters , and and such that , , and .

The structure of has been the subject of many previous studies. It is known that is a full dimensional polyhedron. Furthermore, for every , the constraints , and are facet defining inequalities of and we call them boolean facets [15]. We will denote by the system of linear inequalities corresponding to boolean facets.

The rank constraint is always valid for and defines a facet if and only if is not a multiple of [15]. In [7] the authors obtained another family of facet-defining inequalities for associated with circulant minors.

Lemma 2.2.

[7] Let such that , and let . Then, the inequality

(1)

is a valid inequality for . Moreover, if , and , this inequality defines a facet of .

Observe that the set uniquely determines the set , and hence the minor associated with it. The authors termed (1) as the minor inequality corresponding to . Moreover, the authors showed that every non boolean facet-defining inequality of (whose facetial structure had been previously characterized in [9]) is either the rank constraint or a minor inequality. Similarly, and are completely described by boolean facets and minor inequalities, for any [8].

3 Computing the first Chvátal closure

In our attempt at finding a linear description of the first Chvátal closure of , we use the following well-known result from integer programming:

Lemma 3.1.

Let be a nonempty polyhedron with integral and totally dual integral. Then, .

As we shall see below, if all vertices of are known then a totally dual integral system describing the polyhedron can be computed from this information. This is the case for , whose vertices have been completely characterized by Argiroffo and Bianchi [2].

Lemma 3.2 ([2]).

Let be a vertex of . Then one of the following statements holds:

  1. is integral.

  2. There exists with and such that

In order to express via a totally dual integral system of linear inequalities, we use the method described below (see, e.g., [6, Ch. 8]). Given a polyhedral cone , consider the points in the lattice . An integral generating set for is a set having the property that every can be written as a linear combination of some elements with integral non negative coefficients .

The method consists in adding redundant inequalities to the original system until the following property is verified: If is the set of linear inequalities satisfied with equality by a vertex and is the cone generated by the set of vectors , then is an integral generating set for

This idea leads to the following procedure for computing :

  1. Let .

  2. For every vertex of do

    1. Compute an integral generating set of .

    2. For all , let and add the inequality to .

  3. Return as a linear description of .

Observe that the inequality at step 2.2 is valid for , since minimizes over this polyhedron for any . Moreover, if is integral, then the new inequality added to the system is redundant, as . Therefore, new inequalities for may only arise from integer generating sets related to fractional vertices belonging to one of the two latter clases described in Lemma 3.2.

Firstly, we analyze all inequalities arising from the vertex . The point is known to be a vertex of if and only if . In this case, are the inequalities of the original system satisfied at equality by . In order to find an integral generating set for we need the following result.

Lemma 3.3.

Let be two vectors such that and , with . Then there exists such that and .

Proof.

Observe that is the solution of the linear system

Subtracting each equation from the previous one, and the first from the last, we obtain , for . Moreover, from it follows and, since is integral, we must have . As a consequence, for some .

On the other hand, as , we can write and substitute this expression in the original system. Since , the matrix is invertible and we obtain . Finally, from we have . ∎

With this result we can compute an integral generating set for .

Theorem 3.4.

Let be a circulant matrix such that and consider the vertex of . Then an integral generating set for is given by

Proof.

Let . Then there exists a non negative vector such that . Moreover, as the (unordered) sets of row and column vectors of a circulant matrix are equal, this is equivalent to saying that there exists with . Let . We have and

Since is integral, applying Lemma 3.3 yields , for some . Hence,

and thus can be written as a linear combination of the elements of where all coefficients are non negative integers. ∎

When applying step 2.2. of the procedure described at the beginning of this section, the vectors yield the inequalities from , while for the last vector we obtain the rank constraint of :

Corollary 3.5.

If , then the inequality is valid for .

On the other hand, given a vertex corresponding to a circulant minor , the task of finding an integral generating set for turns out to be more complicated. We present here a partial result which is however sufficient for deriving a new class of facet-defining inequalities for .

Consider a circulant minor of , and let be the corresponding vertex of , defined as in Lemma 3.2(iii). From Lemma 2.1 and Lemma 2.4 in [1] it follows that can be obtained from by deleting each column with and each row with . Hence, holds for and satisfies the following inequalities with equality:

(2)
(3)

Observe that there might be other inequalities from satisfied tightly by . In the following we denote by the subcone of spanned by the normal vectors of the left-hand sides of (2) and (3).

