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The set covering problem on circulant matrices: polynomial instances and the relation with the dominating set problem on webs
Electronic Notes in Discrete Mathematics 36 (2010) 1185–1192 www.elsevier.com/locate/endm
The set covering problem on circulant matrices: polynomial instances and the relation with the dominating set problem on webs 1 S. Bianchi, G. Nasini, P. Tolomei.
2
Departamento de Matem´ atica Universidad Nacional de Rosario and CONICET Rosario, Argentina
Abstract The dominating set polyhedron of a web graph Wnk is the set covering polyhedron of a circulant matrix Cn2k+1 . In a previous work we generalize the results by Argiroffo and Bianchi on valid inequalities associated with every circulant minor of a circulant matrix and we conjecture that, for any k, the minor inequalities together with the boolean facets and the rank constraint are enough to describe the set covering polyhedron of Cnk . In this work we prove that the conjecture is true for the family of k with s = 2, 3 and 0 ≤ r ≤ s−1 and give a polynomial separation algorithm for Csk+r inequalities involved in the description. Thus, we prove the polynomiality of the set covering problem on these families. As a consequence we obtain the polynomiality of the minimum weight dominating set problem on webs of the form Wnt , when n = 2st + s + r with s = 2, 3 and 0 ≤ r ≤ s − 1. Keywords: polyhedral combinatorics, dominating set problem, set covering problem, circulant matrices, web graphs.
1 2
Partially supported by grant of ANPCyT-PICT 38036 (2005). Emails: sbianchi,nasini,[email protected]
1571-0653/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2010.05.150
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S. Bianchi et al. / Electronic Notes in Discrete Mathematics 36 (2010) 1185–1192
Introduction
Given a graph G = (V, E), a dominating set is a node subset D ⊂ V such that every node of V \ D is adjacent to at least one node in D. Given w ∈ RV , the Minimum Weight Dominating Set problem (DSP for short) consists in finding a dominating set D such that u∈D wu is minimum. The DSP is NP-hard for general graphs (see [8] page 190). If |V | = n and N [G] is the n × n 0, 1 matrix whose rows are the characteristic vectors of the closed neighborhood of nodes in V , the DSP can be formulated as the integer linear program min{wx : N [G]x ≥ 1, x ∈ {0, 1}V }. In general, given a 0, 1 m × n matrix M a cover of M is a 0, 1 vector x such that M x ≥ 1. Given w ∈ RV , the Minimum Weight Set Covering problem (SCP for short) consists in solving the integer linear program min{wx : M x ≥ 1, x ∈ {0, 1}n }. Clearly, the DSP can be viewed as the SCP on a particular set of instances. When working on SCP we only consider matrices without zero columns and dominating rows. We denote by D(G) the dominating set polyhedron (of G) defined by D(G) = conv({x ∈ {0, 1}V : N [G]x ≥ 1}), and by Q(M ) the set covering polyhedron (of M ) defined as Q(M ) = conv({x ∈ {0, 1}V : M x ≥ 1}). In [6], Bouchakour et al. obtain the complete description of the D(G) by linear inequalities when G is a cycle. The authors also show that the separation problem for this system of inequalities can be solved in polynomial time, concluding that the DSP is polynomial on cycles. It is not hard to see that if G is a web graph Wnk with node set V = {1, . . . , n} and {i, j} ⊂ V an edge if and only if 1 ≤ |i − j| ≤ k(mod n) then N [G] is the matrix Cn2k+1 , i.e. the n × n 0, 1 circulant matrix with 2k + 1 consecutive ones per row. In fact, as cycles are web graphs Wn1 , it follows that Bouchakour et al. in [6] obtain the complete description of Q(Cn3 ) and prove the polynomiality of the SCP on this family of circulant matrices. Actually, for any fixed k the SCP on Cnk is a polynomial problem. Indeed it can be derived (see [4]) from the behavior of the lift and project disjunctive operator defined by Balas et al [3]. In this work we obtain the complete description of the set covering polyhedron and the polynomiality of the SCP for a family of matrices Cnk with a k non bounded value of k. More specifically, we prove that Csk+r with s = 2, 3 and 0 ≤ r ≤ s − 1 are polynomial instances of the SCP and therefor, the DSP is polynomial for the corresponding family of web graphs.
