## Combined Line Chart: Algorithm Performance vs. Problem Size ### Overview The image presents two line charts side-by-side, comparing the performance of different algorithms (CIM-CAC, CIM-CFC, CIM-SFC) against varying problem sizes. The left chart displays "time steps to solution" on a logarithmic scale, while the right chart shows the "ratio to median TTS" (Time To Solution), also on a logarithmic scale. The x-axis for both charts represents "problem size," expressed as the square root of the actual problem size. ### Components/Axes **Left Chart:** * **Title:** time steps to solution * **Y-axis:** time steps to solution (logarithmic scale) * Axis markers: 10^3, 10^4, 10^5, 10^6, 10^7 * **X-axis:** problem size (square root) * Axis markers: √100, √200, √400, √500, √800, √1000 * **Legend (top-left):** * Blue: CIM-CAC * Green: CIM-CFC * Red: CIM-SFC * Shaded regions around each line represent some measure of variance or uncertainty. **Right Chart:** * **Title:** ratio to median TTS * **Y-axis:** ratio to median TTS (logarithmic scale) * Axis markers: 10^0, 10^1, 10^2 * **X-axis:** problem size (square root) * Axis markers: √100, √200, √400, √500, √800, √1000 * **Legend (top-left):** * Black (solid line): 75th percentile * Black (dashed line): 90th percentile ### Detailed Analysis **Left Chart (Time Steps to Solution):** * **CIM-CAC (Blue):** The line slopes upward, indicating that the time steps to solution increase with problem size. * √100: ~3000 * √200: ~20000 * √400: ~45000 * √500: ~60000 * √800: ~200000 * √1000: ~400000 * **CIM-CFC (Green):** The line slopes upward, indicating that the time steps to solution increase with problem size. * √100: ~2500 * √200: ~10000 * √400: ~30000 * √500: ~40000 * √800: ~100000 * √1000: ~250000 * **CIM-SFC (Red):** The line slopes upward, indicating that the time steps to solution increase with problem size. * √100: ~2000 * √200: ~8000 * √400: ~25000 * √500: ~35000 * √800: ~80000 * √1000: ~200000 **Right Chart (Ratio to Median TTS):** * **75th percentile (Black, solid):** The line is relatively flat, suggesting the 75th percentile ratio to median TTS remains relatively constant across problem sizes. * √100: ~1 * √200: ~1.5 * √400: ~1.5 * √500: ~1.3 * √800: ~1.5 * √1000: ~1.5 * **90th percentile (Black, dashed):** The line fluctuates more than the 75th percentile, but generally stays within a small range. * √100: ~2.5 * √200: ~2 * √400: ~2.5 * √500: ~3 * √800: ~2 * √1000: ~2.5 ### Key Observations * In the left chart, all three algorithms (CIM-CAC, CIM-CFC, CIM-SFC) exhibit an increase in "time steps to solution" as the problem size increases. * CIM-CAC consistently requires more time steps to solution than CIM-CFC and CIM-SFC. * In the right chart, the 75th and 90th percentile ratios to median TTS are relatively stable across different problem sizes. * The ratio to median TTS for CIM-SFC is higher than CIM-CAC and CIM-CFC at smaller problem sizes, but converges as the problem size increases. ### Interpretation The left chart demonstrates the expected behavior that larger problems require more computational steps to solve. The relative positioning of the lines suggests that CIM-CAC is generally less efficient in terms of time steps than CIM-CFC and CIM-SFC for the problem set being analyzed. The right chart provides insight into the variability of the algorithms' performance. The relatively stable 75th and 90th percentile lines indicate that the distribution of Time To Solution (TTS) is consistent across different problem sizes. The higher ratio to median TTS for CIM-SFC at smaller problem sizes suggests that it may have more variable performance in those scenarios, potentially due to factors like initialization or search space exploration. As the problem size increases, the algorithms' performance becomes more consistent relative to their median TTS.

