The comparison of tree-sibling time consistent phylogenetic networks is graph isomorphism-complete

TL;DR

This paper investigates the computational complexity of determining isomorphism in tree-sibling time consistent phylogenetic networks, revealing that the problem's difficulty varies significantly depending on network constraints, with some cases being as hard as graph isomorphism.

Contribution

The paper demonstrates that removing the semibinarity condition makes the isomorphism problem for these networks as hard as graph isomorphism, highlighting the limits of polynomial-time solutions.

Findings

Isomorphism problem for semibinary networks is in P.

Removing semibinarity makes the problem as hard as graph isomorphism.

No polynomial-time metric likely exists for all such networks.

Abstract

In a previous work, we gave a metric on the class of semibinary tree-sibling time consistent phylogenetic networks that is computable in polynomial time; in particular, the problem of deciding if two networks of this kind are isomorphic is in P. In this paper, we show that if we remove the semibinarity condition above, then the problem becomes much harder. More precisely, we proof that the isomorphism problem for generic tree-sibling time consistent phylogenetic networks is polynomially equivalent to the graph isomorphism problem. Since the latter is believed to be neither in P nor NP-complete, the chances are that it is impossible to define a metric on the class of all tree-sibling time consistent phylogenetic networks that can be computed in polynomial time.