## 3 Clause Weight in SLS
### Assessment of the Initial Weight Distribution and Its Impact
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Problems** & **nCl** & **MinN** & **MaxN** & **AvgN** & **sCl\&N** & **sCl\&IN** & **lCl\&N** \\ \hline
4 blocks & 47,820 & 3 & 740 & 433 & 26,670 & 21,137 & 0 & 13 \\ ais12 & 5666 & 12 & 143 & 95 & 197 & 3672 & 11 & 12 \\ BMS-499 & 278 & 7 & 23 & 14 & 88 & 190 & 0 & 0 \\ bw\_large.c & 50,457 & 5 & 465 & 69 & 38,780 & 11,636 & 23 & 18 \\ bw\_large.d & 131,973 & 5 & 741 & 86 & 111,741 & 20,178 & 32 & 22 \\ CBS & 403 & 8 & 33 & 21 & 171 & 23 & 0 & 0 \\ f800-hard & 3440 & 9 & 37 & 22 & 1511 & 1929 & 0 & 0 \\ f1600-hard & 6880 & 8 & 41 & 22 & 3067 & 3813 & 0 & 0 \\ f2000 & 8500 & 7 & 41 & 22 & 3797 & 4703 & 0 & 0 \\ fla-barthel & 1720 & 10 & 39 & 22 & 724 & 996 & 0 & 0 \\ flat200-har & 2237 & 3 & 20 & 13 & 641 & 1596 & 0 & 0 \\ gl25.17 & 66,272 & 17 & 149 & 125 & 31,042 & 35,105 & 0 & 125 \\ g125.18 & 70,163 & 18 & 149 & 125 & 32,868 & 37,170 & 0 & 125 \\ huge & 7054 & 5 & 171 & 63 & 3000 & 4037 & 7 & 10 \\ logistics.d & 21,991 & 5 & 235 & 26 & 11,317 & 8824 & 1325 & 525 \\ uf400-hard & 1700 & 9 & 38 & 22 & 768 & 932 & 0 & 0 \\ unif-k5-v200 & 4223 & 214 & 321 & 269 & 2081 & 2142 & 0 & 0 \\ unif-k7-v98 & 8603 & 1957 & 2327 & 2158 & 4179 & 4424 & 0 & 0 \\ unif-k7-v102 & 8955 & 2000 & 2328 & 2158 & 4466 & 4489 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: General structural statistics of each problem in our problem set. On the other hand, clauses of considerable neighborhood size were harder to become unsatisfied for two reasons: first, the possibility of a clause to contain a true literal is high. Second, the score of the move (\(m\)), where \(m\) is to flip a literal within a large unsatisfied clause, is high. Furthermore, Figure 2 illustrates the weights of the small clauses compared to their neighborhood size. It is clearly shown that clauses with smaller neighborhood gained more weight until the solver reached the solution. Also, Figure 2 show that small clauses, even if they have a large neighborhood, generally gained more weights compared to large clauses, which gained a relatively smaller amount of weight until the solver flipped one of their literals. The above clause weight assessment indicates that one of the reasons for the fluctuation status of the small clauses (from satisfied to unsatisfied and vice versa) is because of the initial weight assignment where the clause weighting SLS solvers initially treats the clauses equally (assigning a weight of 1 to all clauses), regardless of their size or the size of their neighborhood. Thus, the gap between the weights of the clauses in the weights' initial assignment is considerable. To overcome this challenge, the clause weighting SLS solvers adjusts the weights of the clauses multiplicatively (e.g., ESG [47], SDF [48], SAPS) or, in most solvers, additively (e.g., PAWS, Sparrow, BalancedZ, YalSAT...etc.). In the next section, we discuss our novel mechanism in assigning the initial clauses weights dynamically, contributing to filling the clauses' initial gap, which appears due to the initial equal assignment of clauses weights. Figure 1: Weight distribution of the clauses of the bw_large problem until a solution is reached. Figure 2: The weight distribution of the small sized clauses with small neighborhood size of the bw_large problem.