Robin Hirsch posed in 1996 the Really Big Complexity Problem: classify the computational complexity of the network satisfaction problem for all finite relation algebras \(\mathbf{A}\). We provide a complete classification for the case that \(\mathbf{A}\) is symmetric and has a flexible atom; the problem is in this case NP-complete or in P. If a finite integral relation algebra has a flexible atom, then it has a normal representation \(\mathfrak{B}\). We can then study the computational complexity of the network satisfaction problem of \(\mathbf{A}\) using the universal-algebraic approach, via an analysis of the polymorphisms of \(\mathfrak{B}\). We also use a Ramsey-type result of Nešetřil and Rödl and a complexity dichotomy result of Bulatov for conservative finite-domain constraint satisfaction problems.
Joint work with Manuel Bodirsky.