Find minimal property set in neighborhood

You are given a list of properties, each Property(id: int, neighborhood: string, capacity: int). Given a targetNeighborhood (string) and groupSize (int), return the IDs of a combination of properties drawn only from targetNeighborhood such that:

• the total capacity is at least groupSize;
• among all feasible combinations, the number of selected properties is minimized;
• if multiple combinations have the same minimal count, choose one whose total capacity is minimal (i.e., as close to groupSize as possible above it);
• if no feasible combination exists, return [].

Specify and implement an optimal algorithm. Describe:

1. the approach and data structures,
2. time and space complexity,
3. how you handle ties deterministically (e.g., smallest total capacity, then lexicographically smallest ID list),
4. edge cases (e.g., duplicate capacities/IDs, very large inputs).

Example input: Properties = [ Property(1, "downtown", 5), Property(2, "downtown", 3), Property(3, "downtown", 1), Property(4, "uptown", 4), Property(5, "uptown", 2) ] targetNeighborhood = "downtown"

• groupSize = 6 -> one valid answer is [1, 3]
• groupSize = 3 -> [2]
• groupSize = 16 -> [] Note: With targetNeighborhood = "downtown" and groupSize = 10, no solution exists because the capacities of downtown properties sum to 9.