With Jan Kára. Accepted for publication in Theory of Computing Systems. A conference version of the paper appeared in the Proceedings of the International Computer Science Symposium in Russia, CSR'06, St.Petersburg, pages 114-126, 2006.
As soon as you want to classify the constraint satisfaction problems of all templates that have a first-order definition in some structure, you have to deal with equality constraint languages.
We apply the algebraic approach to infinite-valued constraint satisfaction to classify the computational complexity of all constraint satisfaction problems with templates that have a highly transitive automorphism group. A relational structure has such an automorphism group if and only if all the constraint types are Boolean combinations of the equality relation, and we call the corresponding constraint languages equality constraint languages . We show that an equality constraint language is tractable if it admits a constant unary or an injective binary polymorphism, and is NP-complete otherwise.
The conference version: in postscript, and pdf
A draft of the journal version: in postscript, and pdf