S. Bianchi et al. / Electronic Notes in Discrete Mathematics 36 (2010) 1185–1192
2
1187
Previous results
Several authors have studied the SCP on circulant matrices, we can mention [1], [2] and [7], among others. We briefly introduce the results we need in order to present the conjecture which motivates this work. By nonboolean inequality we mean any inequality that cannot be written with 0, 1 coefficients. It is known [9] that the rank constraint ni=1 xi ≥ nk is always valid for Q(Cnk ) and it defines a facet if and only if n is not a multiple of k. Given a 0, 1 matrix M , a minor obtained by contracting a set of columns is the submatrix we get after removing the columns as well as the resulting dominating rows. From now on, columns of Cnk will be indexed by Zn = {0, . . . , n − 1} and whenever we consider S ⊂ Zn , the addition between its elements is taken modulo n. In the next theorem we summarize the results on circulant minors presented by Cornu´ejols and Novick and further generalized by Aguilera. Theorem 2.1 [1,7] Let M be a minor of Cnk obtained by contracting a set of columns indexed by elements in N ⊂ Zn . Then, M is isomorphic to Cnk if and only if there exists a triplet of non negative integer numbers {d, n1 , n3 } such that (i) d = gcd(n − n , k − k ), n1 = k − k and dn1 ≤ k − 1, (ii) there exists n2 ∈ Z+ kn2 + (k + 1)n3 and
such that
gcd(n1 , n2 , n3 ) = d and
n1 n =
(iii) N can be partitioned into d subsets Nj such that, for each j = 1 . . . , d, 3 (a) |Nj | = n2 +n , d (b) i ∈ Nj if and only if |Nj ∩ {i − k, i − (k + 1)}| = 1 and (c) denoting Wj = {i ∈ Nj : i − (k + 1) ∈ Nj }, |Wj | = nd3 . Argiroffo and Bianchi obtain in [2] a family of facets defining inequalities for Q(Cnk ) associated with circular minors corresponding to d = 1 according to the notation in Theorem 2.1. This result was generalized for every circulant minor as it is stated in the following
Theorem 2.2 [5] Let Cnk be a minor of Cnk and for j = 1 . . . , d, let Wj be defined as in Theorem 2.1. If W = dj=1 Wj then n (1) 2xi + xi ≥ k i∈W i∈W /
is a valid inequality for
Q(Cnk )
with Chv´ atal-Gomory rank at most one. More-
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over, if n = 1(mod k ), the inequality (1) defines a facet.
It follows that every minor Cnk of Cnk has an associated valid inequality for Q(Cnk ) that we call the Cnk -inequality. Also, given ax ≥ α valid for Q(Cnk ) we say that it is a minor inequality if there exists a minor Cnk of Cnk such that ax ≥ α is a multiple of the Cnk -inequality. In this paper we address ourselves to the following conjecture: k Conjecture 2.3 Every nonboolean facet defining inequality n ) is ei n of nQ(C k ther the rank constraint or it is a Cn -inequality with k > k and n = 1(mod k ).
The results presented in this paper prove that the conjecture holds for k matrices Csk+r with s = 2, 3 and 0 ≤ r ≤ s − 1.
3
k Some facets of Q(Csk )
First, it is known that Q(Cnk ) with k ≥ 2 is full dimensional. Moreover, if ax ≥ α is a non trivial facet defining inequality of Q(Cnk ) with ai = 0 for some i, then ax ≥ α is one of the inequalities in the system Cnk x ≥ 1 (see [2]). From now on, ax ≥ α is a non boolean, non rank facet defining inequality of Q(Cnk ) with integer coefficients, a0 = min{ai : i ∈ Zn }, W = {i ∈ Zn : ai > a0 } and W = {i ∈ Zn : ai = a0 }. After Theorem 2.1 and 2.2 we can state the following observations Remark 3.1 ax ≥ α is a minor inequality of Q(Cnk ) if and only if the following conditions hold •
a0 ≥ 1, W = ∅ and ai = 2a0 for all i ∈ W ,
•
if n3 = |W |, there exists n1 and n2 non negative integers such that n1 n = n2 k + n3 (k + 1) and n1 ≤ k − 1,
2 +n3 ) a0 and α = n−(n k−n1
• •
If d = gcd(n1 , n2 , n3 ), there exists d disjoint subsets of Zn verifying items (a) and (b) in Theorem 2.1 (iii) and such that, if Wj for j = 1, . . . , d are defined as in item (c) then W = ∪dj=1 Wj .
In the following, given i ∈ Zn , we denote C i,k = {i, i+1, . . . , i+k−1} ⊂ Zn . Moreover, we use x to denote not only a cover of Q(Cnk ) but also the subset of Zn whose characteristic vector is x. In this way, for example, by |x| we denote the cardinality of the set x and by x \ {i} the subset x without the element i. Besides x˜ is a root of ax ≥ α if a˜ x = α.
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The following general properties, mostly based on the fact that Q(Cnk ) is full dimensional, are not hard to prove: Lemma 3.2 Given ax ≥ α; linea (i) a0 ≥ 1, W = ∅ and α ≥ a0 (τ + 1), where τ = (ii) For any i ∈ Zn there exists (a) a root x such that i ∈ x, (b) a root x such that i ∈ / x, (c) a root x such that |x ∩ C i,k | ≥ 2.
n k
.
