Note on: N.E. Aguilera, M.S. Escalante, G.L. Nasini, "The disjunctive procedure and blocker duality"

Aguilera et al. [DiscreteAppl.Math. 121 (2002) 1–13] give a generalization of a theorem of Lehman through an extension P̄j of the disjunctive procedure defined by Balas, Ceria and Cornuéjols. This generalization can be formulated as (A) For every clutter C, the disjunctive index of its set covering polyhedron Q(C) coincides with the disjunctive index of the set covering polyhedron of its blocker, Q(b(C)). In Aguilera et al. [Discrete Appl. Math. 121 (2002) 1–3], (A) is indeed a corollary of the stronger result (B) P̄J ([P̄J (Q(C))]B)= [Q(C)]B . Motivated by the work of Gerards et al. [Math. Oper. Res. 28 (2003) 884–885] we propose a simpler proof of (B) as well as an alternative proof of (A), independent of (B). Both of them are based on the relationship between the “disjunctive relaxations” obtained by P̄j and the set covering polyhedra associated with some particular minors of C. © 2005 Elsevier B.V. All rights reserved.