A knapsack intersection hierarchy

We introduce a knapsack intersection hierarchy for strengthening linear programming relaxations of packing integer programs. In level t of the hierarchy, all valid cuts are added for the integer hull of the intersection of all t-row relaxations. This model captures the maximum possible strength of t-row cuts, an approach often used by solvers for small t. We investigate the integrality gap of the strengthened formulations on the all-or-nothing flow problem in trees (also called unsplittable flow on trees).