This paper deals with the restricted longest common subsequence (RLCS) problem, an extension of the well-studied longest common subsequence problem involving two sets of strings: the input strings and the restricted strings. This problem has applications in bioinformatics, particularly in identifying similarities and discovering mutual patterns and motifs among DNA, RNA, and protein molecules. We introduce a general search framework to tackle the RLCS problem. Based on this, we present an exact best-first search algorithm and a meta-heuristic Beam Search algorithm. To evaluate the effectiveness of these algorithms, we compare them with two exact algorithms and two approximate algorithms from the literature along with a greedy approach. Our experimental results show the superior performance of our proposed approaches. In particular, our exact approach outperforms the other exact methods in terms of significantly shorter computation times, often reaching an order of magnitude compared to the second-best approach. Moreover, it successfully solves all problem instances, which was not the case with the other approaches. In addition, Beam Search provides close-to-optimal solutions with remarkably short computation times.