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

V. LEONI, G. NASINI

DISCRETE APPLIED MATHEMATICS

Elsevier Science B.V.

Lugar: Amsterdam; Año: 2005 vol. 150 p. 251 - 255

0166-218X

Aguilera et al.[Discrete Appl. Math. 121 (2002) 1-13] give a generalization of a theorem of Lehman  through and extention P_j of thedisjunctive procedure defined by Balas, Ceria and Cornuéjols. This generalization can be formulated as:  (A) For every clutter C, the disjunctive index of its coveringpolyhedron Q(C) coincides with the disjunctive index of thecovering polyhedron of its blocker, Q(b(C)). In Aguilera et al.[Discrete Appl. Math. 121 (2002) 1-13], (A) is indeed a corollary of the stronger result (B) P_J(P_J(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 betweenthe``disjunctive relaxations´´ obtained by P_j and the set coveringpolyhedra associated with some particular minors of C.