“…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].…”
Section: Discussionmentioning
confidence: 99%
“…Finally, sinceb T ∈ Z n , 0 ≤ y < 1, and 0 ≤ z < 1, we must have y = (1− r k ′ )1, z = 0, and henceb T = ((r+1)1 T , r1 T , r1 T ) = r1 T + j∈W (e j ) T . Then the statement of the theorem follows from (4).…”
Section: Computing the First Chvátal Closurementioning
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.
“…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].…”
Section: Discussionmentioning
confidence: 99%
“…Finally, sinceb T ∈ Z n , 0 ≤ y < 1, and 0 ≤ z < 1, we must have y = (1− r k ′ )1, z = 0, and henceb T = ((r+1)1 T , r1 T , r1 T ) = r1 T + j∈W (e j ) T . Then the statement of the theorem follows from (4).…”
Section: Computing the First Chvátal Closurementioning
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.
“…The class of row family inequalities (rfi) was proposed in [3] as a counterpart to clique family inequalities in the set packing case [25]. We describe them at next, slightly modified to fit in our current notation.…”
“…The boolean facets include the inequalities x ≥ 0 and Ax ≥ 1 defining Q(A) which can be considered as the inequalities corresponding to (i) and (ii) for the set covering polyhedron. More recently, the class of row family inequalities was proposed in [3] as a counterpart of clique family inequalities in the set covering case. In fact, the same ideas used for P * (A) can be extended to show that Q * (A) is completely described by boolean facets and a particular subclass of row family inequalities associated with certain minors [5].…”
Section: Introductionmentioning
confidence: 99%
“…The Chvátal-rank of Q(A) has been addressed in several previous works. Any minor inequality has Chvátal-depth at most one, but it has been observed in [3] that this might not be the case for row family inequalities.…”
We study minor related row family inequalities for the set covering polyhedron of circular matrices. We address the issue of generating these inequalities via the Chvátal-Gomory procedure and establish a general upper bound for their Chvátal-rank. Moreover, we provide a construction to obtain facets with arbitrarily large coefficients and examples of facets having Chvátal-rank strictly larger than one.
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