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Documents authored by Novotný, Petr

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Novotný, Petr

Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2019.102
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, and Mahsa Shirmohammadi

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

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 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.
Cite as

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)

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@InProceedings{balaji_et_al:LIPIcs.ICALP.2019.102,
  author =	{Balaji, Nikhil and Kiefer, Stefan and Novotn\'{y}, Petr and P\'{e}rez, Guillermo A. and Shirmohammadi, Mahsa},
  title =	{{On the Complexity of Value Iteration}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{102:1--102:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.102},
  URN =		{urn:nbn:de:0030-drops-106782},
  doi =		{10.4230/LIPIcs.ICALP.2019.102},
  annote =	{Keywords: Markov decision processes, Value iteration, Formal verification}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2016.10
Stability in Graphs and Games

Authors: Tomas Brazdil, Vojtech Forejt, Antonin Kucera, and Petr Novotny

Published in: LIPIcs, Volume 59, 27th International Conference on Concurrency Theory (CONCUR 2016)

Abstract
We study graphs and two-player games in which rewards are assigned to states, and the goal of the players is to satisfy or dissatisfy certain property of the generated outcome, given as a mean payoff property. Since the notion of mean-payoff does not reflect possible fluctuations from the mean-payoff along a run, we propose definitions and algorithms for capturing the stability of the system, and give algorithms for deciding if a given mean payoff and stability objective can be ensured in the system.
Cite as

Tomas Brazdil, Vojtech Forejt, Antonin Kucera, and Petr Novotny. Stability in Graphs and Games. In 27th International Conference on Concurrency Theory (CONCUR 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 59, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

Copy BibTex To Clipboard

@InProceedings{brazdil_et_al:LIPIcs.CONCUR.2016.10,
  author =	{Brazdil, Tomas and Forejt, Vojtech and Kucera, Antonin and Novotny, Petr},
  title =	{{Stability in Graphs and Games}},
  booktitle =	{27th International Conference on Concurrency Theory (CONCUR 2016)},
  pages =	{10:1--10:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-017-0},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{59},
  editor =	{Desharnais, Jos\'{e}e and Jagadeesan, Radha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2016.10},
  URN =		{urn:nbn:de:0030-drops-61784},
  doi =		{10.4230/LIPIcs.CONCUR.2016.10},
  annote =	{Keywords: Games, Stability, Mean-Payoff, Window Objectives}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2013.487
Solvency Markov Decision Processes with Interest

Authors: Tomás Brázdil, Taolue Chen, Vojtech Forejt, Petr Novotný, and Aistis Simaitis

Published in: LIPIcs, Volume 24, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)

Abstract
Solvency games, introduced by Berger et al., provide an abstract framework for modelling decisions of a risk-averse investor, whose goal is to avoid ever going broke. We study a new variant of this model, where, in addition to stochastic environment and fixed increments and decrements to the investor's wealth, we introduce interest, which is earned or paid on the current level of savings or debt, respectively. We study problems related to the minimum initial wealth sufficient to avoid bankruptcy (i.e. steady decrease of the wealth) with probability at least p. We present an exponential time algorithm which approximates this minimum initial wealth, and show that a polynomial time approximation is not possible unless P=NP. For the qualitative case, i.e. p=1, we show that the problem whether a given number is larger than or equal to the minimum initial wealth belongs to NP \cap coNP, and show that a polynomial time algorithm would yield a polynomial time algorithm for mean-payoff games, existence of which is a longstanding open problem. We also identify some classes of solvency MDPs for which this problem is in P. In all above cases the algorithms also give corresponding bankruptcy avoiding strategies.
Cite as

Tomás Brázdil, Taolue Chen, Vojtech Forejt, Petr Novotný, and Aistis Simaitis. Solvency Markov Decision Processes with Interest. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 487-499, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

Copy BibTex To Clipboard

@InProceedings{brazdil_et_al:LIPIcs.FSTTCS.2013.487,
  author =	{Br\'{a}zdil, Tom\'{a}s and Chen, Taolue and Forejt, Vojtech and Novotn\'{y}, Petr and Simaitis, Aistis},
  title =	{{Solvency Markov Decision Processes with Interest}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)},
  pages =	{487--499},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-64-4},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{24},
  editor =	{Seth, Anil and Vishnoi, Nisheeth K.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.487},
  URN =		{urn:nbn:de:0030-drops-43959},
  doi =		{10.4230/LIPIcs.FSTTCS.2013.487},
  annote =	{Keywords: Markov decision processes, algorithms, complexity, market models.}
}

