## Diagram: Evolution of Step-by-step Construction Heuristics for KP ### Overview The diagram illustrates the evolution of heuristics for the Knapsack Problem (KP) over 1000 evolution steps. It tracks performance (g') against exploration decay (λ₀) and includes nodes representing key heuristic construction milestones. The graph combines a performance curve with annotated decision points and a shaded exploration/exploitation phase. ### Components/Axes - **Y-axis**: Performance (g') - logarithmic scale from 0 to 1000 - **X-axis**: Evolution Steps (0 to 1000) with λ₀ parameter (exploration decay) - **Legend**: - Blue rectangles: Expansion events (MCTS links) - Orange circles: MCTS nodes - Shaded orange area: Top 10 Heuristics performance range - **Key Elements**: - Nodes with timestamps (t=414, 580, 694, 892) - Action labels (e1, e2, s1) - Performance values (g'=40.17773, 40.13461, 40.17621, 40.17971) - Code snippets and descriptions at each node ### Detailed Analysis 1. **Initial State (t=0)**: - MCTS-Root node at performance g'=0 - λ₀=1 (full exploration) 2. **First Expansion (t=414)**: - Action: e1 - Performance: g'=40.17773 - Code: Contains initialization of MCTS links and node structures 3. **Second Expansion (t=580)**: - Action: e2 - Performance: g'=40.13461 (temporary dip) - Code: Implements node expansion with virtual root 4. **Third Expansion (t=694)**: - Action: s1 - Performance: g'=40.17621 - Code: Shows heuristic selection logic with pruning 5. **Final Expansion (t=892)**: - Action: e2 - Performance: g'=40.17971 (peak) - Code: Final heuristic refinement with MCTS integration 6. **Exploration/Exploitation Transition**: - Shaded area spans λ₀=0 to 1000 - Exploration dominates early steps (λ₀=1) - Exploitation phase begins after t=500 (λ₀=0.5) ### Key Observations - Performance plateaus around g'=40.17-40.18 after initial exploration - Temporary performance dip at t=580 suggests suboptimal action - MCTS nodes appear at critical decision points (t=0, 414, 694) - Top 10 Heuristics range remains consistent (g'=40.13-40.18) - Exploration decay follows exponential curve (λ₀=1 → 0) ### Interpretation The diagram demonstrates a hybrid approach combining MCTS-guided exploration with heuristic refinement. The initial exploration phase (λ₀=1) allows broad search space coverage, while subsequent exploitation focuses on promising heuristics. The performance plateau suggests convergence to near-optimal solutions, with MCTS nodes acting as checkpoints for heuristic validation. The temporary dip at t=580 highlights the importance of action selection in evolutionary algorithms. The final performance value (g'=40.17971) represents the best heuristic found after 892 steps, indicating effective balance between exploration and exploitation.