Augmenting Packing Dynamic Programs to Handle (Many) Additional Budget Constraints

In a packing problem, we are given a collection $I$ of $n$ items, each one with a given profit. Our goal is to compute a maximum profit subset of these items that satisfies a given set of packing constraints which depend on the problem at hand. Several approximation algorithms for well-studied NP-hard packing problems are based on a reduction to an auxiliary (packing) problem which is then solved with a dynamic program (DP). Examples for this include approximation algorithms for Knapsack, Geometric Knapsack, Independent Set of Rectangles, and Maximum Throughput Scheduling.