(iii) For every minimum cover x, |x ∩ W | ≥ 1. Let x be a root of ax ≥ α such that {i, i + s} ⊂ x with s ≤ k − 1. Since x \ {i} ∪ {i + s − k} and x \ {i + s} ∪ {i + k} are covers of Cnk then aj ≥ ai when i + s − k ≤ j ≤ i and aj ≥ ai+s when i + s ≤ j ≤ i + k. This fact, together with Lemma 3.2 (ii)(c) allow us to prove that, for any i ∈ W , there exists s ≤ k − 1 such that i + s ∈ W . Since W = ∅, by applying iteratively this argument we have Lemma 3.3 For every i ∈ Zn , |C i,k ∩ W | ≥ 2. Finally, from previous results it can be proved that Lemma 3.4 For every i ∈ W , ai ≤ 2a0 . From now on, we focus on circulant matrices with n = sk and s ≥ 2. We consider k ≥ 4 since the case k = 3 has already been studied in [6]. Moreover, for k = 4 we take s ≥ 3 since Q(C84 ) is described by boolean inequalities k (see [7]). Given i = 1, . . . , k, we denote by xi,k the minimum cover of Csk , {i + rk, r = 0, . . . , s − 1} ⊂ Zsk . k ) with α = We will prove that every facet defining inequality of Q(Csk 0 (s + 1)a is a minor inequality. Then, in what remains of this section we k consider facet defining inequality of Q(Csk ) with α = (s + 1)a0 . Clearly, every root x verifies |x| ≤ s + 1 and |x| = s + 1 if and only if x ∩ W = ∅. Therefore, for each i ∈ W , i belongs to only one root that is just xi,k . Moreover, Lemma 3.5 For every i ∈ Zsk , |xi,k ∩ W | = 1. The proof is based on the fact that if j = i+rk ∈ W for some 0 ≤ r ≤ s−1, the inequality should have the same roots as the hyperplane xi − xj = 0, a contradiction. All the previous results allow us to conclude: Theorem 3.6 For every i ∈ W , ai = 2a0 .
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So we have proved that every facet defining inequality ax ≥ (s + 1)a0 can be written as 2xi + xi ≥ s + 1 i∈W /
i∈W
where W ⊂ Zsk such that |W ∩ x | = 1 for every i = 1, . . . , k. In order to prove that this inequality is a minor inequality, we only need to prove that W verifies conditions in the remark 3.1. Denoting by ij the element in xj,k ∩ W , is not hard to see that, for every j, ij+1 = ij + nj2 k + (k + 1) where nj2 is integer and 0 ≤ nj2 ≤ s − 2. Defining n2 = kj=1 nj2 and considering Lemma 3.3, we obtain i,k
k ) is a Theorem 3.7 Every facet defining inequality ax ≥ (s + 1)a0 of Q(Csk minor inequality.
Moreover, we can prove the following Theorem 3.8 For every s ≥ 2, facet defining inequalities ax ≥ (s + 1)a0 of k Q(Csk ) can be separated in polynomial time. In the proof, we make use of the following ideas. Given a point x ˆ, deciding if it violates a minor inequality with α = (s + 1)a0 is equivalent to decide if there exists a subset W satisfying Lemma 3.5 such that i∈W xˆi < s + 1 − ni=1 xˆi . Defining Vi = {i, i+k, . . . , i+(s−1)k} for i = 1, . . . , k, V = (∪ki=1 Vi )∪{r, t}, A = ∪k−1 i=1 (Vi × Vi+1 ) ∪ ({r} × V1 ) ∪ (Vk × {t}), the separation problem can be reduced to the shortest rt-path problem on D = (V, A) with length xˆj for each arc (i, j) for j ∈ {1, 2, . . . , sk} and 0 for arcs of the form (j, t).
4
k k The SCP on C2k and C3k matrices
k In this section we first prove that the Conjecture 2.3 holds for Csk when s = 2 or s = 3. The key of our proof relies on the following k ) Theorem 4.1 Let W ⊂ Zsk correspond to a facet defining inequality of Q(Csk k with s = 2 or s = 3. Then, there exists a cover x of Csk such that x ⊂ W and |x| = s + 1.
Actually, if we prove the result above and apply Lemma 3.2 (i), we have that, for any facet defining inequality, α = (s + 1)a0 . This fact together with Theorem 3.7 imply that the Conjecture 2.3 holds in these cases. Moreover, Theorem 3.8 allows us to conclude that the cases s = 2, 3 correspond to polynomial instances of the SCP.