## Charts: Performance Comparison of CIM Algorithms ### Overview The image presents two charts comparing the performance of three Constraint Integer Modeling (CIM) algorithms: CIM-CAC, CIM-CFC, and CIM-SFC. The left chart displays the time steps to solution as a function of problem size, while the right chart shows the ratio to the median Time To Solution (TTS) for the 75th and 90th percentiles. Both charts use a logarithmic scale for the y-axis. ### Components/Axes **Left Chart:** * **X-axis:** Problem Size (labeled with values 100, 200, √400/500, 800, 1000). The x-axis is logarithmic. * **Y-axis:** Time steps to solution (labeled with values 10^3, 10^4, 10^5, 10^6, 10^7). The y-axis is logarithmic. * **Data Series:** * CIM-CAC (Blue line with circles) * CIM-CFC (Green line with triangles) * CIM-SFC (Red line with diamonds) * Each line has a shaded region around it, representing a confidence interval or variance. The shading is light blue for CIM-CAC, light green for CIM-CFC, and light red for CIM-SFC. **Right Chart:** * **X-axis:** Problem Size (labeled with values 100, 200, √400/500, 800, 1000). The x-axis is logarithmic. * **Y-axis:** Ratio to median TTS (labeled with values 10^0, 10^1, 10^2). The y-axis is logarithmic. * **Data Series:** * 75th percentile (Black line with circles) * 90th percentile (Black line with dashed circles) * **Legend:** Located in the top-right corner, clearly labeling the data series. ### Detailed Analysis or Content Details **Left Chart:** * **CIM-CAC (Blue):** The line slopes upward, indicating that the time steps to solution increase with problem size. * At problem size 100: ~1500 time steps. * At problem size 200: ~2500 time steps. * At problem size 500: ~7000 time steps. * At problem size 800: ~15000 time steps. * At problem size 1000: ~30000 time steps. * **CIM-CFC (Green):** The line also slopes upward, but generally remains below CIM-CAC. * At problem size 100: ~1000 time steps. * At problem size 200: ~1500 time steps. * At problem size 500: ~4000 time steps. * At problem size 800: ~8000 time steps. * At problem size 1000: ~15000 time steps. * **CIM-SFC (Red):** The line slopes upward, and is generally the lowest of the three algorithms. * At problem size 100: ~500 time steps. * At problem size 200: ~800 time steps. * At problem size 500: ~2000 time steps. * At problem size 800: ~4000 time steps. * At problem size 1000: ~8000 time steps. **Right Chart:** * **75th percentile (Black):** The line fluctuates, initially decreasing then increasing. * At problem size 100: ~10. * At problem size 200: ~5. * At problem size 500: ~7. * At problem size 800: ~10. * At problem size 1000: ~15. * **90th percentile (Black dashed):** The line also fluctuates, generally higher than the 75th percentile. * At problem size 100: ~20. * At problem size 200: ~10. * At problem size 500: ~15. * At problem size 800: ~20. * At problem size 1000: ~30. ### Key Observations * All three algorithms (CIM-CAC, CIM-CFC, CIM-SFC) exhibit an increasing trend in time steps to solution as the problem size increases (Left Chart). * CIM-SFC consistently requires the fewest time steps to solution across all problem sizes. * The ratio to median TTS increases with problem size for both the 75th and 90th percentiles (Right Chart), indicating greater variability in solution time for larger problems. * The 90th percentile consistently has a higher ratio to median TTS than the 75th percentile, as expected. ### Interpretation The data suggests that CIM-SFC is the most efficient algorithm among the three tested, consistently requiring fewer time steps to reach a solution. The logarithmic scales on both charts emphasize the exponential growth in computational effort as problem size increases. The right chart highlights that while the median solution time may be relatively stable, the tail of the distribution (75th and 90th percentiles) experiences increasing variability with larger problem sizes. This suggests that for critical applications, it may be necessary to consider the worst-case performance of these algorithms, and CIM-SFC offers the most consistent performance. The fluctuations in the ratio to median TTS could be due to the inherent stochasticity of the algorithms or variations in the problem instances used for testing. The √400/500 label on the x-axis suggests that the data points were taken at problem sizes of approximately 20 and 22.36, which is unusual and may indicate a specific experimental setup.

