Discrete Applied Mathematics 154 (2006) 1324 – 1334 www.elsevier.com/locate/dam
Clutter nonidealness G. Argiroffo, S. Bianchi, G. Nasini Depto. de Matemática. Facultad de Ciencias Exactas, Ingeniería y Agrimensura, Universidad Nacional de Rosario, Argentina Received 15 September 2003; received in revised form 21 May 2004; accepted 6 May 2005 Available online 3 February 2006
Abstract Several key results for set packing problems do not seem to be easily or even possibly transferable to set covering problems, although the symmetry between them. The goal of this paper is to introduce a nonidealness index by transferring the ideas used for the imperfection index defined by Gerke and McDiarmid [Graph imperfection, J. Combin. Theory Ser. B 83 (2001) 58–78]. We found a relationship between the two indices and the strength of facets defined in [M. Goemans, Worst-case comparison of valid inequalities for the TSP, mathematical programming, in: Fifth Integer Programming and Combinatorial Optimization Conference, Lecture Notes in Computer Science, vol. 1084, Vancouver, Canada, 1996, pp. 415–429; M. Goemans, L.A. Hall, The strongest facets of the acyclic subgraph polytope are unknown, in: Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, vol. 1084, Springer, Berlin, 1996, pp. 415–429]. We prove that a clutter is as nonideal as its blocker and find some other properties that could be transferred from the imperfection index to the nonidealness index. Finally, we analyze the behavior of the nonidealness index under some clutter operations. © 2006 Elsevier B.V. All rights reserved. Keywords: Set covering; Clutters; Nonidealness index
1. Introduction The set covering problem (SCP) can be stated as min{cx: Ax 1, x ∈ {0, 1}n }, where A is a 0–1 matrix and c ∈ Rn . Reversing the inequality in the definition of SCP we obtain the set packing problem (SPP) max{cx: Ax 1, x ∈ {0, 1}n }. Both problems are NP-complete problems for general 0–1 matrices and they seem to be “symmetric” in some way but in spite of this symmetry, the knowledge on SCP is significantly smaller than the one on SPP although many combinatorial problems of both theoretical and practical interest can be formulated as covering problems. A typical application concerns facility location. There are many other applications of this type, including assigning customers to delivery routes, bus, railway and airline crews to flights and workers to shifts, logical data analysis and political districting among them. E-mail addresses:
[email protected] (G. Argiroffo),
[email protected] (S. Bianchi),
[email protected] (G. Nasini). 0166-218X/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.dam.2005.05.032
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Why the interest on SPP has been greater than on SCP? A possible explanation to this could be the relationship between the SPP and the stable set problem in a graph, which caught more attention because of its relation with perfect graph theory. On the other hand, several key results on SPP do not seem to be easily or even possibly transferable to SCP. For instance, there is no characterization of ideal clutters through forbidden minors since a complete list of the minimally nonideal clutters is already unknown. The polyhedron K = {x 0: Ax 1} has only integral extreme points if and only if A is the maximal clique-node matrix of a perfect graph. In fact, if A is the clique-node matrix of an arbitrary graph G, then K is known as QSTAB(G), the clique-node relaxation of the stable set polytope of G. This last gave the idea of defining a measure of the imperfection of a graph by considering how “far” the extreme points of QSTAB(G) are from being integral. In this way Aguilera et al. defined in [2] an imperfection index of a graph G through the behavior of the disjunctive operator [3] on