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The proof of Theorem 4.1 is based on the following ideas. Recalling that |W ∩ C i,k | ≥ 2 for every i ∈ Zsk , it is clear that W always contains a minimal k cover of Csk . When considering the case s = 2 it is not hard to see that every k minimal cover of C2k has cardinality 3. Hence, Theorem 4.1 is proved when s = 2. For the case s = 3, the cardinality of minimal covers is not always 4. From k now on, let C k be the set of all minimal covers of C3k and let W ⊂ Z3k be k associated with a facet defining inequality of Q(C3k ). By using Lemma 3.3, it is not hard to see that Theorem 4.1 holds when i,k |x ∩ W | = 1 for some i ∈ Z3k . Moreover, it can also be proved that, if |xi,k ∩ W | = 3 for some i ∈ Z3k the proof can be reduced to the case k−1 W ⊂ Z3(k−1) associated with a facet defining inequality of Q(C3(k−1) ). k Then, it remains to prove Theorem 4.1 for W ∈ W , where W k = {W ⊂ Z3k : |W ∩ C i,k | ≥ 2 ∧ |W ∩ xi,k | = 2 ∀i ∈ Z3k }. First, it can be proved that, the case W ∈ W k can always be reduced to the case W ∈ W k−1 when |W | = 2k < k4 (k − 2) and this condition holds for k ≥ 12. For the cases k ≤ 11, we use the following result: Lemma 4.2 Let W ∈ W k and suppose that every x ∈ C k such that |x| = 4 satisfies x ∩ W = ∅. Therefore, if i ∈ W and i + 1 ∈ W then {i + 2, i + k, i + 2k, i + k − 1, i + 2k + 1} ⊂ W . By exploring all the possible configurations of subsets W ⊂ Z3k , with k ≤ 11, satisfying that if i ∈ W and i + 1 ∈ W then {i + 2, i + k, i + 2k, i + k − 1, i + 2k + 1} ⊂ W , we get some i ∈ Z3k such that |W ∩ C i,k | ≤ 1, i.e. a contradiction on W ∈ W k . Hence, Theorem 4.1 can be proved by induction on k, obtaining that the k Conjecture 2.3 is true for Csk with s = 2, 3. Moreover, it is not hard to see that r(k+1) k k any circulant matrix Csk+r with r < s is a minor of Csk with k = k + s−r . k k k This allows to prove that Q(C2k+1 ), Q(C3k+1 ) and Q(C3k+2 ) are described by means of boolean facets and the rank constraint. In summary we can state that: k for s = 2, 3. In addition, Theorem 4.3 The Conjecture 2.3 holds for Csk k Csk+r with s = 2, 3 and 0 ≤ r ≤ s − 1 are polynomial instances of the SCP.
Then, given s = 2, 3 and 0 ≤ r ≤ s − 1 and k = 2t + 1, since the SCP on t can be also formulated as the DSP on W2st+s+r , we can state
k Csk+r
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t Corollary 4.4 The DSP is polynomial for the web graphs W2st+s+r with s = 2, 3 and 0 ≤ r ≤ s − 1.
The results obtained reinforce the conjecture on minor inequalities. We k believe that working on matrices Csk , with s ≥ 4 is a promising topic for further research. In particular, if we consider s = 4, we know that the only possible right hand side values of minor inequalities are just s + 1 and s + 2. However, considering the right hand side equal to s+2, the characterization of subsets W associated with these inequalities becomes much harder to handle.
References [1] Aguilera, N., Arithmetic relations in the set covering polyhedron of circulant clutters, Electronic Notes in Discrete Mathematics 30 (2008), 123–128. [2] Argiroffo, G. and S. Bianchi, On the set covering polyhedron of circulant matrices, Discrete Optimization 6 (2009), 162–173. [3] E. Balas, G. Cornujols and S. Ceria, A lift-and-project cutting plane algorithm for mixed 0-1 programs, Mathematical Programming 58 (1993), 295–324. [4] Bianchi, S. and M. Escalante, The N-rank of circulant matrices, Proceedings of VI ALIO/EURO Workshop on Applied Combinatorial Optimization (2008), ISBN: 978-950-29-1116-8. http://combinatoria.fceia.unr.edu.ar/pdf/NrankBE.pdf [5] Bianchi, S., G. Nasini and P. Tolomei, On the dominating set polytope of web graphs, Electronic Notes in Discrete Mathematics, (2009), 121–126. [6] 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. [7] Cornu´ejols, G. and B. Novick Ideal 0 − 1 Matrices, Journal of Combinatorial Theory B 60 (1994), 145–157. [8] Garey, M. and D. Johnson, Computers and Intractability. A Guide to the Theory of NP-Completeness, W.H. Freeman and Company, New York, 1979. [9] Sassano, A., On the facial structure of the set covering polytope, Mathematical Programming 44 (1989), 181–202.