Novotny, Petr

Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2019.102
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, and Mahsa Shirmohammadi

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

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 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.
Cite as

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)

Copy BibTex To Clipboard

@InProceedings{balaji_et_al:LIPIcs.ICALP.2019.102,
  author =	{Balaji, Nikhil and Kiefer, Stefan and Novotn\'{y}, Petr and P\'{e}rez, Guillermo A. and Shirmohammadi, Mahsa},
  title =	{{On the Complexity of Value Iteration}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{102:1--102:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.102},
  URN =		{urn:nbn:de:0030-drops-106782},
  doi =		{10.4230/LIPIcs.ICALP.2019.102},
  annote =	{Keywords: Markov decision processes, Value iteration, Formal verification}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2016.10
Stability in Graphs and Games

Authors: Tomas Brazdil, Vojtech Forejt, Antonin Kucera, and Petr Novotny

Published in: LIPIcs, Volume 59, 27th International Conference on Concurrency Theory (CONCUR 2016)

Abstract
We study graphs and two-player games in which rewards are assigned to states, and the goal of the players is to satisfy or dissatisfy certain property of the generated outcome, given as a mean payoff property. Since the notion of mean-payoff does not reflect possible fluctuations from the mean-payoff along a run, we propose definitions and algorithms for capturing the stability of the system, and give algorithms for deciding if a given mean payoff and stability objective can be ensured in the system.
Cite as

Tomas Brazdil, Vojtech Forejt, Antonin Kucera, and Petr Novotny. Stability in Graphs and Games. In 27th International Conference on Concurrency Theory (CONCUR 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 59, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

Copy BibTex To Clipboard

@InProceedings{brazdil_et_al:LIPIcs.CONCUR.2016.10,
  author =	{Brazdil, Tomas and Forejt, Vojtech and Kucera, Antonin and Novotny, Petr},
  title =	{{Stability in Graphs and Games}},
  booktitle =	{27th International Conference on Concurrency Theory (CONCUR 2016)},
  pages =	{10:1--10:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-017-0},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{59},
  editor =	{Desharnais, Jos\'{e}e and Jagadeesan, Radha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2016.10},
  URN =		{urn:nbn:de:0030-drops-61784},
  doi =		{10.4230/LIPIcs.CONCUR.2016.10},
  annote =	{Keywords: Games, Stability, Mean-Payoff, Window Objectives}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2013.487
Solvency Markov Decision Processes with Interest

Authors: Tomás Brázdil, Taolue Chen, Vojtech Forejt, Petr Novotný, and Aistis Simaitis

Published in: LIPIcs, Volume 24, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)

Abstract
Solvency games, introduced by Berger et al., provide an abstract framework for modelling decisions of a risk-averse investor, whose goal is to avoid ever going broke. We study a new variant of this model, where, in addition to stochastic environment and fixed increments and decrements to the investor's wealth, we introduce interest, which is earned or paid on the current level of savings or debt, respectively. We study problems related to the minimum initial wealth sufficient to avoid bankruptcy (i.e. steady decrease of the wealth) with probability at least p. We present an exponential time algorithm which approximates this minimum initial wealth, and show that a polynomial time approximation is not possible unless P=NP. For the qualitative case, i.e. p=1, we show that the problem whether a given number is larger than or equal to the minimum initial wealth belongs to NP \cap coNP, and show that a polynomial time algorithm would yield a polynomial time algorithm for mean-payoff games, existence of which is a longstanding open problem. We also identify some classes of solvency MDPs for which this problem is in P. In all above cases the algorithms also give corresponding bankruptcy avoiding strategies.
Cite as

Tomás Brázdil, Taolue Chen, Vojtech Forejt, Petr Novotný, and Aistis Simaitis. Solvency Markov Decision Processes with Interest. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 487-499, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

Copy BibTex To Clipboard

@InProceedings{brazdil_et_al:LIPIcs.FSTTCS.2013.487,
  author =	{Br\'{a}zdil, Tom\'{a}s and Chen, Taolue and Forejt, Vojtech and Novotn\'{y}, Petr and Simaitis, Aistis},
  title =	{{Solvency Markov Decision Processes with Interest}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)},
  pages =	{487--499},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-64-4},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{24},
  editor =	{Seth, Anil and Vishnoi, Nisheeth K.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.487},
  URN =		{urn:nbn:de:0030-drops-43959},
  doi =		{10.4230/LIPIcs.FSTTCS.2013.487},
  annote =	{Keywords: Markov decision processes, algorithms, complexity, market models.}
}
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