On the Complexity of Value Iteration (Track B: Automata, Logic, Semantics, and Theory of Programming)

Authors Nikhil Balaji, Stefan Kiefer, Petr Novotný , Guillermo A. Pérez , Mahsa Shirmohammadi

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Author Details

Nikhil Balaji
  • University of Oxford, UK
Stefan Kiefer
  • University of Oxford, UK
Petr Novotný
  • Masaryk University, Brno, Czech Republic
Guillermo A. Pérez
  • University of Antwerp, Belgium
Mahsa Shirmohammadi
  • CNRS, Paris, France
  • IRIF, Paris, France


We thank James Worrell for helpful comments on early version of this work.

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Nikhil Balaji, Stefan Kiefer, Petr Novotný, Guillermo A. Pérez, and Mahsa Shirmohammadi. On the Complexity of Value Iteration (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 102:1-102:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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 EXPTIME-complete, thus resolving an open problem that goes back to the seminal 1987 paper on the complexity of MDPs by Papadimitriou and Tsitsiklis. To obtain this main result, we develop several stepping stones that yield results of an independent interest. For instance, we show that it is EXPTIME-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. We also provide new complexity results for the bounded halting problem in linear-update counter machines.

Subject Classification

ACM Subject Classification
  • Theory of computation → Probabilistic computation
  • Theory of computation → Logic and verification
  • Theory of computation → Markov decision processes
  • Markov decision processes
  • Value iteration
  • Formal verification


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