## Line Graph: Percentage of Problems Solved vs. Number of Solutions per Problem ### Overview The graph illustrates the performance of five algorithms in solving problems as the number of solutions per problem (N) increases. The y-axis represents the percentage of problems solved (40–60%), while the x-axis shows N in powers of 2 (2² to 2⁶). Each algorithm is represented by a distinct line with markers and shaded confidence intervals. ### Components/Axes - **X-axis**: "N = number of solutions per problems" (logarithmic scale: 2², 2³, 2⁴, 2⁵, 2⁶). - **Y-axis**: "% Problems Solved" (linear scale: 40–60%). - **Legend**: Located in the bottom-right corner, mapping colors/markers to algorithms: - Green squares: Majority Vote - Red triangles: +OmegaPRM - Purple circles: +PRM800K - Blue circles: +Shepherd - Purple diamonds: +Shepherd (ours) ### Detailed Analysis 1. **Majority Vote (Green Squares)**: - Starts at ~39.5% (N=2²) and rises to ~55% (N=2⁶). - Shows a steady upward trend but remains the lowest-performing algorithm. 2. **+OmegaPRM (Red Triangles)**: - Consistently the highest-performing algorithm. - Begins at ~47.5% (N=2²) and reaches ~58% (N=2⁶). - Shaded area indicates moderate variability. 3. **+PRM800K (Purple Circles)**: - Starts at ~46% (N=2²) and climbs to ~57% (N=2⁶). - Outperforms +Shepherd (ours) but lags behind +OmegaPRM and +Shepherd. 4. **+Shepherd (Blue Circles)**: - Begins at ~44% (N=2²) and reaches ~57.5% (N=2⁶). - Shows the second-strongest performance, closely following +OmegaPRM. 5. **+Shepherd (ours) (Purple Diamonds)**: - Starts at ~39.5% (N=2²) and rises to ~55% (N=2⁶). - Overlaps with +PRM800K at higher N values but underperforms at lower N. ### Key Observations - **Performance Trends**: All algorithms improve as N increases, with diminishing returns at higher N values. - **Outliers**: +OmegaPRM maintains a consistent lead, while Majority Vote remains the weakest. - **Confidence Intervals**: Shaded regions suggest variability in performance, but trends are clear. ### Interpretation The data demonstrates that increasing the number of solutions per problem (N) enhances algorithmic performance across all methods. +OmegaPRM is the most effective, likely due to its robust solution generation or prioritization strategy. +Shepherd and +PRM800K show strong performance, with +Shepherd (ours) closely matching +PRM800K at higher N. Majority Vote’s lower performance highlights its limitations in complex problem-solving scenarios. The shaded areas indicate that while confidence intervals widen slightly at higher N, the overall trends remain stable. This suggests that scaling N is a viable strategy for improving problem-solving efficiency, with algorithmic design playing a critical role in determining the ceiling of performance.