Existence of a complexity hierarchy analogous to the W-hierarchy for fixed-parameter sub-exponential algorithms

Determine whether there exists a parameterized complexity hierarchy, analogous to the W-hierarchy, for problems that are solvable in sub-exponential time when a parameter is fixed (or for each constant value of the parameter).

Background

The paper proposes exploring a class of problems solvable via fixed-parameter sub-exponential algorithms, motivated by cases where, for fixed parameter values, running times become sub-exponential in the input size even for problems that are hard in general. The authors discuss how this perspective relates to established parameterized complexity classes.

They explicitly raise uncertainty about whether an overarching hierarchy akin to the W-hierarchy exists for such problems, highlighting a gap in current understanding of how to classify fixed-parameter sub-exponential tractability.