The talk is about the parameterized complexity of MinCSP for equality constraint languages, which are languages over an infinite domain with relations defined by first-order formulas whose only predicate is =. This is a natural class of optimization problems on undirected graphs, including problems such as Multicut, Steiner Multicut, and (under singleton expansion) Multiway Cut. Our main contribution is a classification of MinCSP for all finite equality constraint languages under the natural parameterization. We'll show that these problems are either solvable in FPT time, are W[1]-hard but admit a constant-factor FPT-approximation, or do not admit a constant-factor FPT-approximation unless FPT=W[2]. On the positive side, we obtain an FPT algorithm for a generalization of Multicut with deletable triple cut requests, and show a constant-factor FPT-approximation for Disjunctive Multicut, a generalization where the cut requests come as disjunctions of constant width. The talk is based on joint work with Magnus Wahlström.