Theorem 3.6.

Let be a vertex of associated with a minor , and let . An integral generating set for is given by

Proof.

Let be the square coefficient matrix of the system (2)-(3). Reordering the columns, we may assume that has the following block form:

where the first columns correspond to indices in , the next columns correspond to indices in and the last columns correspond to indices not in . Moreover, , and are identity and zero matrices of the appropriate sizes.

Let . Then there exist , , and with . Now consider the vectors , , , and define:

(4)

Observe that and . From Lemma 3.3 it follows that and , for some . Moreover, if then we must have and from (4) we conclude that is an integral conic combination of the row vectors of , which finishes the proof.

Now assume . For , we have . From the proof of Theorem 3 in [7], it follows that each column of has support equal to and each column of has support equal to . Thus,

Finally, since , , and , we must have , , and hence . Then the statement of the theorem follows from (4). ∎

Vectors in the first two sets of last theorem give rise to boolean inequalities after applying step 2.2. of the procedure from the beginning of this section. For the third set, we have , and hence we obtain:

Corollary 3.7.

Let be such that and . The inequalities

(5)

with , are valid for .

For , these inequalities are the minor inequalities described in [7]. Accordingly, we have called (5) as generalized -minor inequalities. In some cases, generalized -minor inequalities with can be obtained from the addition of (classical) minor inequalities and the rank constraint, and are thus redundant for . However, this is not true in general. For instance, consider and . One can verify that and the corresponding inequality (5) for has the form . Moreover, it can be shown that this inequality defines a facet of . As a consequence we have the following result.

Theorem 3.8.

There are circulant matrices for which minor inequalities, boolean facets, and the rank constraint are not enough to describe .

4 Separation algorithms for generalized minor inequalities

A polynomial time algorithm to separate 1-minor inequalities associated with particular classes of circulant minors has been proposed by S. Bianchi et al. in [7]. In this section we extend some of their results to the case of generalized minor inequalities. More precisely, we address the separation problem for -minor inequalities corresponding to circulant minors having parameters .

Following the ideas presented in [7], let us first prove a technical lemma that will be required for our separation procedure.

Lemma 4.1.

Let , and be the parameters associated with a circulant minor of such that with . Then

Proof.

Let be the nonnegative integer such that . Since we have that

From and it follows that

and the proof is complete. ∎

Observe that if is a multiple of , then the corresponding -minor inequality is redundant for , as the value on the right-hand side is integer. Otherwise, if defines a minor of with parameters , and , where and , the previous lemma implies that the corresponding -minor inequality can be written as

where

or, equivalently,

(6)

Given and two integer numbers with and , we define the function on by and the function on by .

Then, inequality (6) can be written as

(7)

Following the same notation introduced in [7], let be the family of sets defining minors with parameters . We are interested in the separation of generalized -minor inequalities corresponding to sets .

To this end, given let be the digraph with set of nodes

where and set of arcs defined as follows: first consider in the arcs

  • for all such that and (mod ),

then consider in a recursive way for :

  • for each , add whenever is such that and (mod ),

and finally,

  • for each , add whenever is such that and (mod ).

Note that, by construction, is acyclic. In Figure 1 we sketch the digraph where only the arcs corresponding to a particular -path are drawn.

Figure 1: A -path in the digraph .

For the proof of the next lemma, we need the following observation.

Remark 4.2.

Let with , then defines a circulant minor with parameters if and only if (mod ) and , for all .

Now we can state our result.

Lemma 4.3.

There is a one-to-one correspondence between -paths in and subsets with .

Proof.

Let and assume that with .

Then, by Remark 4.2, (mod ) and for all . Thus, the set of nodes with

induces a -path in .

Conversely, let be a -path in . By construction, there exists a nonnegative integer such that for all and for . Then, .

Now, if we define

we have and from Remark 4.2 it follows that . ∎

Theorem 4.4.

Given and , the separation problem for -minor inequalities corresponding to minors with parameters and can be polynomially reduced to at most shortest path problems in an acyclic digraph.

Proof.

Let . We will show that the problem of deciding if, given , there exists with and such that violates the inequality (7) can be reduced to a shortest path problem. W.l.o.g we set .

