O(n) Procedures for identifying maximal cliques and non-dominated extensions of consecutive minimal covers and alternates

Abstract

Cliques, Covers, Extensions, Alternates, Knapsack constraints, 0–1 programs, Tightening,

An enumerative procedure for identifying maximal covers

“…In this section we review some types of covers given in the literature, see [1,4,8,9] among many others. We also state some results concerning these types of covers; their proofs can be found in [4,8].…”

“…Section 3 presents an enumerative procedure for identifying all maximal covers from the set of covers implied by a knapsack constraint. Section 4 proves that the non-dominated extensions considered in [1] are maximal covers, and it presents an improvement on the procedure given in [1] for identifying them. Section 5 reports some computational results for the procedures proposed in Sections 3 and 4.…”

On using an automatic scheme for obtaining the convex hull defining inequalities of a Weismantel 0–1 knapsack constraint

“…By using our scheme for identifying maximal cliques, see Dietrich et al (1994), the following cliques axe obtained: Cj = {j, 6} for j = 1,2,..., 5.…”

“…Our scheme for identifying minimal covers and its extensions and alternates, see Dietrich et al (1994), gives the following covers:…”

“…The scope of this brief note is to show how the full set of inequalities presented by Weismantel (1997) to define the convex hull of the 0-1 L.F. Escudero, A. Garln and G. Pdrez points satisfying a 0-1 knapsack constraint (1.2), can be derived by using successively Our set of procedures for: (1) identifying maximal cliques and non-dominated extensions of consecutive minimal covers and alternates, see Escudero et al (1995); (2) coefficient increase based tightening cover induced inequalities, see Dietrich et al (1994) and Escudero and Mufioz (1998); and (c) coefficient reduction based tightening general 0-1 knapsack constraints.…”