QSTAB(G). In other direction, Gerke and McDiarmid defined in [9] an imperfection index from the quotient between the fractional chromatic number and the clique number in a weighted graph. Although it does not seem to exist any relationship between these indices, they both satisfy that a graph is as imperfect as its complement, giving a generalization of the Perfect Graph Theorem. When working on SCP, if K = {x 0: Ax 1} has only integral extreme points, the matrix A is ideal. It seems natural to define a nonidealness index following the same pattern used for an imperfection index. Aguilera et al. [1] transferred the ideas used for their imperfection index to a nonidealness index defined from an extension of the disjunctive procedure on blocking-type polyhedra whose vertices belong to [0, 1]n . With this index a clutter becomes as nonideal as its blocker and this property can be thought as a generalization of Lehman’s Theorem. The goal of this paper is to introduce a new nonidealness index by transferring the ideas used for the imperfection index defined by Gerke and McDiarmid. Although several polyhedral properties could have been transferred, some difficulties have been observed when dealing with properties related to the combinatorial parameters. This difference could be attributed to the fact that from the combinatorial point of view the “symmetric concept” of perfection should be associated to the packing property (see [4]). This paper is organized as follows. In Section 2 we present the notation, definitions and results we need to define the nonidealness index. In Section 3 we prove that the nonidealness index has more tractable equivalent definitions. In Section 4 we explore more properties for this nonidealness index and finally, in Section 5 we study how this index behaves when clutters are composed under several clutter operations. 2. Notation and definitions Given a polyhedron K ⊂ Rn we denote by V(K) the set of its extreme points and K ∗ = conv(K ∩ Zn ), the convex hull of the integral points of K. Following [6], K is a blocking-type polyhedron if y x ∈ K, then y ∈ K. It is known that K is a blocking-type polyhedron if and only if K = {x ∈ Rn+ : Ax 1}, where A is a nonnegative matrix with no zero rows. The blocker of K, K B , is also a blocking-type polyhedron, defined by K B = { ∈ Rn+ : x 1
for all x ∈ K}.
Since (K B )B = K, we can refer to K and K B as a blocking pair of polyhedra. Moreover, if B is a |V(K)| × n matrix whose rows are the extreme points of K, then K B = { ∈ Rn+ : B 1}. A clutter C is a pair (V (C), E(C)), where V (C) is a finite set and E(C) is a family of subsets of V (C) none of which is included in another. The elements of V (C) and E(C) are the vertices and the edges of C, respectively. A clutter C is trivial if it has no edge or if ∅ is its unique edge. In the following, whenever the meaning is clear from the context, we denote V and E instead of V (C) and E(C), respectively. In general we consider V = {1, . . . , n} and |E| = m. A transversal or vertex cover in a clutter C = (V , E) is a set of vertices B that intersects all the edges, that is, B ⊂ V such that |B ∩ A|1, for all A ∈ E. The vertex covering number denoted by (C), is the minimum cardinality of a vertex cover in C. The blocker of C is the clutter b(C) such that V (b(C)) = V and E(b(C)) is the set of the minimal vertex covers in C. In [5] it is shown that b(b(C)) = C for any clutter C. Given j ∈ V , the clutter C/j obtained by contraction of j is the clutter defined by V (C/j ) = V − {j } and E(C/j ), the set of minimal elements of {S − {j }: S ∈ E}. The clutter C\j obtained by deletion of j is the clutter such that