Consider the digraph and associate the cost with every arc , and the cost with every arc .

Clearly, if is the subset of corresponding to a -path in , the length of is equal to .

Then, there exists with and such that violates the inequality (7), if and only if the length of the shortest -path in is less than . ∎

Since is acyclic, computing each shortest path from the last theorem can be accomplished in time (see, e.g., [10, Theorem 2.18]). Moreover, from the definition of it follows that each node has outdegree of order and that the graph contains nodes. Hence, each shortest path computation requires time, and the separation problem of -minor inequalities can be solved in time for fixed . Repeating this procedure for each possible value of and , we obtain the following result.

Theorem 4.5.

For a fixed , the separation problem for -minor inequalities corresponding to minors of with parameters can be solved in polynomial time.

5 Conclusions

We have presented a new class of valid inequalities for whose Chvátal rank is at most one. These inequalities strictly generalize the class of minor inequalities described in [7]. Moreover, some of these inequalities give rise to new facets of , as shown by the example at the end of Section 3. Hence, despite of the results obtained for in [9] and in [8], (classic) minor inequalities, together with boolean facets and the rank constraint are not sufficient to provide a complete linear description of in general.

An apparently weaker problem consists in finding a complete linear description for the first Chvátal closure of . However, as far as we are aware from literature, no circulant matrix is known for which the Chvátal rank of is strictly larger than one. To complete the characterization of following the path presented here, more work is still needed in order to fully characterize integral generating sets for the cones associated with the fractional vertices defined in Lemma 3.2(iii). All computational experiments we have conducted so far support the conjecture that generalized minor inequalities, together with boolean facets and the rank constraint are sufficient for describing .

Besides of the search for complete linear descriptions of and , one line of future research could be the study of necessary and sufficient conditions for a generalized minor inequality to define a facet of the integer polytope, similar to the conditions presented in [7] for classic minor inequalities. Moreover, the separation problem for generalized minor inequalities corresponding to minors with parameters or is an issue that requires further investigation.

Another possible line of research involves establishing analogies between set packing and set covering polyhedra. As mentioned in the introduction, a complete linear description for set packing polytope related to circulant matrices (in fact, to the more general class of circular matrices) has been reported in [13]. The authors show that this polytope is described by nonnegativity constraints, clique inequalities and so-termed clique family inequalities. The first two families can be regarded as counterparts of boolean facets for the set covering polyhedron. Determining whether there is also an analogous class to clique family inequalities in the case of , and how this class is related to the class of generalized minor inequalities could shed more light on the structure of this polyhedron. For instance, similarly to generalized minor inequalities, clique family inequalities have only two different coefficients, which are consecutive integers. First steps in this direction have been undertaken in [3, 4].

References

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  • [2] Argiroffo, G. and S. Bianchi, On the set covering polyhedron of circulant matrices, Discrete Optimization 6 (2009), 162–173.
  • [3] Argiroffo, G. and S. Bianchi, Row family inequalities for the set covering polyhedron, Electronic Notes in Discrete Mathematics 36 (2010), 1169–1176.
  • [4] Argiroffo, G. and A. Wagler, Generalized row family inequalities for the set covering polyhedron. In: Proceedings of the 10th Cologne-Twente Workshop 2011. (2011), 60–63.
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  • [9] Bouchakour, M., T.M. Contenza, C.W. Lee and A.R. Mahjoub, On the dominating set polytope, European Journal of Combinatorics 29 (3) (2008), 652–661.
  • [10] Cook, W.J., W.H. Cunningham, W.R. Pulleyblank and A. Schrijver, Combinatorial Optimization, Wiley-Interscience, New York, (1998).
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  • [12] Cornuéjols, G. and B. Novick, Ideal Matrices, Journal of Combinatorial Theory B 60 (1994), 145–157.
  • [13] Eisenbrand, F., G. Oriolo, G. Stauffer and P. Ventura, The Stable Set Polytope of Quasi-Line Graphs, Combinatorica 28 (1) (2008), 45–67.
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  • [16] Tolomei, P. and L. M. Torres, On the first Chvátal closure of the set covering polyhedron related to circulant matrices, Electronic Notes in Discrete Mathematics 44 (2013), 377–383.
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