Let $\chi'_f(G)$ be the fractional chromatic index.
For which graph classes $\lceil \chi'_f(G) \rceil = \chi'(G)$?
Since $\chi'_f(G)$ is computable in polynomial time, solving this gives tractable graph classes.
Experiments with Sage suggest this is true for perfect graphs on up to $8$ vertices, though Sage's implementation is not inefficient.