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V (C\j ) = V − {j } and E(C\j ) = {S ∈ E: j ∈ / S}. A minor of C is any clutter obtained from C by a sequence of deletions and contractions. It is straightforward to check that for j, k ∈ V , (C/j )\k = (C\k)/j . Moreover, if V1 and V2 are disjoint sets of vertices in V, contracting all vertices in V1 and deleting all vertices in V2 can be performed sequentially, and the resulting clutter does not depend on the order of the operations or vertices. Therefore, we can denote such clutter as C/V1 \V2 . Furthermore, b(C/V1 \V2 ) = b(C)\V1 /V2 . If C is a nontrivial clutter, M(C) is the 0–1 matrix with columns indexed by V and whose rows are the characteristic vectors of edges in E. Clearly, given x ∈ {0, 1}n , x is the incidence vector of a vertex cover in C if and only if M(C)x 1 and Q(C) = {x ∈ Rn+ : M(C)x 1} is a blocking type polyhedron with extreme points in [0, 1]n . Then, x ∈ {0, 1}n is the incidence vector of a vertex cover in C if and only if x is an integral extreme point of Q(C), so Q∗ (C) = conv(Q(C) ∩ Zn ) is called the set covering polyhedron of C. From the above definitions and results, it follows that (Q∗ (C))B = Q(b(C)) and then, Q∗ (C) and Q(b(C)) is a blocking pair of polyhedra. Clearly (C) = min{1x: x ∈ Q∗ (C)} and we denote by f (C) the fractional covering number of C, defined by f (C) = min{1x: x ∈ Q(C)} = min{1x: M(C)x 1}. Given a weight vector w ∈ Rn+ , we denote w (C) = min{wx: x ∈ Q∗ (C)} and w f (C) = min{wx: x ∈ Q(C)}. w (C), for all w ∈ Zn or equivalently, if and only if Q(C) = Q∗ (C). If (C) = A clutter C is ideal if and only if w + f C is ideal, its blocker also is. A clutter is minimally nonideal if it is not ideal but all its proper minors are. If a clutter is minimally nonideal, so is its blocker (see [10]). Let us present some minimally nonideal clutters, a complete list of whom is not known. For further references, we remit the interested reader to [4] or [13]. • A circulant clutter Cnk is defined by V (Cnk ) = Zn and E(Cnk ) = {{i, i + 1, . . . i + k − 1}, i ∈ Zn }. If n3 and n odd, Cn2 (and their blockers) are minimally nonideal. The only minimally nonideal circulant clutters for k 3 are 3 3 3 4 6 7 C53 , C83 , C11 , C14 , C17 , C74 , C11 , C95 , C11 , C13 .
• F7 , the clutter with 7 vertices and 7 edges corresponding to points and lines of the finite projective geometry on 7 points. In this case b(F7 ) = F7 . • OK5 and its blocker, where OK5 denotes the clutter whose vertices are the edges of K5 , the complete graph on 5 nodes, and whose edges are the odd cycles of K5 . • The degenerate projective planes Jn , n 3, where V (Jn ) = {0, 1, 2 . . . , n} and E(Jn ) = {{1, 2, . . . , n}, {0, 1}, . . . , {0, n}}. In this case b(Jn ) = Jn . Minimally nonideal clutters different to Jn , have a certain regularity. If C is such a clutter, then Q(C) has a unique fractional vertex, namely 1/(b(C))1, and f (C) = n/(b(C)). Moreover, there exists d ∈ Z+ such that n + d = (b(C))(C) (see [11]). In the case of Jn , the unique fractional vertex of Q(Jn ) is ((n − 1)/n, 1/n, . . . , 1/n) and (Jn ) = 2. For a clutter C and j ∈ V , the clutter obtained by duplication of j is the clutter C ∗ j such that V (C ∗ j ) is obtained by adding a new node j to V (C) and E(C ∗ j ) = E ∪ {(B − {j }) ∪ {j }: B ∈ E and j ∈ B}. A clutter obtained from C by a sequence of deletions and duplications is a parallelization of C, and it is easy to check that the order in which these operations are performed is irrelevant. Then, parallelizations of C can be associated with vectors w ∈ Zn+ in the following way: C w is the clutter obtained by deletion of vertices i with wi = 0 and duplicating w wi − 1 times any vertex with wi 1. It is easy to see that w (C) = (C w ) and w f (C) = f (C ). A matching in a clutter C is a set of pairwise disjoint edges and the matching number (C) is the maximum cardinality of a matching in C. Clearly, it holds that (C) (C) and (C) = max{y1: yM(C)1, y ∈ Zm + }.
