## 5 The Labeling Algorithm for Four-Label Enumeration Stable Extension
### Four-Label Algorithm Description
([[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), and Crusti_g2 io (32) ([[https://helda.helsinki.fi/items/7c4a0](https://helda.helsinki.fi/items/7c4a0) bad-0604-44c5-b5 cb-1a6a487 fa8 af]([https://helda.helsinki.fi/items/7c4a0](https://helda.helsinki.fi/items/7c4a0) bad-0604-44c5-b5 cb-1a6a487 fa8 af), accessed on 3 August 2024). These three types of instances share two common characteristics that may have led to suboptimal solving performance, namely, the issues of randomness and community structure. Our algorithm leverages label characteristics to directly perform tree search on the problem, while SAT-based reduction methods encode the original problem into a SAT problem and then call a SAT solver. SAT-based methods may obscure some of the original problem's information, so directly incorporating SAT techniques into the tree search for the original problem could be a key direction for future improvements. * **ArgTools_3 lab:** The basic Argtools solver which uses three labels. * **ArgTools_3 lab+:** Argtools solver with the preprocessing strategy applied. * **ArgTools_3 lab++(ArgTools_4 lab):** Argtools solver with the proposed four-label method applied. * **ArgSemSAT:** Stable extension solving algorithm based on the reduction model. Figure 3 shows the results of the ArgTools_3 lab+ and ArgTools_3 lab solvers on the instances of the main track of ICCMA 2023, which include a total of 330 instances. Figure 3 displays the cactus plot for these two solvers on the above instances. Each point \((x,y)\) in the figure corresponds to the solving time for a specific instance, where the \(x\)-axis represents the runtime, and the \(y\)-axis represents the number of instances solved. From this, it can be concluded that preprocessing operations for solving instances, such as prioritizing initial and self-attacking arguments, can speed up labeling propagation efficiency. Figure 3(b) shows the scatter plot for the instance families in the main track of ICCMA 2023. These instances are grouped into 17 families, with each family consisting of instances with similar names. A point \((x,y)\) in the figure represents an instance family, where \(x\) (\(y\)) denotes the average solving time of ArgTools_3 lab+ (ArgTools_3 lab) for each instance in the respective family. From Figure 3(b), we can see that the preprocessing strategy of the four-label algorithm is effective across all instance families. Similar to Figure 3, Figure 4 presents a comparison of the solving performance between the ArgTools_4 lab and ArgTools_3 lab+ solvers on the instances of the main track of ICCMA 2023. From the figure, it is evident that the four-label method is more effective

Figure 3: (a) Cactus plots comparing the number of solved instances by ArgTools_3 lab+ and ArgTools_3 lab on the benchmark of the main track of ICCMA 2023. (b) Scatter plots comparing the runtime of ArgTools_3 lab+ (\(x\)-axis) and ArgTools_3 lab (\(y\)-axis) on 17 benchmark families of the main track of ICCMA 2023. If the point is on the dashed line, then the solving effects of both solvers are the same; If the point is above the dashed line, then the solver ArgTools_3 lab+ performs better; conversely, if it is below the line, the performance is the opposite. in propagating argument labelings compared to the two-label and three-label methods. Additionally, the four-label labeling rules are generally effective across all instance families. To further demonstrate that the four-label method proposed in this paper is not only applicable to the ArgTools solver mentioned in the article but also performs well in other labeling algorithms, we selected the DREDD solver by [PERSON] et al.