My first instinct was a greedy sort on one dimension and I ran with it way too long before realizing it breaks on the second constraint.
First, validate that no machine exceeds the rack capacity limits; if any does, return -1. Then, model the problem as a 2D bin packing problem and propose a greedy heuristic (e.g., sort machines by one resource and use a best-fit strategy) to minimize the number of racks. Discuss the NP-hardness and potential exact solutions for small N, but focus on practical heuristics for large N.
Pro tip: Acknowledge the NP-hard nature of 2D bin packing and emphasize that while exact solutions are impractical for large N, well-designed heuristics can achieve near-optimal results; this shows you understand real-world trade-offs.
Check if any machine's resource consumption exceeds the rack's capacity in either dimension. If so, return -1 immediately.
Recognize this as a 2D bin packing problem where machines are items and racks are bins with two capacity constraints. Note that it is NP-hard.
Choose an appropriate approach based on N: for small N, consider exact methods like branch-and-bound; for large N, use heuristics like First-Fit Decreasing or Best-Fit with sorting.
Design a greedy heuristic: sort machines by one resource (e.g., CPU) descending, then place each machine into the first rack that can accommodate both resources, possibly using a best-fit strategy to minimize waste.
Analyze time and space complexity of the chosen approach, and discuss trade-offs between optimality and efficiency, mentioning that heuristics may not always yield the minimum but are practical.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.