Facets of the Knapsack Polytope From Minimal Covers

In this paper we give easily computable best upper and lower bounds on the coefficients of facets of the knapsack polytope associated with minimal covers. For some coefficients the upper bounds are equal to the lower bounds; for the others the two bounds differ by 1. We give a necessary and sufficient condition for all lower bounds to be equal to the corresponding upper bounds, i.e. for the facet associated with the given minimal cover to be unique. Also, we define a partial order on the set of minimal covers and show that all facets associated with minimal covers can be obtained from weak covers; but that each facet obtainable from several ordered minimal covers is easiest to compute from the strongest one.Further, we characterize the class of all facets associated with minimal covers, and show that the facets obtainable by Padberg’s sequential lifting procedure are precisely those members of the class which have integer coefficients for a certain right-hand side. We then give a procedure for generating ...