  We develop the novel machinery of smooth approximations, and apply it to
confirm the CSP dichotomy conjecture for first-order reducts of the random
tournament, various homogeneous graphs including the random graph, and for
expansions of the order of the rationals. Apart from obtaining these dichotomy
results, we show how our new proof technique allows to unify and significantly
simplify the previous results from the literature. For all but the last
structure, we moreover characterize those CSPs which are solvable by local
consistency methods, again using the same machinery.
