  The betweenness centrality of a vertex v is an important centrality measure
that quantifies how many optimal paths between pairs of other vertices visit v.
Computing betweenness centrality in a temporal graph, in which the edge set may
change over discrete timesteps, requires us to count temporal paths that are
optimal with respect to some criterion. For several natural notions of
optimality, including foremost or fastest temporal paths, this counting problem
reduces to #Temporal Path, the problem of counting all temporal paths between a
fixed pair of vertices; like the problems of counting foremost and fastest
temporal paths, #Temporal Path is #P-hard in general. Motivated by the many
applications of this intractable problem, we initiate a systematic study of the
prameterised and approximation complexity of #Temporal Path. We show that the
problem presumably does not admit an FPT-algorithm for the feedback vertex
number of the static underlying graph, and that it is hard to approximate in
general. On the positive side, we proved several exact and approximate
FPT-algorithms for special cases.
