Shortest common supersequence with applications

Abstract

The shortest common supersequence problem (SCS) is a classical problem that was first studied by Maier in 1978. Recently, some research on the tree-child network inference problem, developed from the inference of phylogenetic networks, which involves a variant of SCS, has been studied. The variant of the shortest common supersequence problem on permutation strings was introduced to solve the tree-child network problem. In this thesis, we first provide the sketch of the proof of the NP-completeness of the SCS problem on the permutation strings from the feedback vertex set problem. From the reduction, we continue analyzing the inapproximability of this new variation. Then, we study the FPT tractability of SCS using different parameters. With the new reduction developed, which corrected and simplified the former reduction provided by Maier from the Vertex Cover problem, we extended the proof to the non-existence of the FPT algorithm parameterized by the number of the inserting positions in the longest input sequence even if it is fixed to be 3. On the other hand, an FPT algorithm parameterized by the solution size has been explored. Furthermore, we extended our proof on the NP-completeness and W[1] hardness of the k-disjoint-SCS problem as another variant of the SCS. Lastly, a new heuristic algorithm has been constructed for practical use by using the concept of maximum increasing substrings, which can be applied to both permutation strings and non-permutation strings. The empirical results show that both algorithms perform similarly in producing the shortest common supersequence, while our algorithm takes longer runtime on average due to the more complex computation during the process.