Approximation Schemes for Packing Problems with ℓ p -norm Diversity Constraints.

We consider the problem of packing a set of items, each of them from a specific category, to obtain a solution set of high total profit, respecting the capacities, and exhibiting a good balance in terms of the categories represented by the chosen solution. Formally, this diversity constraint is captured by including a general family of ℓ p -norm constraints. These constraints make the problem considerably harder, and, in particular, the relaxation of the feasible region of the optimization problem we get is no longer convex. We show first that approximating this family of problems up to any extent is hard, and then we design two types of approximation schemes for them, depending on whether we are willing to violate the capacity or the ℓ p -norm constraints by a negligible amount. As a corollary, we get approximation schemes for Packing problems with constraints on the Hill diversity of the solution, which is a classical measure to quantify the diversity of a set of categorized elements.