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Defining f (C), the fractional matching number of C as f (C) = max{y1: yM(C) 1, y ∈ Rm + }, by linear programming duality we have f (C) = f (C). Therefore, given w ∈ Zn+ we have that 0 < f (C w )/(C w ) 1. Moreover, by definition C is ideal if and only if f (C w )/(C w ) = 1 for all w ∈ Zn+ . Equivalently, C is not ideal if and only if there exists some w ∈ Zn+ for which f (C w )/(C w ) < 1. The nonidealness index of a clutter C is defined as follows. Definition 2.1. Given a nontrivial clutter C, the nonidealness index of C, denoted by ini(C) is f (C w ) n ini(C) = inf for all w ∈ Z , w = 0 . + (C w ) 2.1. The relation with Gerke and McDiarmid’s imperfection index Let us show the analogy between the above nonidealness index and the imperfection index of a graph G = (V , E) defined by Gerke and McDiarmid in [9]. This imperfection index is defined by taking the minimum among ratios between the weighted fractional chromatic number, f (G, x) = min yS : yS xv ; v ∈ V , yS 0; S stable set in G , S⊂V
v∈S
and (G, x) the clique number of Gx , the graph obtained by replicating each vertex v of G by a clique of size xv ∈ Z+ . Let us now consider the clutter C of maximal cliques of G. We have that a(C), the antiblocker of C, is the clutter of maximal stable sets of G, i.e. a(C) is the clutter of the maximal cliques of the complement of G and a(a(C)) = C. Now, we can restate (G, x) and f (G, x) as |V |
(G, x) = max{xy: M(a(C))y 1, y ∈ Z+ } and f (G, x) = min{y1: yM(a(C))x, y 0}. Since the imperfection index is the same for a graph and its complement, it could have been defined on G , the complement of G, by taking the minimum among ratios between f (G , x) = min{y1: yM(C)x, y 0} and |V |
(G , x) = max{xy: M(C)y 1, y ∈ Z+ }. Recalling the nonidealness index definition as the minimum among ratios between w f (b(C)) = max{y1: yM(C)w, y 0} and |V |
w (b(C)) = min{wx: M(C)x 1, x ∈ Z+ }, the symmetry between both indices is evident. In next section we follow [9] in order to find which results on the imperfection index can be transferred to the nonidealness index.
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3. Equivalent definitions and properties In order to study the nonidealness index of clutters, it will be useful to have alternative definitions. Given w ∈ Rn+ , let us first define, m w f (C) = max{y1: yM(C) w, y ∈ R+ }. w w w It is easy to see that if w ∈ Zn+ , then w f (C) = f (C ) and that for any t ∈ R+ , f (C ) and (C ) satisfy w tw f (C) = tf (C ),
(1)
tw (C) = t(C w ).
(2)
Moreover, it allows us to extend the definition of f (C w ) and (C w ) for rational w, and by continuity, for real w. Besides, we can restate ini(C) in the following way: f (C w ) n ini(C) = inf , w = 0 . for all w ∈ R + (C w ) Let us now observe that, by definition, (C w ) t if and only if wx t for all x ∈ Q∗ (C), or equivalently, w/t ∈ (Q∗ (C))B = Q(b(C)). Then we have (C w )t if and only if w ∈ tQ(b(C)).
(3)
Analogously, f (C w )r if and only if wx r for all x ∈ Q(C) or w/r ∈ (Q(C))B = Q∗ (b(C)). So, f (C w )r if and only if w ∈ rQ∗ (b(C)).
(4)
We state now the following equivalences: Lemma 3.1. For any nontrivial clutter C, the following statements are equivalent: (i) (ii) (iii) (iv)
ini(C)r, f (C w )r for all w ∈ Q(b(C)), if w ∈ Q(b(C)) then w ∈ rQ∗ (b(C)), wz r for all w ∈ Q(b(C)) and z ∈ Q(C).
