Nearly cubic hardness of weighted tree edit distance

Establish whether the weighted version of tree edit distance requires almost cubic time, as suggested by its equivalence to the All-Pairs Shortest Paths problem and the conjectured near-cubic hardness of that problem.

Background

The paper distinguishes unweighted TED, which is the main focus, from weighted TED, where edit-operation costs depend on node labels. Weighted TED has been shown to be equivalent to the All-Pairs Shortest Paths (APSP) problem.

Because APSP is conjectured to require almost cubic time, the paper presents this as evidence that weighted TED may inherently require superquadratic, potentially near-cubic, time. The statement remains a conjectural complexity claim rather than an established lower bound.