On the Complexity of Value Iteration

TL;DR

This paper proves that computing optimal policies via value iteration for Markov Decision Processes with a binary horizon is EXP-complete, resolving a long-standing open problem in the computational complexity of MDPs.

Contribution

It establishes the EXP-completeness of the value iteration problem for finite-horizon MDPs, a fundamental result in understanding the algorithm's computational limits.

Findings

Computing optimal policies with value iteration is EXP-complete.

It is EXP-complete to compute the n-fold iteration of certain functions.

The result resolves an open problem from 1987 about MDP complexity.

Abstract

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.