Servicios Personalizados
Revista
Articulo
Indicadores
- Citado por SciELO
Links relacionados
- Similares en SciELO
Compartir
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 linear programs, where the set of feasible solutions may be seen as the set of integral solutions in a polyhedron , i.e. .
In spite of optimizing a linear function over is equivalent to do it over
in the general case, the complete description of 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 will be called a relaxation of if . Given two different relaxations and of such that , the bound obtained by optimizing over is tighter than the one obtained by optimizing over and we say that the relaxation is weaker than .
This fact motivates the definition of operators such that, applied on a relaxation of , in each step they obtain a new relaxation , arriving to 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 () defined by Balas, Ceria and Cornuéjols [4].
In [1] and [2] the authors analyze the performance of the 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 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 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 obtained from 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 . After a lift-and-project iteration of the operator on , they obtain
Applying iteratively the operator on a subset of indices , they proved that
For any subset , will be called a disjunctive relaxation of .
Clearly, . This last property allows us to talk about the minimum number of iterations needed to find , which is called the disjunctive index of . It is also clear that if and only if the disjunctive index of is equal to zero.
Given a graph with as set of nodes, let us recall that is a clique in if in every pair in , its nodes are adjacent. Moreover, 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 of is the convex hull of the incidence vector of its stable sets. The clique relaxation of is defined as
In [1], the authors study the behaviour of the disjunctive operator on polyhedra .
Following Lovász's Perfect Graph Theorem in [12], if a graph is perfect then its complementary graph is also perfect. From the polyhedral characterization of perfect graphs given by Chvátal [5], i.e is perfect if and only if , we have that the disjunctive index of is zero if and only if the disjunctive index of is zero. Moreover, it is known that if is minimally non perfect (mnp), then is also mnp and has only one fractional extreme point. It holds that if a graph is , the disjunctive indices of 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 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 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 , .
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 , called the vertex set.
Given a clutter over , the blocker of , , is the clutter of minimal vertex covers of , i.e. minimal subsets of satisfying
We will denote by the matrix (with entries 0 or 1) whose rows are the characteristic vectors of the edges of . Clearly, has no dominating rows.
Given a clutter , is the incidence vector of a vertex cover of if and only if . The set covering polyhedron associated to is defined as
where is called the original relaxation of .
In this case, considering the blocker of a polyhedron defined as
and the blocker 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 of the disjunctive operator, as
where . They showed that preserves the main properties of . In particular, for any ,
The disjunctive index is also well defined, since
Ideal clutters, defined by Lehman, are those for which 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 is zero if and only if the disjunctive index of is zero. Moreover, it is known that if is minimally non ideal (), then is also mni and has only one fractional extreme point. It holds that if a clutter is , the disjunctive indices of and 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 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 , .
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 be a polyhedron and consider for ,
and
If is a fixed subset of , and , let us define
It is not hard to see [1] that
and, if is the set of extreme points of ,
Given a graph with and considering , it is clear that is not empty if and only if is a stable set in . Moreover, denoting by is adjacent to some , if , for all . Therefore, calling , we have
Given , we write any as , with . Therefore, denoting by (respectively , 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 .
This lemma is the key of the proof in [13] of Theorem 6 below.
Theorem 6. [13] For any , if and only if 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 and , denotes the clutter obtained from by the contraction of , i.e. the clutter defined over , whose edges are the minimal elements of . Also, denotes the clutter obtained from by the deletion of , i.e. the clutter defined over , whose edges are the edges of not containing . Given two disjoints subsets , of , denotes the minor of obtained by the contraction of nodes in and deletion of nodes in . It is known that .
In this way, in [10] it is proved that
Theorem 7. [10] For any , if and only if is ideal for every .
Let us notice that, meanwhile the integrality of involves the perfection of only the subgraph induced by (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 . This difference should be found in the characterization of the extreme points of the disjunctive relaxations of . We present here an scheme of the proof of Theorem 7 in [10].
In the following, will be a fixed subset of . For with , let us consider the polyhedron
where
It can be easily seen that
Then, every extreme point of is an extreme point of , for some . Moreover, writing any as where and denoting by the characteristic vector of , we have
Lemma 8. Given and , is an extreme point of if and only if and is an extreme point of .
The difference with the "stable set case" appears noting that it is not true that for any , every extreme point of is an extreme point of . 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 .
Now, we derive Theorem 7, the "symmetric" of Theorem 6:
Proof. (of Theorem 7)
Suppose that for some , is not ideal, i.e. that has some fractional extreme point . Then, is a fractional extreme point of . From Lemma 9, there exists such that is a fractional extreme point of , and then is not integral. Therefore, .
To see the converse, if is ideal for every , from Lemma 8 follows that is an integral polyhedron for every . Therefore, 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 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 is a subset of edges where any two of them are incident to a same node of . The matching polytope is defined as the convex hull of the incidence vector of the matchings in . The natural original relaxation of is
It is known that if and only if is bipartite. In [3], it is proved that the disjunctive index of 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 of . However, it cannot be seen as a particular case of Theorem 6 because the relaxation of is, in general, weaker than .
In a more general context, in a set packing problem we are given a clutter over a set and optimize over , the convex hull of vectors in .
If we consider the associated graph of defined by a set of nodes equal to , the vertex set of , and where two nodes in are adjacent if there exists an edge of that contains both of them, it is not hard to see that and is a relaxation, weaker than .
