# Dynamic Initial Weight Assignment for MaxSAT
## 4 Experimental Results and Analyses
### Weight Distribution  
The weights and their distribution were the core component of this study. Therefore, we extensively studied the weights and their distributions for all problems under consideration. This was done through an experimental study of the distribution of the weights, which began by recording the weights and their changes for all the problem clauses. Then, we compared the clauses' weights and whether those clauses were located in a neighborhood of small, medium, or large size. This comparison aimed to determine whether the clause weights correlated with the size of the neighborhoods in which they were located. As we expected, this appeared clearly in all the problems under discussion, as follows:  
**pigeon hole problem set:** Figure 1 illustrates the clauses' weight distribution compared to their neighborhood size for all six instances of the pigeon hole problem. It was indicated that most weighted clauses had a small neighborhood size. Moreover, the statistics showed that 91% of the clauses had less than the average weight where the rest of the clauses, 9%, were weighted the most.  
**AIM problem set:** Similar to the pigeon hole problems, the most weighted clauses were in small neighborhoods. Eighty-six percent of the least weighted clauses had a neighborhood size greater than the average, where 14% had a smaller than average neighborhood size, as illustrated in Figure 2.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Problem** & **Instance** & **nCls** & **AvgNeighbS** & **\% \textless{AvgNeighbS**} & **\% \textless{AvgNeighbS**} & **\% \textless{AvgNeighbS**}** \\ \hline \multirow{3}{*}{aim} & aim-50 & 80 & 7.91 & 0.156 & 0.44 & 0.41 \\  & aim-100 & 160 & 7.63 & 0.14 & 0.48 & 0.3725 \\  & aim-200 & 320 & 7.67 & 0.15 & 0.23 & 0.61 \\ \hline \multirow{6}{*}{uuf} & uuf-01 & 217 & 21.91 & 0.37 & 0.11 & 0.52 \\  & uuf-10 & 217 & 22.40 & 0.39 & 0.11 & 0.5 \\ \cline{1-1}  & uuf-100 & 217 & 22.42 & 0.4 & 0.12 & 0.48 \\ \cline{1-1}  & uuf-200 & 217 & 22.32 & 0.45 & 0.00 & 0.47 \\ \cline{1-1}  & uuf-300 & 217 & 22.26 & 0.43 & 0.11 & 0.45 \\ \cline{1-1}  & uuf-410 & 217 & 22.29 & 0.45 & 0.1 & 0.45 \\ \cline{1-1}  & uuf-800 & 217 & 21.70 & 0.39 & 0.1 & 0.51 \\ \hline \hline \end{tabular}
\end{table}
Table 1: _Cont._  
Figure 1: The figure illustrates the most weighted clauses and their corresponding neighborhood size for the pigeon hole problem set (six instances), where the overall average of the neighborhood size is \(\approx\)16.  
**jnh problem set:** jnh was of particular interest as the correlation between the clause weights and their corresponding neighborhood sizes was very explicit. Figure 3 illustrates the correlation, as most weighted clauses, specifically 21% of the clauses, had a small neighborhood size 79% of the clauses were the least weighted clauses with a large neighborhood size.  
**uuf problem set:** Figure 4 illustrates an overview of the uniform unsatisfiable problem set, weight distribution where 83% of the clauses were least weighted with the large neighborhood and 17% of the clauses were most weighted and had a small neighborhood size.  
Figure 3: The figure illustrates the most weighted clauses and their corresponding neighborhood size for the jnh problem set (20 instances), where the overall average of the neighborhood size is \(\approx\)107.  
Figure 2: The figure illustrates the most weighted clauses and their corresponding neighborhood size for the aim problem set (30 instances), where the overall average of the neighborhood size is \(\approx\)7.5.