We study and provide efficient algorithms for multi-objective model checking problems for Markov Decision Processes (MDPs). Given an MDP, M , and given multiple linear-time (ω-regular or LTL) properties ϕi, and probabilities ri ∈ [0, 1], i = 1, . . . , k, we ask whether there exists a strategy σ for the controller such that, for all i, the probability that a trajectory of M controlled by σ satisfies ϕi is at least ri. We provide an algorithm that decides whether there exists such a strategy and

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... if so produces it, and which runs in time polynomial in the size of the MDP. Such a strategy may require the use of both randomization and memory. We also consider more general multi-objective ω-regular queries, which we motivate with an application to assume-guarantee compositional reasoning for probabilistic systems. Note that there can be trade-offs between different properties: satisfying property ϕ1 with high probability may necessitate satisfying ϕ2 with low probability. Viewing this as a multi-objective optimization problem, we want information about the "trade-off curve" or Pareto curve for maximizing the probabilities of different properties. We show that one can compute an approximate Pareto curve with respect to a set of ω-regular properties in time polynomial in the size of the MDP. Our quantitative upper bounds use LP methods. We also study qualitative multiobjective model checking problems, and we show that these can be analysed by purely graph-theoretic methods, even though the strategies may still require both randomization and memory.