I knew the special cases right away, K=1 means sum everything, K >= number of tasks means just take the max.
Recognize this as the identical parallel machine scheduling problem (P||Cmax), which is NP-hard. Propose a binary search on the makespan combined with a feasibility check using a greedy algorithm like Longest Processing Time (LPT) or a bin-packing heuristic. Discuss the trade-off between optimality and efficiency, and mention that for small inputs exact methods like DP or branch-and-bound can be used.
Pro tip: Acknowledge that the problem is NP-hard and that practical solutions often use approximation algorithms with proven bounds (e.g., LPT guarantees a makespan at most 4/3 - 1/(3K) times optimal). This shows you understand both theory and real-world constraints.
Confirm that tasks are non-preemptive, workers are identical, and each task must be assigned to exactly one worker. Ask about input size and whether an exact or approximate solution is preferred.
State that this is the identical parallel machine scheduling problem (P||Cmax), which is NP-hard. Mention that no polynomial-time exact algorithm is known unless P=NP.
For exact solutions on small inputs, suggest binary search on makespan with a backtracking feasibility check, or dynamic programming. For large inputs, recommend approximation algorithms like LPT or binary search with a greedy feasibility check.
Compare exact vs. approximate methods in terms of time complexity and solution quality. Highlight that LPT runs in O(n log n) and guarantees a makespan within 4/3 of optimal, while binary search with greedy check may not guarantee optimality but is fast.
Explain how to implement the chosen algorithm, including data structures (e.g., priority queue for LPT) and handling of edge cases like K >= n or very large durations.
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The 4/3 bound is one of those things I half-remembered from school and had to reconstruct on the spot.
Start by formally defining the problem and the LPT algorithm, then prove its approximation ratio using a tight example, and finally analyze its time complexity. Emphasize the intuition behind why LPT works and discuss practical implications.
Pro tip: Mention that LPT is a 4/3-approximation and that this bound is tight, but also note that in practice it often performs much better; this shows you understand both theory and real-world performance.
Clearly state the makespan minimization problem on identical machines and describe the LPT algorithm: sort jobs in non-increasing order of processing time and assign each to the machine with the smallest current load.
Show that LPT achieves a makespan at most (4/3 - 1/(3m)) times the optimal, where m is the number of machines. Use a proof by contradiction or exchange argument, focusing on the last job assigned to the critical machine.
Provide a tight example that achieves the 4/3 bound (e.g., three machines with jobs of sizes 3,3,2,2,2) to show the bound cannot be improved.
Explain that sorting takes O(n log n) and assigning jobs using a min-heap takes O(n log m), so overall O(n log n + n log m) = O(n log n) since m ≤ n.
Mention that LPT is simple, fast, and often near-optimal in practice, but for exact solutions one might use dynamic programming or branch-and-bound for small instances.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.