On the computational complexity of stochastic controller optimization in POMDPs

We show that the problem of finding an optimal stochastic 'blind' controller in a Markov decision process is an NP-hard problem. The corresponding decision problem is NP-hard, in PSPACE, and SQRT-SUM-hard, hence placing it in NP would imply a breakthrough in long-standing open problems in computer science. Our optimization result establishes that the more general problem of stochastic controller optimization in POMDPs is also NP-hard. Nonetheless, we outline a special case that is solvable to arbitrary accuracy in polynomial time via semidefinite or second-order cone programming.