On the 0/1 knapsack polytope

Given a set N of items and a capacity b 2 IN, and let N j be the set of items with weight j, 1 j b. The 0/1 knapsack polytope is the convex hull of all 0/1 vectors that satisfy the inequality b X j=1 X i2N j jx i b: In this paper we rst present a complete linear description of the 0/1 knapsack polytope for two special cases: (a) N j = ; for all 1 < j b b 2 c and (b) N j = ; for all 1 < j b b 3 c and N j = ; for all j b b 2 c + 1. It turns out that the inequalities that are needed for the complete description of these special polytopes are derived by means of some \reduction principle". This principle is then generalized to yield valid and in many cases facet deening inequalities for the general 0/1 knapsack polytope. The separation problem for this class of inequalities can be solved in pseudo polynomial time via dynamic programming techniques.