Exact complexity of directedness for NFAs over variable alphabets

Determine the exact computational complexity of deciding directedness for regular languages given by nondeterministic finite automata (NFAs) when the alphabet size is part of the input; that is, given an NFA A over a variable alphabet Σ, ascertain the precise complexity class of deciding whether L(A) is directed under the scattered subword ordering.

Background

The paper proves that deciding directedness is NL-complete for NFAs over fixed alphabets and in AC^1 for general NFAs, but it does not pin down the exact complexity for the variable-alphabet case. The authors explicitly note that despite their upper bound, the precise classification remains unknown.

This open problem concerns the core decision task studied in the paper: whether the language accepted by an NFA is directed (i.e., every pair of words has a common scattered superword in the language). Resolving the exact complexity for variable alphabets would complete the complexity picture begun by the paper.