We study dominant strategy incentive compatible (DIC) and deterministic mechanisms in a social choice setting with several alternatives. The agents are privately informed about their preferences, and have single-crossing utility functions. Monetary transfers are not feasible. We use an equivalence between deterministic, DIC mechanisms and generalized median voter schemes to construct the constrained-efficient, optimal mechanism for an utilitarian planner. Optimal schemes for other welfare criteria such as, say, a Rawlsian maximin can be analogously obtained.