The set covering problem on circulant matrices: polynomial instances and the relation with the dominating set problem on webs 1 S. Bianchi, G. Nasini, P. Tolomei.
2
Departamento de Matem´ atica Universidad Nacional de Rosario and CONICET Rosario, Argentina
Abstract The dominating set polyhedron of a web graph Wnk is the set covering polyhedron of a circulant matrix Cn2k+1 . In a previous work we generalize the results by Argiroffo and Bianchi on valid inequalities associated with every circulant minor of a circulant matrix and we conjecture that, for any k, the minor inequalities together with the boolean facets and the rank constraint are enough to describe the set covering polyhedron of Cnk . In this work we prove that the conjecture is true for the family of k with s = 2, 3 and 0 ≤ r ≤ s−1 and give a polynomial separation algorithm for Csk+r inequalities involved in the description. Thus, we prove the polynomiality of the set covering problem on these families. As a consequence we obtain the polynomiality of the minimum weight dominating set problem on webs of the form Wnt , when n = 2st + s + r with s = 2, 3 and 0 ≤ r ≤ s − 1. Keywords: polyhedral combinatorics, dominating set problem, set covering problem, circulant matrices, web graphs.
1 2
Partially supported by grant of ANPCyT-PICT 38036 (2005). Emails: sbianchi,nasini,[email protected]
1571-0653/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2010.05.150
1186
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S. Bianchi et al. / Electronic Notes in Discrete Mathematics 36 (2010) 1185–1192
Introduction
Given a graph G = (V, E), a dominating set is a node subset D ⊂ V such that every node of V \ D is adjacent to at least one node in D. Given w ∈ RV , the Minimum Weight Dominating Set problem (DSP for short) consists in finding a dominating set D such that u∈D wu is minimum. The DSP is NP-hard for general graphs (see [8] page 190). If |V | = n and N [G] is the n × n 0, 1 matrix whose rows are the characteristic vectors of the closed neighborhood of nodes in V , the DSP can be formulated as the integer linear program min{wx : N [G]x ≥ 1, x ∈ {0, 1}V }. In general, given a 0, 1 m × n matrix M a cover of M is a 0, 1 vector x such that M x ≥ 1. Given w ∈ RV , the Minimum Weight Set Covering problem (SCP for short) consists in solving the integer linear program min{wx : M x ≥ 1, x ∈ {0, 1}n }. Clearly, the DSP can be viewed as the SCP on a particular set of instances. When working on SCP we only consider matrices without zero columns and dominating rows. We denote by D(G) the dominating set polyhedron (of G) defined by D(G) = conv({x ∈ {0, 1}V : N [G]x ≥ 1}), and by Q(M ) the set covering polyhedron (of M ) defined as Q(M ) = conv({x ∈ {0, 1}V : M x ≥ 1}). In [6], Bouchakour et al. obtain the complete description of the D(G) by linear inequalities when G is a cycle. The authors also show that the separation problem for this system of inequalities can be solved in polynomial time, concluding that the DSP is polynomial on cycles. It is not hard to see that if G is a web graph Wnk with node set V = {1, . . . , n} and {i, j} ⊂ V an edge if and only if 1 ≤ |i − j| ≤ k(mod n) then N [G] is the matrix Cn2k+1 , i.e. the n × n 0, 1 circulant matrix with 2k + 1 consecutive ones per row. In fact, as cycles are web graphs Wn1 , it follows that Bouchakour et al. in [6] obtain the complete description of Q(Cn3 ) and prove the polynomiality of the SCP on this family of circulant matrices. Actually, for any fixed k the SCP on Cnk is a polynomial problem. Indeed it can be derived (see [4]) from the behavior of the lift and project disjunctive operator defined by Balas et al [3]. In this work we obtain the complete description of the set covering polyhedron and the polynomiality of the SCP for a family of matrices Cnk with a k non bounded value of k. More specifically, we prove that Csk+r with s = 2, 3 and 0 ≤ r ≤ s − 1 are polynomial instances of the SCP and therefor, the DSP is polynomial for the corresponding family of web graphs.
