  We discuss the computational complexity of the perturbative evaluation of
scattering amplitudes, both by the Caravaglios-Moretti algorithm and by direct
evaluation of the individual diagrams. For a self-interacting scalar theory, we
determine the complexity as a function of the number of external legs. We
describe a method for obtaining the number of topologically inequivalent
Feynman graphs containing closed loops, and apply this to one- and two-loop
amplitudes. We also compute the number of graphs weighted by their symmetry
factors, thus arriving at exact and asymptotic estimates for the average
symmetry factor of diagrams. We present results for the asymptotic number of
diagrams up to 10 loops, and prove that the average symmetry factor approaches
unity as the number of external legs becomes large.
