On the Complexity of Value Iteration

D 2019

On the Complexity of Value Iteration

BALAJI, Nikhil, Stefan KIEFER, Petr NOVOTNÝ, Guillermo A. PÉREZ, Mahsa SHIRMOHAMMADI et. al.

Basic information

Original name

On the Complexity of Value Iteration

Authors

BALAJI, Nikhil (356 India), Stefan KIEFER (276 Germany), Petr NOVOTNÝ (203 Czech Republic, guarantor, belonging to the institution), Guillermo A. PÉREZ (340 Honduras) and Mahsa SHIRMOHAMMADI (364 Islamic Republic of Iran)

Edition

Dagstuhl, Germany, Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), p. "102:1"-"102:15", 15 pp. 2019

Publisher

Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik

Other information

Language

English

Type of outcome

Stať ve sborníku

Field of Study

10200 1.2 Computer and information sciences

Country of publisher

Germany

Confidentiality degree

není předmětem státního či obchodního tajemství

Publication form

electronic version available online

RIV identification code

RIV/00216224:14330/19:00107669

Organization unit

Faculty of Informatics

ISBN

978-3-95977-109-2

ISSN

DOI

http://dx.doi.org/10.4230/LIPIcs.ICALP.2019.102

Keywords in English

Markov decision processes; probabilistic verification; value iteration

Abstract

V originále

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.

Links

GA19-15134Y, interní kód MU

Name: Verifikace a analýza pravděpodobnostních programů

Investor: Czech Science Foundation

GJ19-15134Y, research and development project

Name: Verifikace a analýza pravděpodobnostních programů

Citovat

BALAJI, Nikhil, Stefan KIEFER, Petr NOVOTNÝ, Guillermo A. PÉREZ and Mahsa SHIRMOHAMMADI. On the Complexity of Value Iteration. Online. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, Stefano Leonardi. Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Dagstuhl, Germany: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2019, p. "102:1"-"102:15", 15 pp. ISBN 978-3-95977-109-2. Available from: https://dx.doi.org/10.4230/LIPIcs.ICALP.2019.102.

@inproceedings{1561197,
   author = {Balaji, Nikhil and Kiefer, Stefan and Novotný, Petr and Pérez, Guillermo A. and Shirmohammadi, Mahsa},
   address = {Dagstuhl, Germany},
   booktitle = {Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
   doi = {http://dx.doi.org/10.4230/LIPIcs.ICALP.2019.102},
   editor = {Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, Stefano Leonardi},
   keywords = {Markov decision processes; probabilistic verification; value iteration},
   howpublished = {elektronická verze "online"},
   language = {eng},
   location = {Dagstuhl, Germany},
   isbn = {978-3-95977-109-2},
   pages = {"102:1"-"102:15"},
   publisher = {Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik},
   title = {On the Complexity of Value Iteration},
   year = {2019}
}

TY  - JOUR
ID  - 1561197
AU  - Balaji, Nikhil - Kiefer, Stefan - Novotný, Petr - Pérez, Guillermo A. - Shirmohammadi, Mahsa
PY  - 2019
TI  - On the Complexity of Value Iteration
PB  - Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik
CY  - Dagstuhl, Germany
SN  - 9783959771092
KW  - Markov decision processes
KW  - probabilistic verification
KW  - value iteration
N2  - 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.
ER  -

BALAJI, Nikhil, Stefan KIEFER, Petr NOVOTNÝ, Guillermo A. PÉREZ and Mahsa SHIRMOHAMMADI. On the Complexity of Value Iteration. Online. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, Stefano Leonardi. \textit{Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}. Dagstuhl, Germany: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2019, p.~''102:1''-''102:15'', 15 pp. ISBN~978-3-95977-109-2. Available from: https://dx.doi.org/10.4230/LIPIcs.ICALP.2019.102.

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