S. Bianchi et al. / Electronic Notes in Discrete Mathematics 36 (2010) 1185–1192
2
1187
Previous results
Several authors have studied the SCP on circulant matrices, we can mention [1], [2] and [7], among others. We briefly introduce the results we need in order to present the conjecture which motivates this work. By nonboolean inequality we mean any inequality that cannot be written with 0, 1 coefficients. It is known [9] that the rank constraint ni=1 xi ≥ nk is always valid for Q(Cnk ) and it defines a facet if and only if n is not a multiple of k. Given a 0, 1 matrix M , a minor obtained by contracting a set of columns is the submatrix we get after removing the columns as well as the resulting dominating rows. From now on, columns of Cnk will be indexed by Zn = {0, . . . , n − 1} and whenever we consider S ⊂ Zn , the addition between its elements is taken modulo n. In the next theorem we summarize the results on circulant minors presented by Cornu´ejols and Novick and further generalized by Aguilera. Theorem 2.1 [1,7] Let M be a minor of Cnk obtained by contracting a set of columns indexed by elements in N ⊂ Zn . Then, M is isomorphic to Cnk if and only if there exists a triplet of non negative integer numbers {d, n1 , n3 } such that (i) d = gcd(n − n , k − k ), n1 = k − k and dn1 ≤ k − 1, (ii) there exists n2 ∈ Z+ kn2 + (k + 1)n3 and
such that
gcd(n1 , n2 , n3 ) = d and
n1 n =
(iii) N can be partitioned into d subsets Nj such that, for each j = 1 . . . , d, 3 (a) |Nj | = n2 +n , d (b) i ∈ Nj if and only if |Nj ∩ {i − k, i − (k + 1)}| = 1 and (c) denoting Wj = {i ∈ Nj : i − (k + 1) ∈ Nj }, |Wj | = nd3 . Argiroffo and Bianchi obtain in [2] a family of facets defining inequalities for Q(Cnk ) associated with circular minors corresponding to d = 1 according to the notation in Theorem 2.1. This result was generalized for every circulant minor as it is stated in the following
Theorem 2.2 [5] Let Cnk be a minor of Cnk and for j = 1 . . . , d, let Wj be defined as in Theorem 2.1. If W = dj=1 Wj then n (1) 2xi + xi ≥ k i∈W i∈W /
is a valid inequality for
Q(Cnk )
with Chv´ atal-Gomory rank at most one. More-
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over, if n = 1(mod k ), the inequality (1) defines a facet.
It follows that every minor Cnk of Cnk has an associated valid inequality for Q(Cnk ) that we call the Cnk -inequality. Also, given ax ≥ α valid for Q(Cnk ) we say that it is a minor inequality if there exists a minor Cnk of Cnk such that ax ≥ α is a multiple of the Cnk -inequality. In this paper we address ourselves to the following conjecture: k Conjecture 2.3 Every nonboolean facet defining inequality n ) is ei n of nQ(C k ther the rank constraint or it is a Cn -inequality with k > k and n = 1(mod k ).
The results presented in this paper prove that the conjecture holds for k matrices Csk+r with s = 2, 3 and 0 ≤ r ≤ s − 1.
3
k Some facets of Q(Csk )
First, it is known that Q(Cnk ) with k ≥ 2 is full dimensional. Moreover, if ax ≥ α is a non trivial facet defining inequality of Q(Cnk ) with ai = 0 for some i, then ax ≥ α is one of the inequalities in the system Cnk x ≥ 1 (see [2]). From now on, ax ≥ α is a non boolean, non rank facet defining inequality of Q(Cnk ) with integer coefficients, a0 = min{ai : i ∈ Zn }, W = {i ∈ Zn : ai > a0 } and W = {i ∈ Zn : ai = a0 }. After Theorem 2.1 and 2.2 we can state the following observations Remark 3.1 ax ≥ α is a minor inequality of Q(Cnk ) if and only if the following conditions hold •
a0 ≥ 1, W = ∅ and ai = 2a0 for all i ∈ W ,
•
if n3 = |W |, there exists n1 and n2 non negative integers such that n1 n = n2 k + n3 (k + 1) and n1 ≤ k − 1,
2 +n3 ) a0 and α = n−(n k−n1
• •
If d = gcd(n1 , n2 , n3 ), there exists d disjoint subsets of Zn verifying items (a) and (b) in Theorem 2.1 (iii) and such that, if Wj for j = 1, . . . , d are defined as in item (c) then W = ∪dj=1 Wj .
In the following, given i ∈ Zn , we denote C i,k = {i, i+1, . . . , i+k−1} ⊂ Zn . Moreover, we use x to denote not only a cover of Q(Cnk ) but also the subset of Zn whose characteristic vector is x. In this way, for example, by |x| we denote the cardinality of the set x and by x \ {i} the subset x without the element i. Besides x˜ is a root of ax ≥ α if a˜ x = α.
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The following general properties, mostly based on the fact that Q(Cnk ) is full dimensional, are not hard to prove: Lemma 3.2 Given ax ≥ α; linea (i) a0 ≥ 1, W = ∅ and α ≥ a0 (τ + 1), where τ = (ii) For any i ∈ Zn there exists (a) a root x such that i ∈ x, (b) a root x such that i ∈ / x, (c) a root x such that |x ∩ C i,k | ≥ 2.
n k
.
