## Chart: Runtime for solving OneMinMax ### Overview The image is a line chart comparing the runtime (measured in fitness evaluations) of different algorithms for solving the OneMinMax problem. The x-axis represents 'n', and the y-axis represents 'Fitness Evaluations' on a logarithmic scale. The chart compares the performance of GSEMO and NSGA-II with varying parameters. Error bars are present on each data point. ### Components/Axes * **Title:** Runtime for solving OneMinMax * **X-axis:** * Label: n * Scale: Linear * Markers: 100, 200, 300, 400 * **Y-axis:** * Label: Fitness Evaluations * Scale: Logarithmic (base 10) * Markers: 10^5, 10^6 * **Legend:** Located in the top-left corner. * GSEMO (Black) * NSGA-II, N=1.5(n+1) (Yellow) * NSGA-II, N=2(n+1) (Light Green) * NSGA-II, N=4(n+1) (Orange) * NSGA-II, N=8(n+1) (Dark Green) ### Detailed Analysis * **GSEMO (Black):** * Trend: The line slopes upward. * Data Points: * n=100: Fitness Evaluations ≈ 80,000 +/- 20,000 * n=200: Fitness Evaluations ≈ 250,000 +/- 50,000 * n=300: Fitness Evaluations ≈ 1,000,000 +/- 200,000 * n=400: Fitness Evaluations ≈ 1,800,000 +/- 200,000 * **NSGA-II, N=1.5(n+1) (Yellow):** * Trend: The line slopes upward. * Data Points: * n=100: Fitness Evaluations ≈ 40,000 +/- 10,000 * n=200: Fitness Evaluations ≈ 80,000 +/- 20,000 * n=300: Fitness Evaluations ≈ 400,000 +/- 100,000 * n=400: Fitness Evaluations ≈ 1,000,000 +/- 200,000 * **NSGA-II, N=2(n+1) (Light Green):** * Trend: The line slopes upward. * Data Points: * n=100: Fitness Evaluations ≈ 60,000 +/- 10,000 * n=200: Fitness Evaluations ≈ 150,000 +/- 30,000 * n=300: Fitness Evaluations ≈ 500,000 +/- 100,000 * n=400: Fitness Evaluations ≈ 1,200,000 +/- 200,000 * **NSGA-II, N=4(n+1) (Orange):** * Trend: The line slopes upward. * Data Points: * n=100: Fitness Evaluations ≈ 100,000 +/- 20,000 * n=200: Fitness Evaluations ≈ 300,000 +/- 50,000 * n=300: Fitness Evaluations ≈ 1,100,000 +/- 200,000 * n=400: Fitness Evaluations ≈ 1,600,000 +/- 200,000 * **NSGA-II, N=8(n+1) (Dark Green):** * Trend: The line slopes upward. * Data Points: * n=100: Fitness Evaluations ≈ 80,000 +/- 20,000 * n=200: Fitness Evaluations ≈ 400,000 +/- 100,000 * n=300: Fitness Evaluations ≈ 700,000 +/- 100,000 * n=400: Fitness Evaluations ≈ 2,500,000 +/- 500,000 ### Key Observations * All algorithms show an increase in fitness evaluations as 'n' increases. * NSGA-II with N=8(n+1) (Dark Green) generally requires the highest number of fitness evaluations. * NSGA-II with N=1.5(n+1) (Yellow) generally requires the lowest number of fitness evaluations. * The error bars indicate the variability in the runtime of each algorithm. ### Interpretation The chart illustrates the performance of different algorithms in solving the OneMinMax problem. The number of fitness evaluations required to solve the problem increases with 'n' for all algorithms. The choice of algorithm and its parameters (N in NSGA-II) significantly impacts the runtime. NSGA-II with smaller N values (e.g., 1.5(n+1)) appears to be more efficient in terms of fitness evaluations compared to larger N values (e.g., 8(n+1)). GSEMO's performance is in the middle range. The error bars suggest that the runtime can vary significantly for each algorithm, indicating the stochastic nature of these optimization processes.