When 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 coincides with the minimum number of nodes we must delete from in order to obtain a graph , such that for some and .
Given a family C of clutters and a graph , let us say that is a C-graph if for some .
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 is in C.
Definition 11. Given an hereditary family of clutters C and a C-graph such that with , is C-perfect if and only .
Condition 1 in Definition 10 implies that C-perfection in Definition 11 is well defined. Besides, it is not hard to see that . 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,
and it is known that C-perfect graphs are bipartite graphs.
This definition also includes as a relaxation of . In this case, the elements of C have vertex set equal to the set of edges of a graph , and the edges are the subsets of incident in to a node in . Besides, and . It is known that 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 is C-perfect.
To prove Theorem 12, it only remains to recall that
where and notice that again, is not empty if and only if is a stable set in . Moreover, if , for all in . Therefore, calling , we have
It is not difficult to see that with is an extreme point of if and is an extreme point of . Moreover, .
Proof. (of Theorem 12)
Clearly, if and only is an integral polyhedron. Equivalently, from the previous observations, has to be integral, for any . Equivalently, . Finally, if and only if is C-perfect for all . Since is a node induced subgraph of , the result follows.
Now, for any graph , the results for and may be seen as particular cases of the previous theorem. We also have that the disjunctive index of 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 . Moreover, is an extreme point of if and only if is an extreme point of .
Theorem 4 easily follows from the previous result.
In this section, we will find similar relationships between the antiblocker of a disjunctive relaxation of and some particular subgraphs of a graph .
In the following, will be denote a fixed subset of . Let with and for some .
From the observations made by Gerards, Maróti and Schrijver in [8], we know that if and (), then is an edge of In particular, since for all ; is a clique in . Moreover, if and , then is a node of .
Now, let be an extreme point of , for some and . Since , . Then, we have and when is not a node of In the following theorem, we prove that the converse also holds, i.e.
Theorem 14. Let . If
and when is not a node of then . Moreover, if is an extreme point of then is an extreme point of .
Proof. Let be an extreme point of with and . Let us define . We need to show that . Since , clearly if . If , then and it only remains to prove that
Let and with . Since is an edge of , obtaining that .
For the second part, let us assume that is a convex combination of two points and in . Defining and from the previous results we have for and is a convex combination of and .
From this characterization follows an easy new alternative proof for Theorem 2.
Proof. (of Theorem 2)
Since , it suffices to prove that the polyhedron is integral. If is any extreme point of it, then is an extreme point of . Moreover, for some From Theorem 14, we have that is an extreme point of 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 and , let
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 ;
- with is an extreme point of if and only if there exists such that and is an extreme point of .
- 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 ;
- is an extreme point of if and only if there exists such that and is an extreme point of .
References
[1] N. Aguilera, M. Escalante and G. Nasini. A Generalization of the Perfect Graph Theorem under the Disjunctive Index, Mathematics of Operations Research 27 (2002), pp. 460–469. [ Links ]
[2] N. Aguilera, M. Escalante and G. Nasini. The Disjunctive Procedure and Blocker Duality, Discrete Applied Mathematics 121/1-3 (2002), pp. 1–13. [ Links ]
[3] N. Aguilera, S. Bianchi and G. Nasini. Lift and Project Relaxations for the Matching and Related Polytopes, Discrete Applied Mathematics 134 (2004), pp. 193–212. [ Links ]
[4] E. Balas, G. Cornuéjols and S. Ceria. A Lift-and-Project Cutting Plane Algorithm for Mixed 0-1 programs, Mathematical Programming 58 (1993), pp. 295–324. [ Links ]
[5] V. Chvátal. On certain Polytopes Associated with Graphs, Journal of Combinatoriaal Theory Series B 18 (1975), pp. 138–154. [ Links ]
[6] M. Escalante, G. Nasini and M. Varaldo. Note on Lift-and- Project ranks and Antiblocking Duality, Electronic Notes on Discrete Mathematics 18 (2004), pp. 115–119. [ Links ]
[7] D. Fulkerson. Blocking and Antiblocking Pairs of Polyhedra, Mathematical Programming 1 (1971), pp. 168–194. [ Links ]
[8] A. Gerards, M. Maróti and A. Schrjiver. Note on : N. E. Aguilera, M. S. Escalante and G. L. Nasini, "A Generalization of the Perfect Graph Theorem under the Disjunctive Index", Mathematics of Operations Research 28 4 (2003), pp. 884–885. [ Links ]
[9] A. Lehman. On the With-Length Inequality, Mathematical Programming 17 (1979), pp. 403–417. [ Links ]
[10] V. Leoni and G. Nasini. Note on: N.E. Aguilera, M.S. Escalante, G.L. Nasini, "The disjunctive procedure and Blocker Duality", Discrete Applied Mathematics 150 (2005), pp. 251-255. [ Links ]
[11] L. Lipták and L. Tunçel. Lift-and-Project Ranks and Antiblocker Duality, Research Report: CORR 2003-16. [ Links ]
[12] L. Lovász and A. Schrijver. Cones of matrices and set-functions and 0-1 optimization, SIAM J. Optim. 1 (1991), pp. 166–190. [ Links ]
[13] G. Nasini. El índice de imperfección de un grafo y su complemento, Anales JAIIO-SIO 01 (2001), pp. 101–106. [ Links ]
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