Sequential and Simultaneous Liftings of Minimal Cover Inequalities for Generalized Upper Bound Constrained Knapsack Polytopes

Abstract

A family of facets for the GUB (Generalized Upper Bound) constrained knapsack polytope that are obtainable through a lifting procedure is characterized. The sequential lifting procedure developed herein computes lifted coefficients of the variables in each GUB set simultaneously, in contrast with the usual sequential lifting procedure that lifts only one variable at a time. Moreover, this sequential lifting procedure can be implemented in polynomial time of complexity $O(nm),$ where $n$ is the number of variables and $m(\leq n)$ is the number of GUB sets. In addition, a characterization of the facets obtainable through a simultaneous lifting procedure is derived. This characterization enables us to deduce lower and upper bounds on the lifted coefficients. In particular, for the case of the ordinary knapsack polytope, a known lower bound on the coefficients of lifted facets derived from minimal covers was further tightened.