11 Search Results for "Gaubert, Stéphane"

Document
DOI: 10.4230/LIPIcs.STACS.2026.31
Lower Bounds for Ranking-Based Pivot Rules

Authors: Yann Disser, Georg Loho, Matthew Maat, and Nils Mosis

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
The existence of a polynomial pivot rule for the simplex method for linear programming, policy iteration for Markov decision processes, and strategy improvement for parity games each are prominent open problems in their respective fields. While numerous natural candidates for efficient rules have been eliminated, all existing lower bound constructions are tailored to individual or small sets of pivot rules. We introduce a unified framework for formalizing classes of rules according to the information about the input that they rely on. Within this framework, we show lower bounds for ranking-based classes of rules that base their decisions on orderings of the improving pivot steps induced by the underlying data. Our first result is a superpolynomial lower bound for strategy improvement, obtained via a family of sink parity games, which applies to memory-based generalizations of Bland’s rule that only access the input by comparing the ranks of improving edges in some global order. Our second result is a subexponential lower bound for policy iteration, obtained via a family of Markov decision processes, which applies to memoryless rules that only access the input by comparing improving actions according to their ranks in a global order, their reduced costs, and the associated improvements in objective value. Both results carry over to the simplex method for linear programming.
Cite as

Yann Disser, Georg Loho, Matthew Maat, and Nils Mosis. Lower Bounds for Ranking-Based Pivot Rules. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 31:1-31:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{disser_et_al:LIPIcs.STACS.2026.31,
  author =	{Disser, Yann and Loho, Georg and Maat, Matthew and Mosis, Nils},
  title =	{{Lower Bounds for Ranking-Based Pivot Rules}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{31:1--31:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.31},
  URN =		{urn:nbn:de:0030-drops-255207},
  doi =		{10.4230/LIPIcs.STACS.2026.31},
  annote =	{Keywords: lower bounds, Markov decision processes, parity games, pivot rules, policy iteration, simplex method}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2025.145
Reducing Stochastic Games to Semidefinite Programming

Authors: Manuel Bodirsky, Georg Loho, and Mateusz Skomra

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
We present a polynomial-time reduction from max-average constraints to the feasibility problem for semidefinite programs. This shows that Condon’s simple stochastic games, stochastic mean payoff games, and in particular mean payoff games and parity games can all be reduced to semidefinite programming.
Cite as

Manuel Bodirsky, Georg Loho, and Mateusz Skomra. Reducing Stochastic Games to Semidefinite Programming. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 145:1-145:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bodirsky_et_al:LIPIcs.ICALP.2025.145,
  author =	{Bodirsky, Manuel and Loho, Georg and Skomra, Mateusz},
  title =	{{Reducing Stochastic Games to Semidefinite Programming}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{145:1--145:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.145},
  URN =		{urn:nbn:de:0030-drops-235224},
  doi =		{10.4230/LIPIcs.ICALP.2025.145},
  annote =	{Keywords: Mean-payoff games, stochastic games, semidefinite programming, max-average constraints, max-atom problem}
}
Document
DOI: 10.4230/LIPIcs.STACS.2025.8
Tropical Proof Systems: Between R(CP) and Resolution

