Computational complexity of the rooted NNI tree-basedness proximity measure

Determine the computational complexity of the rooted nearest-neighbor-interchange tree-basedness proximity measure for rooted binary phylogenetic networks.

Background

The paper introduces the rooted NNI tree-basedness proximity measure, which is the minimum number of rooted nearest-neighbor-interchange operations needed to transform a rooted binary phylogenetic network into a tree-based network. It provides an exact neighborhood-search algorithm whose running time is exponential in the number of reticulation vertices, while noting that polynomial-time computation is possible for networks sufficiently close to being tree-based. The general computational complexity of this proximity measure is explicitly left unresolved.

References

While the complexity of the NTBP is an open question, we know from \cref{prop::algo} that for a sufficiently small $k_2$ we can compute $\delta_{\text{rNNI}$ in polynomial time because $\eta*(N)$ can be computed efficiently.

— The nearest neighbor interchange proximity measure for the tree-basedness of rooted binary phylogenetic networks  (2610.23426 - Frohn et al., 16 Oct 2026) in Section 5, “Conclusion and future research”