## Line Chart: Runtime for solving OneMinMax ### Overview This chart displays the runtime, measured in Fitness Evaluations, for solving the OneMinMax problem using different algorithms and parameter settings. The x-axis represents the problem size 'n', and the y-axis represents the number of Fitness Evaluations on a logarithmic scale. Error bars are included for each data series, indicating the variability in runtime. ### Components/Axes * **Title:** "Runtime for solving OneMinMax" (positioned at the top-center) * **X-axis Label:** "n" (positioned at the bottom-center) * **Y-axis Label:** "Fitness Evaluations" (positioned vertically along the left side) - Logarithmic scale. * **Legend:** Located at the top-right corner of the chart. * GSEMO (Black) * NSGA-II (Orange) * NSGA-II, N=1.5(n+1) (Light Green) * NSGA-II, N=2(n+1) (Red) * NSGA-II, N=4(n+1) (Magenta) * NSGA-II, N=8(n+1) (Dark Green) ### Detailed Analysis The chart shows six data series, each representing a different algorithm or parameter configuration. All lines exhibit an upward trend, indicating that runtime increases with problem size 'n'. The error bars show the variance in runtime for each configuration. * **GSEMO (Black):** The line slopes upward consistently. * At n ≈ 100, Fitness Evaluations ≈ 2.5 x 10⁵, with an error bar extending from approximately 1.5 x 10⁵ to 3.5 x 10⁵. * At n ≈ 400, Fitness Evaluations ≈ 8.0 x 10⁵, with an error bar extending from approximately 6.0 x 10⁵ to 1.0 x 10⁶. * **NSGA-II (Orange):** The line slopes upward consistently. * At n ≈ 100, Fitness Evaluations ≈ 3.0 x 10⁵, with an error bar extending from approximately 2.0 x 10⁵ to 4.0 x 10⁵. * At n ≈ 400, Fitness Evaluations ≈ 9.0 x 10⁵, with an error bar extending from approximately 7.0 x 10⁵ to 1.1 x 10⁶. * **NSGA-II, N=1.5(n+1) (Light Green):** The line slopes upward consistently. * At n ≈ 100, Fitness Evaluations ≈ 1.5 x 10⁵, with an error bar extending from approximately 1.0 x 10⁵ to 2.0 x 10⁵. * At n ≈ 400, Fitness Evaluations ≈ 5.0 x 10⁵, with an error bar extending from approximately 4.0 x 10⁵ to 6.0 x 10⁵. * **NSGA-II, N=2(n+1) (Red):** The line slopes upward consistently. * At n ≈ 100, Fitness Evaluations ≈ 2.0 x 10⁵, with an error bar extending from approximately 1.5 x 10⁵ to 2.5 x 10⁵. * At n ≈ 400, Fitness Evaluations ≈ 7.0 x 10⁵, with an error bar extending from approximately 5.0 x 10⁵ to 9.0 x 10⁵. * **NSGA-II, N=4(n+1) (Magenta):** The line slopes upward consistently. * At n ≈ 100, Fitness Evaluations ≈ 3.0 x 10⁵, with an error bar extending from approximately 2.0 x 10⁵ to 4.0 x 10⁵. * At n ≈ 400, Fitness Evaluations ≈ 1.0 x 10⁶, with an error bar extending from approximately 8.0 x 10⁵ to 1.2 x 10⁶. * **NSGA-II, N=8(n+1) (Dark Green):** The line slopes upward consistently. * At n ≈ 100, Fitness Evaluations ≈ 1.0 x 10⁵, with an error bar extending from approximately 7.0 x 10⁴ to 1.3 x 10⁵. * At n ≈ 400, Fitness Evaluations ≈ 6.0 x 10⁵, with an error bar extending from approximately 5.0 x 10⁵ to 7.0 x 10⁵. ### Key Observations * The NSGA-II algorithm with N=1.5(n+1) consistently exhibits the lowest runtime across all tested values of 'n'. * The GSEMO algorithm generally performs worse than NSGA-II, with higher fitness evaluation counts. * As 'n' increases, the error bars tend to widen, suggesting greater variability in runtime for larger problem sizes. * NSGA-II with N=8(n+1) and N=4(n+1) show the highest runtimes at n=400. ### Interpretation The chart demonstrates the scalability of different algorithms for solving the OneMinMax problem. The results suggest that NSGA-II, particularly with the parameter setting N=1.5(n+1), is the most efficient algorithm for this problem, requiring the fewest fitness evaluations to reach a solution. The logarithmic scale of the y-axis highlights the exponential growth in runtime as the problem size increases. The error bars indicate that the runtime can vary significantly, even for the same algorithm and problem size, potentially due to factors such as random initialization or the inherent stochasticity of the algorithms. The widening error bars with increasing 'n' suggest that the runtime becomes more unpredictable as the problem becomes more complex. The data suggests a polynomial relationship between problem size and runtime for all algorithms tested. The choice of 'N' parameter in NSGA-II significantly impacts performance, with N=1.5(n+1) being optimal in this case.

