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Documents authored by Loho, Georg

Document
DOI: 10.4230/LIPIcs.CSL.2024.24
The Worst-Case Complexity of Symmetric Strategy Improvement

Authors: Tom van Dijk, Georg Loho, and Matthew T. Maat

Published in: LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

Abstract
Symmetric strategy improvement is an algorithm introduced by Schewe et al. (ICALP 2015) that can be used to solve two-player games on directed graphs such as parity games and mean payoff games. In contrast to the usual well-known strategy improvement algorithm, it iterates over strategies of both players simultaneously. The symmetric version solves the known worst-case examples for strategy improvement quickly, however its worst-case complexity remained open. We present a class of worst-case examples for symmetric strategy improvement on which this symmetric version also takes exponentially many steps. Remarkably, our examples exhibit this behaviour for any choice of improvement rule, which is in contrast to classical strategy improvement where hard instances are usually hand-crafted for a specific improvement rule. We present a generalized version of symmetric strategy iteration depending less rigidly on the interplay of the strategies of both players. However, it turns out it has the same shortcomings.
Cite as

Tom van Dijk, Georg Loho, and Matthew T. Maat. The Worst-Case Complexity of Symmetric Strategy Improvement. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 24:1-24:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{vandijk_et_al:LIPIcs.CSL.2024.24,
  author =	{van Dijk, Tom and Loho, Georg and Maat, Matthew T.},
  title =	{{The Worst-Case Complexity of Symmetric Strategy Improvement}},
  booktitle =	{32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)},
  pages =	{24:1--24:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-310-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{288},
  editor =	{Murano, Aniello and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.24},
  URN =		{urn:nbn:de:0030-drops-196672},
  doi =		{10.4230/LIPIcs.CSL.2024.24},
  annote =	{Keywords: Parity game, Mean payoff game, Symmetric strategy improvement, Strategy improvement, Worst-case complexity}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2022.63
Beyond Value Iteration for Parity Games: Strategy Iteration with Universal Trees

Authors: Zhuan Khye Koh and Georg Loho

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Abstract
Parity games have witnessed several new quasi-polynomial algorithms since the breakthrough result of Calude et al. (STOC 2017). The combinatorial object underlying these approaches is a universal tree, as identified by Czerwiński et al. (SODA 2019). By proving a quasi-polynomial lower bound on the size of a universal tree, they have highlighted a barrier that must be overcome by all existing approaches to attain polynomial running time. This is due to the existence of worst case instances which force these algorithms to explore a large portion of the tree. As an attempt to overcome this barrier, we propose a strategy iteration framework which can be applied on any universal tree. It is at least as fast as its value iteration counterparts, while allowing one to take bigger leaps in the universal tree. Our main technical contribution is an efficient method for computing the least fixed point of 1-player games. This is achieved via a careful adaptation of shortest path algorithms to the setting of ordered trees. By plugging in the universal tree of Jurdziński and Lazić (LICS 2017), or the Strahler universal tree of Daviaud et al. (ICALP 2020), we obtain instantiations of the general framework that take time O(mn²log nlog d) and O(mn²log³ n log d) respectively per iteration.
Cite as

Zhuan Khye Koh and Georg Loho. Beyond Value Iteration for Parity Games: Strategy Iteration with Universal Trees. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 63:1-63:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{koh_et_al:LIPIcs.MFCS.2022.63,
  author =	{Koh, Zhuan Khye and Loho, Georg},
  title =	{{Beyond Value Iteration for Parity Games: Strategy Iteration with Universal Trees}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{63:1--63:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.63},
  URN =		{urn:nbn:de:0030-drops-168619},
  doi =		{10.4230/LIPIcs.MFCS.2022.63},
  annote =	{Keywords: parity games, strategy iteration, value iteration, progress measure, universal trees}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.24
Signed Tropical Convexity

Authors: Georg Loho and László A. Végh

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract
We establish a new notion of tropical convexity for signed tropical numbers. We provide several equivalent descriptions involving balance relations and intersections of open halfspaces as well as the image of a union of polytopes over Puiseux series and hyperoperations. Along the way, we deduce a new Farkas' lemma and Fourier-Motzkin elimination without the non-negativity restriction on the variables. This leads to a Minkowski-Weyl theorem for polytopes over the signed tropical numbers.
Cite as

Georg Loho and László A. Végh. Signed Tropical Convexity. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 24:1-24:35, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{loho_et_al:LIPIcs.ITCS.2020.24,
  author =	{Loho, Georg and V\'{e}gh, L\'{a}szl\'{o} A.},
  title =	{{Signed Tropical Convexity}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{24:1--24:35},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.24},
  URN =		{urn:nbn:de:0030-drops-117097},
  doi =		{10.4230/LIPIcs.ITCS.2020.24},
  annote =	{Keywords: tropical convexity, signed tropical numbers, Farkas' lemma}
}
Document
Multimedia Contribution
DOI: 10.4230/LIPIcs.SoCG.2017.66
MatchTheNet - An Educational Game on 3-Dimensional Polytopes (Multimedia Contribution)

Authors: Michael Joswig, Georg Loho, Benjamin Lorenz, and Rico Raber

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Abstract
We present an interactive game which challenges a single player to match 3-dimensional polytopes to their planar nets. It is open source, and it runs in standard web browsers.
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Michael Joswig, Georg Loho, Benjamin Lorenz, and Rico Raber. MatchTheNet - An Educational Game on 3-Dimensional Polytopes (Multimedia Contribution). In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 66:1-66:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{joswig_et_al:LIPIcs.SoCG.2017.66,
  author =	{Joswig, Michael and Loho, Georg and Lorenz, Benjamin and Raber, Rico},
  title =	{{MatchTheNet - An Educational Game on 3-Dimensional Polytopes}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{66:1--66:5},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.66},
  URN =		{urn:nbn:de:0030-drops-72435},
  doi =		{10.4230/LIPIcs.SoCG.2017.66},
  annote =	{Keywords: three-dimensional convex polytopes, unfoldings}
}
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