## 5 The Labeling Algorithm for Four-Label Enumeration Stable Extension
### Four-Label Algorithm Description
All experiments run on Ubuntu 22.04 LTS (64 bit) with Intel Core i7-11700 @ 2.50 GHz CPU and 32 GB of RAM. The three methods above were tested on the ICCMA 2019, ICCMA 2021, and ICCMA 2023 datasets from the International Competition on Computational Models of Argument (ICCMA) ([[https://argumentationcompetition.org/2023/tracks.html](https://argumentationcompetition.org/2023/tracks.html)]([https://argumentationcompetition.org/2023/tracks.html](https://argumentationcompetition.org/2023/tracks.html)), accessed on 3 August 2024). To be consistent with the testing standards of ICCMA 2023, we set the time limit as 1200 s for stable extensions of every AF instance. The result of each instance was presented as success (if the problem was solved) or failure (including crashes or timeouts). The source code has been made available ([[https://github.com/bear-eng/label-algorithm.git](https://github.com/bear-eng/label-algorithm.git)]([https://github.com/bear-eng/label-algorithm.git](https://github.com/bear-eng/label-algorithm.git)), accessed on 3 August 2024). In the analysis of the experiment results, this paper primarily considers two metrics: the number of solved instances and PAR-2. PAR-2 represents an absolute performance measure, where the run time for problems that could not be solved within the allotted time is set to twice the cutoff time, while the run time for other instances is the actual run time. ### Experiment Results and Analysis

Table 1 presents a comparative analysis of the performance of the two-label algorithm (ArgTools_2 lab), the three-label algorithm (ArgTools_3 lab), and the four-label algorithm (ArgTools_4 lab) on the benchmarks of the main track of ICCMA 2019, ICCMA 2021, and ICCMA 2023. From Table 1, it can be concluded that the four-label algorithm outperforms the other two algorithms across all three benchmarks. In other words, this method improves algorithm efficiency by introducing new labels to differentiate between the types of arguments and by refining the labeling rules based on the relationships among arguments. Next, we conducts three sets of ablation experiments to analyze the reasons for the efficiency improvements and demonstrates that the method is also effective in other solvers. Overall, the four-label algorithm has improved efficiency by 17.4% compared to the two-label algorithm and by 13.5% compared to the three-label algorithm. In the ablation experiments of the preprocessing strategy and the four-label transition strategy, we configured these two strategies separately in the ArgTools solver and then ran the configured solvers on the main track of the ICCMA 2023 benchmark. Additionally, to better illustrate the algorithm's performance on each instance and on each instance family, we have created cactus plots for individual instances and scatter plots for instance families. Note that instance family refers to a set of instances with similar names. Based on the solution results in Table 2, there are three main families with difficult-to-solve problems: SccGenerator (25) ([[https://argumentationcompetition.org/2017/benchmark_selection_iccma2017.pdf](https://argumentationcompetition.org/2017/benchmark_selection_iccma2017.pdf)]([https://argumentationcompetition.org/2017/benchmark_selection_iccma2017.pdf](https://argumentationcompetition.org/2017/benchmark_selection_iccma2017.pdf)), accessed on 3 August 2024), StableGenerator (25) \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Instances** & **Solver** & **\# Solved** & **PAR-2 (s)** \\ \hline ICCMA 2019 & ArgTools\_2 lab & 292 & 260.3 \\ (326) & ArgTools\_3 lab & 298 & 217.13 \\  & ArgTools\_4 lab & **326** & **7.36** \\ \hline ICCMA 2021 & ArgTools\_2 lab & 132 & 1994.39 \\ (587) & ArgTools\_3 lab & 136 & 1963.9 \\  & ArgTools\_4 lab & **171** & **1729.37** \\ \hline ICCMA 2023 & ArgTools\_2 lab & 190 & 1073.15 \\ (329) & ArgTools\_3 lab & 201 & 956.82 \\  & ArgTools\_4 lab & **224** & **752.19** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experiment results on instances of the main track of ICCMA 2023 and ICCMA 2019.