Robust Markov decision processes (RMDPs) extend standard Markov decision processes (MDPs) to account for uncertainty in the transition probabilities. RMDPs have an uncertainty set that defines a set of possible transition functions, each of which induces a standard MDP. The natural objective in an RMDP is to optimize the discounted cumulative reward under the worst-case transition function in the uncertainty set. We study the complexity of the associated threshold problem for RMDPs with polytopic uncertainty sets in halfspace representation. Previous results focused on approximating the optimum or restricted attention to specific subclasses of RMDPs, such as interval MDPs or $L_\infty$-RMDPs. Our contributions are threefold: (1) For (s,a)-rectangular RMDPs, we prove that robust policy evaluation is in P via robust linear programming, and that the threshold problem is in NP. As a corollary, robust policy iteration is a polynomial-time algorithm for these RMDPs when the discount factor is fixed. (2) For $s$-rectangular RMDPs, we show that the threshold problem is in PSPACE via the first-order theory of the reals. (3) We establish lower bounds by reducing both parity games and bisimulation metrics between MDP states to the RMDP threshold problem. A polynomial-time algorithm for the threshold problem would resolve the long-standing open question of whether parity games can be solved in polynomial time. The reduction from bisimulation metrics also yields a practical benefit: it allows us to apply robust policy iteration as a more efficient alternative to the standard fixed-point iteration, as our empirical evaluation demonstrates.

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