Proof. Let C a clutter and 0 < r 1 such that ini(C) r. By definition we have that f (C w )/(C w ) r or, equivalently f (C w )r(C w ), for all w ∈ Zn+ , w = 0. Now given w ∈ Q(b(C)) by (3), (C w ) 1 and it follows that f (C w )r(C w ) r. Conversely, let w ∈ Rn+ , w = 0. Taking t = (C w ) in (3), it follows that w = w/(C w ) ∈ Q(b(C)). Then, by the hypothesis on w and (1) we have that f (C w ) = f (C w ) r, (C w ) and then ini(C)r, proving that (i) is equivalent to (ii). The equivalence between (ii) and (iii) follows immediately from (4). Finally, using once more the equality (Q(C))B = Q∗ (b(C)) the statement (iii) can be rewritten as: if w ∈ Q(b(C)) then wz r for all z ∈ Q(C), and (iii) is equivalent to (iv). Let us observe that in the previous lemma we can restrict w and z to be extreme points of Q(b(C)) and Q(C), respectively. This remark allows us to state the following equivalent definitions for the nonidealness index.
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Corollary 3.2. For any nontrivial clutter C, ini(C) = min{f (C w ): w ∈ Q(b(C))} = max{r ∈ R: Q(b(C)) ⊂ rQ∗ (b(C))} = min{wz: w ∈ Q(b(C)), z ∈ Q(C)}.
(5) (6) (7)
Applying the above corollary, we are able to prove that the nonidealness index has the following properties. Proposition 3.3. For any nontrivial clutter C, (i) (ii) (iii) (iv) (v)
0 < ini(C)1. ini(C) = 1 if and only if C is ideal. For any w ∈ Zn , w 1, ini(C w ) = ini(C). ini(C) = ini(b(C)). n ini(C) . (b(C))(C) of C, ini(C) ini(C). (vi) For every minor C
Proof. The first four properties follow immediately from the equivalent definitions of the nonidealness index. Property (v) follows from the definition and the fact that for any clutter f (C) = f (C) n/(b(C)). In order to prove property (vi), it is enough to show that it holds for minors obtained by deletion or contraction of one vertex. We only need to observe that x ∈ Q(C\j ) (respectively, Q(C/j )) if and only if (1, x ) ∈ Q(C) (respectively, (0, x ) ∈ Q(C)). By (7) and the fact that b(C\j ) = b(C)/j the property easily follows. 3.1. The relation with Goemans’s strength of facets Given a relaxation P of a blocking type polyhedron Q and ax b, a facet defining inequality of Q, Goemans and Hall defined in [8], the strength of the facet with respect to P as b , min{ax: x ∈ P } and proved that the maximum strength of the facet defining inequalities for Q is min{ ∈ R+ : P ⊂ Q}. Then, if Q = Q∗ (C) and P = Q(C), the maximum strength with respect to Q(C) of the facet defining inequalities for Q∗ (C) is 1/ini(C). Similarly, for an antiblocking type polyhedron Q and a relaxation P of Q, Goemans in [7] defined the strength with respect to P of ax b, a facet defining inequality for Q, as max{ax: x ∈ P } b and proved that min{ ∈ R+ : P ⊂ Q} is the maximum strength with respect to P of the facet defining inequalities for Q. Given a graph G, if Q = STAB(G) and P = QSTAB(G) then, the imperfection index of the graph G is the maximum strength with respect to QSTAB(G) of the facet defining inequalities for STAB(G). 4. Further properties The imperfection index defined in [9] is unbounded in the sense that, given any rational number r greater than 1, there exists a graph whose imperfection index is r. In the case of the nonidealness index, we will see that we can find a clutter as nonideal as possible.
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Proposition 4.1. Given ε > 0, there exists a clutter C such that ini(C) < ε. Proof. Let Cn.k be the clutter on n vertices {1, 2, . . . , n} whose edges are the subsets of k elements. It is immediate to check that b(Cn,k ) = Cn,n−k+1 and then (Cn.k ) = n − k + 1 and (b(Cn.k )) = k. Then by Proposition 3.3, ini(Cn.k )
n . k(n − k + 1)
In particular considering k = n/2 for n even, the proposition follows.
