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

  • Authors:

  • V. Leoni;G. Nasini

  • Affiliations:

  • Depto. de Matemática, Facultad de Ciencias Exactas, Ingeniería y Agrimensura, Rosario, Argentina and Fundación Antorchas;Depto. de Matemática, Facultad de Ciencias Exactas, Ingeniería y Agrimensura, Rosario, Argentina

  • Venue:
  • Discrete Applied Mathematics - Special issue: Max-algebra
  • Year:
  • 2005

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Abstract

Aguilera et al. [Discrete Appl. Math. 121 (2002) 1-13] give a generalization of a theorem of Lehman through an extension P-j 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 2(C) coincides with the disjunctive index of the set covering polyhedron of its blocker, 2(b(C)).In Aguilera et al. [Discrete Appl. Math. 121 (2002) 1-3], (A) is indeed a corollary of the stronger result (B) P-j ([P-j(2(C))]B) = [2(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 P-j and the set covering polyhedra associated with some particular minors of C.