Simon Knäuer

# Network satisfaction for symmetric relation algebras with a flexible atom

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.