## Line Chart: Runtime for solving OneMinMax ### Overview The image is a line chart comparing the computational runtime (measured in fitness evaluations) of five different algorithms or algorithm configurations for solving the "OneMinMax" problem. The runtime is plotted against the problem size, denoted by `n`. All data series show a clear increasing trend, with runtime growing as `n` increases. The chart uses a logarithmic scale for the y-axis. ### Components/Axes * **Title:** "Runtime for solving OneMinMax" (centered at the top). * **X-Axis:** * **Label:** `n` (centered below the axis). * **Scale:** Linear scale with major tick marks and labels at `100`, `200`, `300`, and `400`. * **Y-Axis:** * **Label:** "Fitness Evaluations" (rotated 90 degrees, left side). * **Scale:** Logarithmic scale (base 10). Major tick marks and labels are at `10^5` and `10^6`. Minor tick marks are present between these major values. * **Legend:** Located in the top-left corner of the plot area. It contains five entries, each with a colored line segment, a marker, and a text label. 1. **Label:** `GSEMO` | **Color/Marker:** Dark blue line with a plus (`+`) marker. 2. **Label:** `NSGA-II, N=1.5(n+1)` | **Color/Marker:** Yellow-orange line with a plus (`+`) marker. 3. **Label:** `NSGA-II, N=2(n+1)` | **Color/Marker:** Light green line with a plus (`+`) marker. 4. **Label:** `NSGA-II, N=4(n+1)` | **Color/Marker:** Orange-red line with a plus (`+`) marker. 5. **Label:** `NSGA-II, N=8(n+1)` | **Color/Marker:** Dark green line with a plus (`+`) marker. * **Data Series:** Five lines, each with error bars (vertical lines with caps) at the data points for `n=100, 200, 300, 400`. ### Detailed Analysis The following table reconstructs the approximate data points for each algorithm. Values are estimated from the logarithmic y-axis. The trend for all series is a roughly linear increase on this log-linear plot, indicating an exponential relationship between `n` and the number of fitness evaluations. | Algorithm (Legend Label) | Color | Approx. Fitness Evaluations at n=100 | Approx. Fitness Evaluations at n=200 | Approx. Fitness Evaluations at n=300 | Approx. Fitness Evaluations at n=400 | Visual Trend Description | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | **GSEMO** | Dark Blue | ~9 x 10^4 | ~4 x 10^5 | ~1.2 x 10^6 | ~2.5 x 10^6 | Steep, consistent upward slope. | | **NSGA-II, N=1.5(n+1)** | Yellow-Orange | ~5 x 10^4 | ~2 x 10^5 | ~5 x 10^5 | ~1 x 10^6 | Shallowest slope among the series. Lowest runtime at all points. | | **NSGA-II, N=2(n+1)** | Light Green | ~7 x 10^4 | ~3 x 10^5 | ~7 x 10^5 | ~1.2 x 10^6 | Slope is steeper than N=1.5 but less steep than others. | | **NSGA-II, N=4(n+1)** | Orange-Red | ~1 x 10^5 | ~6 x 10^5 | ~1.2 x 10^6 | ~2.5 x 10^6 | Very steep slope, similar to GSEMO. Intersects with GSEMO line near n=300. | | **NSGA-II, N=8(n+1)** | Dark Green | ~2 x 10^5 | ~9 x 10^5 | ~2 x 10^6 | ~4 x 10^6 | Steepest slope overall. Highest runtime at all points. | **Error Bars:** All data points have vertical error bars, indicating variability in the measured runtime. The length of the error bars appears relatively consistent across the series for a given `n`, though they are more pronounced on the logarithmic scale at higher values. ### Key Observations 1. **Clear Hierarchy:** There is a consistent performance hierarchy across all problem sizes (`n`). From fastest (lowest evaluations) to slowest: `NSGA-II, N=1.5(n+1)` < `NSGA-II, N=2(n+1)` < `GSEMO` ≈ `NSGA-II, N=4(n+1)` < `NSGA-II, N=8(n+1)`. 2. **Impact of N:** For the NSGA-II algorithm, increasing the population size parameter `N` (from 1.5(n+1) to 8(n+1)) results in a significant increase in runtime (fitness evaluations). 3. **GSEMO vs. NSGA-II:** The GSEMO algorithm's performance is comparable to NSGA-II with `N=4(n+1)`. Their lines are very close and intersect around `n=300`. 4. **Scaling Behavior:** All algorithms exhibit what appears to be exponential scaling with respect to `n`, as indicated by the roughly straight lines on the log-linear plot. The slope of the line (the exponent) varies between algorithms. ### Interpretation This chart demonstrates the computational cost scaling of different evolutionary algorithms on the OneMinMax benchmark problem. The key insight is the trade-off between algorithm configuration and runtime. * **Population Size Matters:** For NSGA-II, a larger population size (`N`) leads to dramatically higher computational cost. The `N=8(n+1)` variant is approximately an order of magnitude more expensive than the `N=1.5(n+1)` variant at `n=400`. This suggests that while larger populations might offer other benefits (like solution quality or diversity), they come at a steep runtime penalty for this problem. * **Algorithm Comparison:** GSEMO, a multi-objective evolutionary algorithm based on a steady-state genetic algorithm and a archive, shows a runtime profile similar to a moderately configured NSGA-II (`N=4(n+1)`). This provides a point of reference for comparing different algorithmic paradigms. * **Practical Implication:** For large problem sizes (`n`), the choice of algorithm and its parameters has a massive impact on feasibility. Using the most efficient configuration shown (`NSGA-II, N=1.5(n+1)`) could mean solving a problem with `n=400` in roughly 1 million evaluations, whereas the least efficient (`NSGA-II, N=8(n+1)`) would require around 4 million evaluations—a fourfold increase in computational resources. * **Underlying Pattern:** The linear trend on the log plot suggests the runtime `R` can be modeled as `R ≈ a * b^n`, where `b` is a base greater than 1. The different slopes indicate different values of `b` for each algorithm configuration, quantifying their relative scalability.

