not only acknowledge speciation but also non-tree-like processes such as hybridization and lateral gene transfer to be driving forces in the evolution of certain groups of organisms (e.g. bacteria, plants, and fish) [@mallet16; @soucy15], phylogenetic networks become more widely used to represent ancestral histories. A phylogenetic network is a generalization of a rooted phylogenetic tree. More precisely, such a network is a rooted directed acyclic graph whose leaves are labeled [@huson10].

The following optimization problem, which is biologically relevant and mathematically challenging, motivates much of the theoretical work that has been done in reconstructing phylogenetic networks from phylogenetic trees. Given a collection ${{\mathcal P}}$ of rooted binary phylogenetic trees on a set of species such that ${{\mathcal P}}$ correctly represents the tree-like evolution of different parts of the species’ genomes, what is the smallest number of reticulation events that is required to simultaneously embed the trees in ${{\mathcal P}}$ into a phylogenetic network? Here, reticulation events are collectively referring to all non-tree-like events and they are represented by vertices in a phylogenetic network whose in-degree is at least two. Without any structural constraints on a phylogenetic network, it is well-known that ${{\mathcal P}}$ can always be embedded into such a 