Given up to N different menu items, each with a unit price, and a list of combo offers where each offer specifies quantities for some items and a total combo price, write an algorithm that, given the required quantities for each item, returns the minimum cost to satisfy the order without buying extra items. Describe and implement a dynamic programming or memoized search solution, stating the state definition, transitions, and base cases. Analyze time and space complexity in terms of N (item types), Q (maximum needed quantity per item), and S (number of offers). Follow-up: When N = 3, propose and implement an optimization that leverages a fixed 3D state (i, j, k) to reduce overhead; explain any preprocessing such as pruning dominated or irrelevant offers and capping offer counts to needs; provide the resulting complexity. Discuss additional practical optimizations such as skipping offers that exceed current needs or reordering states to enable bottom-up computation.