  Robin Hirsch posed in 1996 the 'Really Big Complexity Problem': classify the
computational complexity of the network satisfaction problem for all finite
relation algebras A. We provide a complete classification for the case that A
is symmetric and has a flexible atom; in this case, the problem is NP-complete
or in P. The classification task can be reduced to the case where A is
integral. If a finite integral relation algebra has a flexible atom, then it
has a normal representation B. We can then study the computational complexity
of the network satisfaction problem of A using the universal-algebraic
approach, via an analysis of the polymorphisms of B. We also use a Ramsey-type
result of Ne\v{s}et\v{r}il and R\"odl and a complexity dichotomy result of
Bulatov for conservative finite-domain constraint satisfaction problems.
