Valid inequalities for 0-1 knapsack polytopes often prove useful when tackling hard 0-1 Linear Programming problems. To use such inequalities effectively, one needs separation algorithms for them, i.e., routines for detecting when they are violated. We show that the separation problems for the so-called extended cover and weight inequalities can be solved exactly in O(nb) time and O((n + a_max)b) time, respectively, where n is the number of items, b is the knapsack capacity and a_max is the largest item weight. We also present fast and effective separation heuristics for the extended cover and lifted cover inequalities. Finally, we present a new exact separation algorithm for the 0-1 knapsack polytope itself, which is faster than existing methods. Extensive computational results are also given.
K. Kaparis & A.N. Letchford (2010) Separation algorithms for 0-1 knapsack polytopes. Math. Program., 124(1-2), 69-91.