## [Chart Type]: Dual-Panel Performance Analysis of CIM Variants ### Overview The image contains two side-by-side line charts comparing the performance of three computational methods (CIM-CAC, CIM-CFC, CIM-SFC) across different problem sizes. The left chart plots the absolute time to solution, while the right chart plots the ratio of time to the median Time To Solution (TTS). Both charts use a logarithmic scale on the y-axis. ### Components/Axes **Left Chart:** * **Title:** None visible. * **X-axis:** Label: `problem size`. Ticks: `√100`, `√200`, `√400`, `√500`, `√800`, `√1000`. * **Y-axis:** Label: `time steps to solution`. Scale: Logarithmic, ranging from `10^3` to `10^7`. * **Legend:** Located in the top-left corner. Contains three entries: * `CIM-CAC` (Blue line with circular markers) * `CIM-CFC` (Green line with circular markers) * `CIM-SFC` (Red line with circular markers) * **Data Series:** Three lines, each with a shaded region representing confidence intervals or variance. **Right Chart:** * **Title:** None visible. * **X-axis:** Label: `problem size`. Ticks: `√100`, `√200`, `√400`, `√500`, `√800`, `√1000`. * **Y-axis:** Label: `ratio to median TTS`. Scale: Logarithmic, ranging from `10^0` to `10^2`. * **Legend:** Located in the top-left corner. Contains two entries: * `75th percentile` (Solid line) * `90th percentile` (Dashed line) * **Data Series:** Six lines total. The colors (Blue, Green, Red) correspond to the same methods (CIM-CAC, CIM-CFC, CIM-SFC) from the left chart. Each method has a solid line (75th percentile) and a dashed line (90th percentile). ### Detailed Analysis **Left Chart - Time Steps to Solution:** * **Trend Verification:** All three lines show a clear, consistent upward trend as problem size increases. The relationship appears roughly linear on this log-log plot, suggesting a power-law relationship between problem size and time steps. * **Data Points (Approximate):** * **CIM-CAC (Blue):** Starts at ~7x10^3 (√100), rises to ~1.2x10^7 (√1000). It is consistently the highest (slowest) method. * **CIM-CFC (Green):** Starts at ~3x10^3 (√100), rises to ~5x10^6 (√1000). It is consistently in the middle. * **CIM-SFC (Red):** Starts at ~2x10^3 (√100), rises to ~3x10^6 (√1000). It is consistently the lowest (fastest) method. * **Spatial Grounding:** The shaded confidence bands are widest for the blue line (CIM-CAC) and narrowest for the red line (CIM-SFC), suggesting greater variance in the performance of CIM-CAC. **Right Chart - Ratio to Median TTS:** * **Trend Verification:** The trends are more complex and non-monotonic. The 90th percentile lines (dashed) are generally above their corresponding 75th percentile lines (solid), as expected. * **Data Points & Cross-Referencing (Approximate):** * **CIM-CAC (Blue):** * 75th %ile (Solid Blue): Hovers between ~1.5 and ~2.5 across all problem sizes. * 90th %ile (Dashed Blue): Starts at ~3 (√100), dips to ~2 (√200), then rises to ~3.5 (√1000). * **CIM-CFC (Green):** * 75th %ile (Solid Green): Starts at ~2.5 (√100), dips to ~1.5 (√200/√400), then rises to ~2 (√1000). * 90th %ile (Dashed Green): Starts at ~4 (√100), dips to ~2.5 (√200), peaks at ~4 (√500), ends at ~3 (√1000). * **CIM-SFC (Red):** * 75th %ile (Solid Red): Starts at ~2 (√100), dips to ~1.5 (√200), then remains relatively flat around ~1.8. * 90th %ile (Dashed Red): Shows the most variation. Starts at ~10 (√100), dips to ~6 (√200), peaks at ~12 (√500), dips to ~3 (√800), and ends at ~11 (√1000). ### Key Observations 1. **Performance Hierarchy:** There is a clear and consistent performance hierarchy in absolute time (left chart): CIM-SFC > CIM-CFC > CIM-CAC. CIM-SFC is the fastest method by a significant margin. 2. **Scaling Behavior:** All methods scale similarly with problem size (similar slopes on the left chart), but with different constant factors. 3. **Variability:** The right chart reveals significant differences in performance variability. CIM-SFC (Red) shows extreme variability at the 90th percentile, with ratios spiking above 10x the median at several points (√100, √500, √1000). This indicates that while CIM-SFC is fastest on average, its worst-case performance can be much worse relative to its median. 4. **Percentile Spread:** The gap between the 75th and 90th percentile lines is largest for CIM-SFC, further highlighting its higher performance variance compared to the other two methods. ### Interpretation This data suggests a classic trade-off between average-case speed and performance consistency. **CIM-SFC is the optimized, high-performance variant**—it delivers the fastest solution times under typical conditions (as shown by the lowest line on the left chart). However, this speed comes at the cost of **predictability and robustness**. The high and volatile 90th percentile ratios (right chart, dashed red line) indicate that CIM-SFC is susceptible to occasional, severe slowdowns where it can take over 10 times longer than its median time. In contrast, **CIM-CAC is the slower but more predictable method**. Its performance bands are tighter, and its percentile ratios are lower and more stable. **CIM-CFC occupies a middle ground**. For a user, the choice depends on the application context. If consistent, reliable completion time is critical (e.g., real-time systems), CIM-CAC or CIM-CFC might be preferable despite their slower average speed. If minimizing average time is the primary goal and occasional long runs are acceptable, CIM-SFC is the best choice. The charts effectively visualize not just "which is faster," but the nuanced risk profile associated with each algorithm's performance.

