Computational complexity of Network Episode Clustering

Determine the computational complexity of the Network Episode Clustering (NetEC) problem for general phylogenetic networks, beyond the linear-time solvable tree case.

Background

The paper defines NetEC as the problem of computing the minimum number of duplication-episode locations needed to explain a collection of rooted gene trees on a phylogenetic network through valid mappings. Although the corresponding problem is solvable in linear time when the species structure is a tree, the authors leave its complexity for general networks unresolved. Their proposed exact optimization algorithm has exponential worst-case complexity, while the feasibility subproblem is solvable in polynomial time. In the Discussion, the authors further conjecture that NetEC is intractable for general networks, but no hardness result is established.