(iii) For every minimum cover x, |x ∩ W | ≥ 1. Let x be a root of ax ≥ α such that {i, i + s} ⊂ x with s ≤ k − 1. Since x \ {i} ∪ {i + s − k} and x \ {i + s} ∪ {i + k} are covers of Cnk then aj ≥ ai when i + s − k ≤ j ≤ i and aj ≥ ai+s when i + s ≤ j ≤ i + k. This fact, together with Lemma 3.2 (ii)(c) allow us to prove that, for any i ∈ W , there exists s ≤ k − 1 such that i + s ∈ W . Since W = ∅, by applying iteratively this argument we have Lemma 3.3 For every i ∈ Zn , |C i,k ∩ W | ≥ 2. Finally, from previous results it can be proved that Lemma 3.4 For every i ∈ W , ai ≤ 2a0 . From now on, we focus on circulant matrices with n = sk and s ≥ 2. We consider k ≥ 4 since the case k = 3 has already been studied in [6]. Moreover, for k = 4 we take s ≥ 3 since Q(C84 ) is described by boolean inequalities k (see [7]). Given i = 1, . . . , k, we denote by xi,k the minimum cover of Csk , {i + rk, r = 0, . . . , s − 1} ⊂ Zsk . k ) with α = We will prove that every facet defining inequality of Q(Csk 0 (s + 1)a is a minor inequality. Then, in what remains of this section we k consider facet defining inequality of Q(Csk ) with α = (s + 1)a0 . Clearly, every root x verifies |x| ≤ s + 1 and |x| = s + 1 if and only if x ∩ W = ∅. Therefore, for each i ∈ W , i belongs to only one root that is just xi,k . Moreover, Lemma 3.5 For every i ∈ Zsk , |xi,k ∩ W | = 1. The proof is based on the fact that if j = i+rk ∈ W for some 0 ≤ r ≤ s−1, the inequality should have the same roots as the hyperplane xi − xj = 0, a contradiction. All the previous results allow us to conclude: Theorem 3.6 For every i ∈ W , ai = 2a0 .
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So we have proved that every facet defining inequality ax ≥ (s + 1)a0 can be written as 2xi + xi ≥ s + 1 i∈W /
i∈W
where W ⊂ Zsk such that |W ∩ x | = 1 for every i = 1, . . . , k. In order to prove that this inequality is a minor inequality, we only need to prove that W verifies conditions in the remark 3.1. Denoting by ij the element in xj,k ∩ W , is not hard to see that, for every j, ij+1 = ij + nj2 k + (k + 1) where nj2 is integer and 0 ≤ nj2 ≤ s − 2. Defining n2 = kj=1 nj2 and considering Lemma 3.3, we obtain i,k
k ) is a Theorem 3.7 Every facet defining inequality ax ≥ (s + 1)a0 of Q(Csk minor inequality.
Moreover, we can prove the following Theorem 3.8 For every s ≥ 2, facet defining inequalities ax ≥ (s + 1)a0 of k Q(Csk ) can be separated in polynomial time. In the proof, we make use of the following ideas. Given a point x ˆ, deciding if it violates a minor inequality with α = (s + 1)a0 is equivalent to decide if there exists a subset W satisfying Lemma 3.5 such that i∈W xˆi < s + 1 − ni=1 xˆi . Defining Vi = {i, i+k, . . . , i+(s−1)k} for i = 1, . . . , k, V = (∪ki=1 Vi )∪{r, t}, A = ∪k−1 i=1 (Vi × Vi+1 ) ∪ ({r} × V1 ) ∪ (Vk × {t}), the separation problem can be reduced to the shortest rt-path problem on D = (V, A) with length xˆj for each arc (i, j) for j ∈ {1, 2, . . . , sk} and 0 for arcs of the form (j, t).
4
k k The SCP on C2k and C3k matrices
k In this section we first prove that the Conjecture 2.3 holds for Csk when s = 2 or s = 3. The key of our proof relies on the following k ) Theorem 4.1 Let W ⊂ Zsk correspond to a facet defining inequality of Q(Csk k with s = 2 or s = 3. Then, there exists a cover x of Csk such that x ⊂ W and |x| = s + 1.
Actually, if we prove the result above and apply Lemma 3.2 (i), we have that, for any facet defining inequality, α = (s + 1)a0 . This fact together with Theorem 3.7 imply that the Conjecture 2.3 holds in these cases. Moreover, Theorem 3.8 allows us to conclude that the cases s = 2, 3 correspond to polynomial instances of the SCP.