Let us now study the nonidealness index of minimally nonideal clutters. Proposition 4.2. For a minimally nonideal clutter C = Jn , is ini(C) =
n , (b(C))(C)
and ini(Jn ) = 1 −
n−1 . n2
Proof. The proof follows from (7) and from the fact that if C is a minimally nonideal clutter, then Q(C) has a unique fractional extreme point. If C = Jn , this extreme point is 1/(b(C))1, and for Jn = b(Jn ), it is ((n − 1)/n, 1/n, . . . , 1/n). As we have mentioned before if C = Jn then (b(C))(C) = n + d, d ∈ N. Moreover, for the known minimally nonideal clutters we have listed, different from F7 , d = 1 and ini(C) = n/(n + 1). In the case of F7 , d = 2 and ini(F7 ) = n/(n + 2). Proposition 4.3. There is an integral weight vector z such that for any positive integer multiple z of z, ini(C) =
(C z ) . (C z )
Proof. We know that ini(C) = min{f (C w ): w ∈ Q(b(C))}, then there exists w in Q(b(C)) such that ini(C) = ) = y1, where y is a rational vector in Rm . Given ∈ N such that y ∈ Zm , and f (C w w ∈ Zn+ we have + + ini(C)
) ) f (C w (C w = . w w (C ) (C )
) 1. Then On the other hand, since w in Q(b(C)) we know that (C w ) ) f (C w f (C w = f (C w ) = ini(C). ) ) (C w (C w
and considering z = w the proof is complete.
Defining rk = min
(C w ) n and k = min{w : i ∈ V } , : w ∈ Z i + (C w )
we can state the following proposition. Proposition 4.4. For any clutter C, ini(C) = limk→∞ rk .
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Proof. Let us call f (C w ) n : w ∈ Z rk = min and k = min{w : i ∈ V } . i + (C w ) Let z ∈ Zn+ such that ini(C) = f (C z )/(C z ) and k = min{zi : i ∈ V }. For any k > k and y = k z , k = min{yi : i ∈ V } rk f (C y )/(C y ).
k z f (C y ) f (C k )
k
Since we can say that = k f (C z ). On the other hand, it can be and k checked that (k − k)z kyand then (C (k−k)z ) (C ky ) or equivalently ((k − k)/k)(C z ) (C y ). Hence for all k > k, rk
f (C y ) (C y )
y kz k
k z k f (C ) (k−k) (C z ) k
=
k (k − k)
ini(C).
Taking limits on both sides we have limk→∞ rk ini(C). Since by definition ini(C) rk , for all k ∈ N we have proved that lim rk = ini(C).
(8)
k→∞
Clearly, rk rk for all k ∈ N and lim rk lim rk .
k→∞
(9)
k→∞
On the other hand, let k ∈ N and k=k1. For any y ∈ Zn+ such that k =min{yi : i ∈ V }, rounding down each coordinate of the fractional optimal solution where f (C y ) is attained, we obtain a feasible solution for the integral problem and then, f (C y ) (C y )f (C y ) − |E|. Next note that (C y ) (C k ) = k(C). Hence for all y such that k = min{yi : i ∈ V }, f (C y ) (C y ) f (C y ) |E| m − − , (C y ) (C y ) (C y ) (C y ) k(C) or rk rk −
m k(C)
for all k ∈ N.
Again, taking limits on both sides lim rk lim rk .
k→∞
k→∞
Summarizing we have that limk→∞ rk = limk→∞ rk = ini(C).