## Line Chart: Runtime for solving OneMinMax ### Overview The chart compares the runtime (measured in fitness evaluations) of five optimization algorithms (GSEMO and NSGA-II variants) as a function of problem size `n` (ranging from 100 to 400). The y-axis uses a logarithmic scale (10⁵ to 10⁷) to accommodate exponential growth trends. ### Components/Axes - **X-axis**: `n` (problem size), labeled with ticks at 100, 200, 300, 400. - **Y-axis**: "Fitness Evaluations" (log scale, 10⁵ to 10⁷). - **Legend**: Located in the top-left corner, with five entries: - **GSEMO** (dark blue, cross markers) - **NSGA-II, N=1.5(n+1)** (orange, plus markers) - **NSGA-II, N=2(n+1)** (green, plus markers) - **NSGA-II, N=4(n+1)** (red, plus markers) - **NSGA-II, N=8(n+1)** (dark green, plus markers) ### Detailed Analysis 1. **GSEMO** (dark blue): - At `n=100`: ~8×10⁴ evaluations. - At `n=200`: ~3×10⁵ evaluations. - At `n=300`: ~1.2×10⁶ evaluations. - At `n=400`: ~2.5×10⁶ evaluations. - **Trend**: Linear increase on the log scale (exponential growth in raw values). 2. **NSGA-II, N=1.5(n+1)** (orange): - At `n=100`: ~5×10⁴ evaluations. - At `n=200`: ~1.5×10⁵ evaluations. - At `n=300`: ~5×10⁵ evaluations. - At `n=400`: ~1.2×10⁶ evaluations. - **Trend**: Slightly steeper than GSEMO but less than higher-N NSGA-II variants. 3. **NSGA-II, N=2(n+1)** (green): - At `n=100`: ~7×10⁴ evaluations. - At `n=200`: ~2×10⁵ evaluations. - At `n=300`: ~8×10⁵ evaluations. - At `n=400`: ~2×10⁶ evaluations. - **Trend**: Intermediate growth rate between GSEMO and higher-N NSGA-II. 4. **NSGA-II, N=4(n+1)** (red): - At `n=100`: ~1×10⁵ evaluations. - At `n=200`: ~5×10⁵ evaluations. - At `n=300`: ~1.5×10⁶ evaluations. - At `n=400`: ~3×10⁶ evaluations. - **Trend**: Steeper than GSEMO and N=2 NSGA-II. 5. **NSGA-II, N=8(n+1)** (dark green): - At `n=100`: ~1.5×10⁵ evaluations. - At `n=200`: ~1×10⁶ evaluations. - At `n=300`: ~3×10⁶ evaluations. - At `n=400`: ~6×10⁶ evaluations. - **Trend**: Steepest slope, indicating the highest computational cost. ### Key Observations - **Scalability**: All algorithms exhibit exponential growth in runtime as `n` increases, but NSGA-II with larger `N` values scales poorly. - **GSEMO vs. NSGA-II**: GSEMO consistently outperforms NSGA-II with `N=1.5(n+1)` but is outperformed by NSGA-II with `N≥2(n+1)`. - **Error Bars**: Small vertical error bars (e.g., ±10⁴ at `n=100`) suggest low variability in evaluations across runs. ### Interpretation The data demonstrates that **NSGA-II with larger `N` values (e.g., 8(n+1))** has the highest runtime, likely due to increased population diversity or computational overhead. **GSEMO** offers a middle-ground performance, suggesting it may be more efficient than NSGA-II for moderate `N` values but less scalable than NSGA-II with smaller `N`. The logarithmic y-axis highlights the exponential nature of the problem, emphasizing the need for algorithm selection based on problem size and resource constraints. Notably, NSGA-II with `N=8(n+1)` doubles its runtime between `n=200` and `n=400`, indicating severe scalability issues for large-scale problems.