  In this paper, we study the relationship of parametrised enumeration
complexity classes defined by Creignou et al. (MFCS 2013). Specifically, we
introduce two hierarchies (IncFPTa and CapIncFPTa) of enumeration complexity
classes for incremental fpt-time in terms of exponent slices and show how they
interleave. Furthermore, we define several parametrised function classes and,
in particular, introduce the parametrised counterpart of the class of
nondeterministic multivalued functions with values that are polynomially
verifiable and guaranteed to exist, TFNP, known from Megiddo and Papadimitriou
(TCS 1991). We show that TF(para-NP) collapsing to F(FPT) is equivalent to
OutputFPT coinciding with IncFPT. This result is in turn connected to a
collapse in the classical function setting and eventually to the collapse of
IncP and OutputP which proves the first direct connection of classical to
parametrised enumeration.

Logic in Computer Science;Computational Complexity