On the Complexity of Value Iteration
This resolves a foundational open problem in computational complexity for MDPs, with implications for algorithm design and analysis in AI and operations research.
The paper settles the computational complexity of value iteration for Markov Decision Processes (MDPs), showing that computing an optimal policy given a horizon in binary is EXP-complete, resolving a long-standing open problem.
Value iteration is a fundamental algorithm for solving Markov Decision Processes (MDPs). It computes the maximal $n$-step payoff by iterating $n$ times a recurrence equation which is naturally associated to the MDP. At the same time, value iteration provides a policy for the MDP that is optimal on a given finite horizon $n$. In this paper, we settle the computational complexity of value iteration. We show that, given a horizon $n$ in binary and an MDP, computing an optimal policy is EXP-complete, thus resolving an open problem that goes back to the seminal 1987 paper on the complexity of MDPs by Papadimitriou and Tsitsiklis. As a stepping stone, we show that it is EXP-complete to compute the $n$-fold iteration (with $n$ in binary) of a function given by a straight-line program over the integers with $\max$ and $+$ as operators.