Polynomial-time computability of the ELRF distance

Determine whether the ELRF (extended labeled Robinson–Foulds) distance between two labeled trees can be computed in polynomial time by either developing a polynomial-time algorithm or establishing computational hardness for the problem.

Background

The ELRF distance extends the Robinson–Foulds framework to labeled trees by allowing labeled contractions, labeled extensions, and label flips. While its unlabeled counterpart is efficiently computable, the label constraints—specifically, that edge operations must preserve labels on both endpoints—complicate algorithmic design. A linear-time variant (LRF) exists by replacing edge operations with node insertions/deletions, but the computational status of ELRF itself remains unresolved.

Clarifying the complexity of ELRF is important for practical comparison of reconciled gene trees where event labels (duplication/speciation) are integral to the evolutionary interpretation.