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The proof of Theorem 4.1 is based on the following ideas. Recalling that |W ∩ C i,k | ≥ 2 for every i ∈ Zsk , it is clear that W always contains a minimal k cover of Csk . When considering the case s = 2 it is not hard to see that every k minimal cover of C2k has cardinality 3. Hence, Theorem 4.1 is proved when s = 2. For the case s = 3, the cardinality of minimal covers is not always 4. From k now on, let C k be the set of all minimal covers of C3k and let W ⊂ Z3k be k associated with a facet defining inequality of Q(C3k ). By using Lemma 3.3, it is not hard to see that Theorem 4.1 holds when i,k |x ∩ W | = 1 for some i ∈ Z3k . Moreover, it can also be proved that, if |xi,k ∩ W | = 3 for some i ∈ Z3k the proof can be reduced to the case k−1 W ⊂ Z3(k−1) associated with a facet defining inequality of Q(C3(k−1) ). k Then, it remains to prove Theorem 4.1 for W ∈ W , where W k = {W ⊂ Z3k : |W ∩ C i,k | ≥ 2 ∧ |W ∩ xi,k | = 2 ∀i ∈ Z3k }. First, it can be proved that, the case W ∈ W k can always be reduced to the case W ∈ W k−1 when |W | = 2k < k4 (k − 2) and this condition holds for k ≥ 12. For the cases k ≤ 11, we use the following result: Lemma 4.2 Let W ∈ W k and suppose that every x ∈ C k such that |x| = 4 satisfies x ∩ W = ∅. Therefore, if i ∈ W and i + 1 ∈ W then {i + 2, i + k, i + 2k, i + k − 1, i + 2k + 1} ⊂ W . By exploring all the possible configurations of subsets W ⊂ Z3k , with k ≤ 11, satisfying that if i ∈ W and i + 1 ∈ W then {i + 2, i + k, i + 2k, i + k − 1, i + 2k + 1} ⊂ W , we get some i ∈ Z3k such that |W ∩ C i,k | ≤ 1, i.e. a contradiction on W ∈ W k . Hence, Theorem 4.1 can be proved by induction on k, obtaining that the k Conjecture 2.3 is true for Csk with s = 2, 3. Moreover, it is not hard to see that r(k+1) k k any circulant matrix Csk+r with r < s is a minor of Csk with k = k + s−r . k k k This allows to prove that Q(C2k+1 ), Q(C3k+1 ) and Q(C3k+2 ) are described by means of boolean facets and the rank constraint. In summary we can state that: k for s = 2, 3. In addition, Theorem 4.3 The Conjecture 2.3 holds for Csk k Csk+r with s = 2, 3 and 0 ≤ r ≤ s − 1 are polynomial instances of the SCP.
Then, given s = 2, 3 and 0 ≤ r ≤ s − 1 and k = 2t + 1, since the SCP on t can be also formulated as the DSP on W2st+s+r , we can state
k Csk+r
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t Corollary 4.4 The DSP is polynomial for the web graphs W2st+s+r with s = 2, 3 and 0 ≤ r ≤ s − 1.
The results obtained reinforce the conjecture on minor inequalities. We k believe that working on matrices Csk , with s ≥ 4 is a promising topic for further research. In particular, if we consider s = 4, we know that the only possible right hand side values of minor inequalities are just s + 1 and s + 2. However, considering the right hand side equal to s+2, the characterization of subsets W associated with these inequalities becomes much harder to handle.
References [1] Aguilera, N., Arithmetic relations in the set covering polyhedron of circulant clutters, Electronic Notes in Discrete Mathematics 30 (2008), 123–128. [2] Argiroffo, G. and S. Bianchi, On the set covering polyhedron of circulant matrices, Discrete Optimization 6 (2009), 162–173. [3] E. Balas, G. Cornujols and S. Ceria, A lift-and-project cutting plane algorithm for mixed 0-1 programs, Mathematical Programming 58 (1993), 295–324. [4] Bianchi, S. and M. Escalante, The N-rank of circulant matrices, Proceedings of VI ALIO/EURO Workshop on Applied Combinatorial Optimization (2008), ISBN: 978-950-29-1116-8. http://combinatoria.fceia.unr.edu.ar/pdf/NrankBE.pdf [5] Bianchi, S., G. Nasini and P. Tolomei, On the dominating set polytope of web graphs, Electronic Notes in Discrete Mathematics, (2009), 121–126. [6] 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. [7] Cornu´ejols, G. and B. Novick Ideal 0 − 1 Matrices, Journal of Combinatorial Theory B 60 (1994), 145–157. [8] Garey, M. and D. Johnson, Computers and Intractability. A Guide to the Theory of NP-Completeness, W.H. Freeman and Company, New York, 1979. [9] Sassano, A., On the facial structure of the set covering polytope, Mathematical Programming 44 (1989), 181–202.