## Line Chart and Scatter Plot: Performance Analysis of CIM Methods ### Overview The image contains two side-by-side graphs analyzing the performance of three computational methods (CIM-CAC, CIM-CFC, CIM-SFC) across varying problem sizes. The left graph shows time steps to solution, while the right graph compares performance variability via percentile ratios. ### Components/Axes **Left Graph (Line Chart):** - **X-axis**: Problem size (√100 to √1000, logarithmic scale) - **Y-axis**: Time steps to solution (10³ to 10⁷, logarithmic scale) - **Legend**: - Blue: CIM-CAC - Green: CIM-CFC - Red: CIM-SFC - **Shaded Areas**: Confidence intervals (10% uncertainty) around each line **Right Graph (Scatter Plot):** - **X-axis**: Problem size (√100 to √1000, logarithmic scale) - **Y-axis**: Ratio to median TTS (10⁰ to 10², logarithmic scale) - **Legend**: - Solid dots: 75th percentile - Dashed dots: 90th percentile ### Detailed Analysis **Left Graph Trends:** 1. **CIM-CAC (Blue)**: - Starts at ~10³.5 steps at √100 - Rises to ~10⁶.5 steps at √1000 - Steepest slope among all methods 2. **CIM-CFC (Green)**: - Begins at ~10³.2 steps at √100 - Reaches ~10⁶ steps at √1000 - Slightly flatter than CIM-CAC 3. **CIM-SFC (Red)**: - Starts at ~10³ steps at √100 - Ends at ~10⁵.5 steps at √1000 - Most gradual increase **Right Graph Trends:** 1. **75th Percentile (Solid Dots)**: - Ranges from 0.5 to 5 across problem sizes - Peaks at √400 (3.2) and √800 (4.1) 2. **90th Percentile (Dashed Dots)**: - Ranges from 1 to 10 across problem sizes - Sharp peak at √1000 (10) - Dips below 1 at √200 (0.6) ### Key Observations 1. **Scalability**: All methods show exponential time growth with problem size, but CIM-SFC scales best (10⁵.5 vs 10⁶.5 for CIM-CAC at √1000). 2. **Variability**: - 90th percentile values are consistently 2-3x higher than 75th percentile - Largest variability occurs at √1000 (ratio reaches 10) 3. **Confidence Intervals**: Shaded areas in left graph widen by 40% between √400 and √1000, indicating increasing uncertainty. ### Interpretation The data demonstrates that: - **CIM-SFC** offers the most efficient scaling but with higher variability (wider shaded area) - **CIM-CAC** provides consistent performance but at a computational cost 10x higher than CIM-SFC at maximum problem size - The 90th percentile ratio suggests significant outlier cases in large problems, potentially indicating edge cases or hardware limitations - Logarithmic scaling reveals exponential growth patterns that would be less apparent in linear plots The shaded confidence intervals and percentile ratios together highlight a tradeoff between average performance and worst-case scenarios across different computational approaches.