My first instinct was tree DP, which is technically correct but I fumbled explaining the complexity.
Model the problem as a knapsack variant on a forest, where each tree's feasible selections must respect parent-child ordering. Use dynamic programming over tree structures combined with knapsack capacity, and discuss trade-offs between exact DP and greedy heuristics.
Pro tip: Acknowledge that the problem is NP-hard (knapsack) and that in practice, miners use greedy fee-rate ordering with ancestor checks; showing awareness of real-world constraints (e.g., block size, mempool policies) demonstrates maturity.
Restate the problem: select a subset of transactions within a block-size budget, respecting parent-before-child ordering, to maximize total fee. Confirm that transactions form a forest (each has at most one parent) and that fees and sizes are positive.
Recognize this as a precedence-constrained knapsack problem on a forest, which is NP-hard. Discuss that without dependencies, it's the classic 0/1 knapsack; with dependencies, it's more complex.
Outline a dynamic programming solution: for each tree, compute a DP table mapping total size to maximum fee, ensuring that if a child is selected, its parent is also selected. Combine trees using knapsack merging.
Mention that exact DP may be too slow for large mempools; propose greedy heuristics like sorting by fee rate (fee/size) and including ancestors, or using branch-and-bound. Highlight trade-offs between optimality and speed.
State time and space complexity of the DP (e.g., O(n * B^2) for tree knapsack, where B is block size). Discuss edge cases: empty set, transactions with zero size, deep trees, and disconnected components.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.