The tree-child network problem and the shortest common supersequences for permutations are NP-hard

Abstract

Reconstructing phylogenetic networks presents a significant and complex challenge within the fields of phylogenetics and genome evolution. One strategy for reconstruction of phylogenetic networks is to solve the phylogenetic network problem, which involves inferring phylogenetic trees first and subsequently computing the smallest phylogenetic network that displays all the trees. This approach capitalizes on exceptional tools available for inferring phylogenetic trees from biomolecular sequences. Since the vast space of phylogenetic networks poses difficulties in obtaining comprehensive sampling, the researchers switch their attention to inferring tree-child networks from multiple phylogenetic trees, where in a tree-child network each non-leaf node must have at least one child that is a tree node (i.e. indegree-one node). We prove that the tree-child network problem for multiple trees remains NP-hard by a reduction from the shortest common supersequnece problem for permuations and proving that the latter is NP-hard.Comment: 3 figures and 11 page