5. The nonidealness index and several clutter operations In this section, we analyze the behavior of the nonidealness index under some clutter operations. The study of how the nonidealness index works with these operations has several advantages, by one side it is possible to generate bigger an more complex clutters, by the other it is possible to decompose a clutter in smaller or simpler ones with known nonidealness indices. In any case it is useful to know how the nonidealness index changes. From Proposition 3.3, we know that the nonidealness index does not change under duplications. Let us study first its “blocking dual” operation: the replication of vertices. For a clutter C, and j ∈ V , the clutter obtained by replication of j , denoted by C ⊕ j , is the clutter such that V (C ⊕ j ) = V ∪ {j } and E(C ⊕ j ) = {B ∪ {j }: B ∈ E, j ∈ B} ∪ {B ∈ E: j ∈ / B}. Note that with the above definition (C ⊕ j )/j = C and it is straightforward to check that b(C ∗ j ) = b(C) ⊕ j and b(C ⊕ j ) = b(C) ∗ j , i.e. duplication and replication interchange when passing to the blocker (as contraction and deletion do). Therefore, it is immediate to see that ini(C ⊕ j ) = ini(b(C ⊕ j )) = ini(b(C) ∗ j ) = ini(C).
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In the following we consider operations between two clutters C1 and C2 , and we denote by C1 , C2 the clutter obtained by applying the corresponding operation. If C1 and C2 are minors of C1 , C2 , we clearly have that ini(C1 , C2 ) min{ini(C1 ), ini(C2 )}.
(10)
In other words, we have that the clutter obtained from the operation is “less ideal” than the original clutters. 5.1. Clique identification Let us first present some definitions and properties following [12]. Given a clutter C, a set A ⊂ V is a cutset of C if there exists a partition (V1 , V2 ) of V − A with V1 , V2 = ∅ and such that every edge of C is either a subset of V1 ∪ A or a subset of V2 ∪ A. A cutset A of a clutter C is a weak cutset if |{B ∩ A : B ∈ b(C)}| |A | + 1
for each A ⊂ A.
If A is a weak cutset of a clutter C, then C arises as weak cutset identification of the clutters C1 = C\V1 and C2 = C\V2 and the linear description of Q∗ (C) is given by the union of the linear descriptions of Q∗ (C1 ) and Q∗ (C2 ). In this case, clearly ini(C) = min{ini(C1 ), ini(C2 )}. A 2-clique of the clutter C is a subset R = {v1 , . . . , vr } of V such that for each pair of elements vi , vj ∈ R, {vi , vj } is an edge of C. When a clutter C has a 2-clique as cutset, it actually is a weak cutset of C. This leads us to define the following clutter operation. Let C1 and C2 two clutters and R1 ={v11 , . . . , vr11 } and R2 ={v12 , . . . , vr22 } maximal 2-cliques of C1 and C2 , respectively. Let us assume that r1 r2 . We choose a subset of R2 , say R2 , of r1 vertices and overlap them to the vertices of R1 . We define the clique identification of C1 and C2 as the clutter C1 , C2 whose vertex set is V1 ∪ V2 − R2 and edge set is the set E1 ∪ {B ∈ E2 : B ∩ {v12 , . . . , vr21 } = ∅} ∪ E2 where E2 are the edges obtained by taking every edge in E2 that contains a vertex in {v12 , . . . , vr21 } and by replacing it by the corresponding vertex in R2 via the overlapping. Then it is clear that R1 is a weak cutset of C1 , C2 and so, in this operation (10) becomes an equality. Moreover, ini(C1 , C2 ) does not depend on the maximal 2-cliques chosen nor in the way the nodes in the 2-cliques are overlapped. 