## 6 General Remarks, Conclusions, and Further Research Directions  
Over the past few decades, the HSTP attracted a great deal of research attention, as evidenced by international competitions, biannual conferences, and multiple publications [2; 3]. In this current research work, we presented a new algorithm based on a local version of PSO for a discrete search space. This algorithm exhibited comparable performance with other published algorithms dealing with the Greek HSTP, improving lower bounds reported so far in the literature in several cases. Specifically, it exhibited generally a better performance than the CSO algorithm, which up to now constituted the state-of-the art among soft computing attempts in terms of the best-case solution. The proposed algorithm performed better than the CSO algorithm in five out of ten test instances, achieving the same results in four out of ten cases, whereas it exhibited an inferior performance only in one out of ten instances. In addition, it compared favorably to the algorithm based on the global version of PSO, which was also the state-of-the art among soft computing attempts, in five out of ten instances. In the remaining instances, our algorithm matched the performance of the global version of PSO. Most importantly, it achieved the proven optima provided by integer programming and, additionally, produced new lower bounds for instances 1, 2, 4, and 7, which the exact method was unable to solve to optimality within a reasonable computational time. In particular, for the hardest instance, the proposed algorithm achieved 45-50% improvement over the lower bound produced by the two most powerful commercial mathematical programming solvers. This fact underlined the potential of the proposed algorithm for harder instances, which may reflect more complex problem settings. We believe that the desirable performance exhibited by the proposed local version of a PSO-based algorithm was due to the use of the backtracking schemas, one used in the main algorithm and the other used in the second refining procedure. The refining schemas prevent the algorithm's entrapment in local optima, and they eliminate the teacher gaps. The use of a local instead of a global version of PSO algorithm can help prevent premature convergence, making this a valuable complement to the existing literature. Additionally, it is well known that the local version of PSO outperforms the global one in the case of a continuous search space most of the time [16]. Based on the \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & **Main Algorithm Score** & **1 st Refining Phase Score** & **2 nd Refining Phase Score** & **Total Refining Improvement (\%)** \\ \hline
1 & 44 & 7 & 7 & 84.1 \\
2 & 41 & 14 & 13 & 68.3 \\
3 & 12 & 5 & 5 & 58.3 \\
4 & 39 & 5 & 3 & 92.3 \\
5 & 6 & 0 & 0 & 100 \\
7 & 59 & 25 & 6 & 89.9 \\
8 & 18 & 11 & 11 & 38.9 \\
9 & 25 & 2 & 2 & 92 \\
10 & 19 & 10 & 10 & 47.4 \\
11 & 32 & 19 & 19 & 40.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Effect of refining schemas on the outcome of the proposed algorithm. current work, and on the relative comparison to previous works where a global version of PSO was used, we experimentally verified the above statement in the case of the discrete search space, too. To the best of our knowledge, this is the first time that a local version of PSO was applied to the school timetabling problem and compared to other soft computing and exact methods. This fact, along with the presentation of new lower bounds for four out of ten given instances of the Beligiannis' data set, constituted the main contribution of our work. Even though the model we presented was based on real-world problem settings in Greece, some special cases were omitted. For instance, in Greek schools, there is often a need for two consecutive teaching hours of Literature or other subjects. Moreover, in Greek Technical High schools, there is a need of allocating several classes to a limited number of laboratory rooms.