5.2. Composition of clutters Let C1 = (V1 , E1 ), C2 = (V2 , E2 ) be two clutters such that V1 ∩ V2 = ∅ and let j ∈ V1 . The clutter obtained by composition of C1 and C2 on the node j is C = (C1 )j ←C2 , such that: V (C) = (V1 − {j }) ∪ V2 , and / E1 } ∪ {(B1 − {j }) ∪ B2 : B1 ∈ E1 , j ∈ B1 , B2 ∈ E2 }. E(C) = {B1 ∈ E1 : j ∈ It is known that C is ideal if and only if C1 and C2 are ideal (see [13]). Moreover, it is easy to check that C/(V1 −{j })=C2 , and C/(V2 − {k}) = C1 for any k ∈ V2 . Then, by (10) we have that ini(C) min{ini(C1 ), ini(C2 )}. Although this bound does not depend on the vertex where the composition is done, we can see in the following example that ini(C) does. Example 5.1. Let C1 = J3 and C2 = C52 . From Proposition 4.2 we have that ini(C1 ) = 79 and ini(C2 ) = 56 . By computing the extreme points of (C1 )j ←C2 and of b((C1 )j ←C2 ) and applying (7), we have: if j = 0 then 3 ini((C1 )0←C2 ) = , 4
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1333
and if j = 1 then ini((C1 )1←C2 ) =
41 . 54
In the sequel, and without loss of generality, we will consider that V (C1 ) = {1, . . . , j }, i.e. the vertex j is the “last” vertex of C1 . We state first the following proposition whose straightforward proof is omitted. Proposition 5.2. Let C1 = (V1 , E1 ), C2 = (V2 , E2 ) be clutters with V1 ∩ V2 = ∅, j ∈ V1 and C = (C1 )j ←C2 , then: (1) b(C) = b(C1 )j ←b(C2 ) . (2) w is an extreme point of Q(C) if and only if there exist extreme points v of Q(C1 ) and u of Q(C2 ) such that w = (v1 , . . . , vj −1 , vj u1 , . . . , vj un ). The following proposition provides a lower bound for ini(C) when C is a clutter obtained from a composition. Proposition 5.3. The composition C of the clutters C1 and C2 satisfies ini(C) ini(C1 )ini(C2 ). Proof. Let x = (v1 , . . . vj −1 , vj u1 , . . . , vj un ) and y = (s1 , . . . , sj −1 , sj t1 , . . . , sj tn ) be the extreme points of Q(C) and Q(b(C)) such that ini(C) = x T y where (v1 , . . . , vj −1 , vj ), (u1 , . . . , un ), (s1 , . . . , sj −1 , sj ) and (t1 , . . . , tn ) are extreme points of Q(C1 ), Q(C2 ), Q(b(C1 )) and Q(b(C2 )), respectively. Then x T y = (v1 , . . . , vj −1 , vj u1 , . . . , vj un )T (s1 , . . . , sj −1 , sj t1 , . . . , sj tn ) =
j −1 i=1 j −1
v i si + v j sj
n
u i ti
i=1
vi si + vj sj ini(C2 )
i=1
⎞ ⎛ j −1 ⎝ vi si + vj sj ⎠ ini(C2 ) i=1
ini(C1 )ini(C2 ).
Remark 5.4. Let us observe that if C1 or C2 are ideal clutters from (10) and the above proposition we have that ini(C1 )ini(C2 ) = ini(C) = min{ini(C1 ), ini(C2 )}. The following example shows that the bound given in (10) is tight in the sense that we can obtain equality even in the case of the composition of nonideal clutters. Example 5.5. Let C1 = C52 and C2 = J3 . As we have seen in Example 5.1, ini(C1 ) = 56 and ini(C2 ) = 79 . In this case we have that ini((C1 )5←C2 ) = 79 = min{ini(C1 ), ini(C2 )} although C1 and C2 are both nonideal clutters. Acknowledgments Supported by grant of Universidad Nacional de Rosario (Argentina). References [1] N. Aguilera, G. Nasini, M. Escalante, The disjunctive procedure and blocker duality, Discrete Appl. Math. 121 (1–3) (2002) 1–13. [2] N. Aguilera, G. Nasini, M. Escalante, A generalization of the perfect graph theorem under the disjunctive index, Math. Oper. Res. 27 (3) (2002) 460–469.
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