Authors: Yaroslav Alekseev, Dima Grigoriev, and Edward A. Hirsch

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
Propositional proof complexity deals with the lengths of polynomial-time verifiable proofs for Boolean tautologies. An abundance of proof systems is known, including algebraic and semialgebraic systems, which work with polynomial equations and inequalities, respectively. The most basic algebraic proof system is based on Hilbert’s Nullstellensatz [Paul Beame et al., 1996]. Tropical ("min-plus") arithmetic has many applications in various areas of mathematics. The operations are the real addition (as the tropical multiplication) and the minimum (as the tropical addition). Recently, [Bertram and Easton, 2017; Dima Grigoriev and Vladimir V. Podolskii, 2018; Joo and Mincheva, 2018] demonstrated a version of Nullstellensatz in the tropical setting. In this paper we introduce (semi)algebraic proof systems that use min-plus arithmetic. For the dual-variable encoding of Boolean variables (two tropical variables x and x ̅ per one Boolean variable x) and {0,1}-encoding of the truth values, we prove that a static (Nullstellensatz-based) tropical proof system polynomially simulates daglike resolution and also has short proofs for the propositional pigeon-hole principle. Its dynamic version strengthened by an additional derivation rule (a tropical analogue of resolution by linear inequality) is equivalent to the system Res(LP) (aka R(LP)), which derives nonnegative linear combinations of linear inequalities; this latter system is known to polynomially simulate Krajíček’s Res(CP) (aka R(CP)) with unary coefficients. Therefore, tropical proof systems give a finer hierarchy of proof systems below Res(LP) for which we still do not have exponential lower bounds. While the "driving force" in Res(LP) is resolution by linear inequalities, dynamic tropical systems are driven solely by the transitivity of the order, and static tropical proof systems are based on reasoning about differences between the input linear functions. For the truth values encoded by {0,∞}, dynamic tropical proofs are equivalent to Res(∞), which is a small-depth Frege system called also DNF resolution. Finally, we provide a lower bound on the size of derivations of a much simplified tropical version of the {Binary Value Principle} in a static tropical proof system. Also, we establish the non-deducibility of the tropical resolution rule in this system and discuss axioms for Boolean logic that do not use dual variables. In this extended abstract, full proofs are omitted.
Cite as

Yaroslav Alekseev, Dima Grigoriev, and Edward A. Hirsch. Tropical Proof Systems: Between R(CP) and Resolution. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{alekseev_et_al:LIPIcs.STACS.2025.8,
  author =	{Alekseev, Yaroslav and Grigoriev, Dima and Hirsch, Edward A.},
  title =	{{Tropical Proof Systems: Between R(CP) and Resolution}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{8:1--8:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.8},
  URN =		{urn:nbn:de:0030-drops-228332},
  doi =		{10.4230/LIPIcs.STACS.2025.8},
  annote =	{Keywords: Cutting Planes, Nullstellensatz refutations, Res(CP), semi-algebraic proofs, tropical proof systems, tropical semiring}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2024.147
Smoothed Analysis of Deterministic Discounted and Mean-Payoff Games

Authors: Bruno Loff and Mateusz Skomra

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We devise a policy-iteration algorithm for deterministic two-player discounted and mean-payoff games, that runs in polynomial time with high probability, on any input where each payoff is chosen independently from a sufficiently random distribution and the underlying graph of the game is ergodic. This includes the case where an arbitrary set of payoffs has been perturbed by a Gaussian, showing for the first time that deterministic two-player games can be solved efficiently, in the sense of smoothed analysis. More generally, we devise a condition number for deterministic discounted and mean-payoff games played on ergodic graphs, and show that our algorithm runs in time polynomial in this condition number. Our result confirms a previous conjecture of Boros et al., which was claimed as a theorem [Boros et al., 2011] and later retracted [Boros et al., 2018]. It stands in contrast with a recent counter-example by Christ and Yannakakis [Christ and Yannakakis, 2023], showing that Howard’s policy-iteration algorithm does not run in smoothed polynomial time on stochastic single-player mean-payoff games. Our approach is inspired by the analysis of random optimal assignment instances by Frieze and Sorkin [Frieze and Sorkin, 2007], and the analysis of bias-induced policies for mean-payoff games by Akian, Gaubert and Hochart [Akian et al., 2018].
Cite as

Bruno Loff and Mateusz Skomra. Smoothed Analysis of Deterministic Discounted and Mean-Payoff Games. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 147:1-147:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{loff_et_al:LIPIcs.ICALP.2024.147,
  author =	{Loff, Bruno and Skomra, Mateusz},
  title =	{{Smoothed Analysis of Deterministic Discounted and Mean-Payoff Games}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{147:1--147:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.147},
  URN =		{urn:nbn:de:0030-drops-202908},
  doi =		{10.4230/LIPIcs.ICALP.2024.147},
  annote =	{Keywords: Mean-payoff games, discounted games, policy iteration, smoothed analysis}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2023.10
Solving Irreducible Stochastic Mean-Payoff Games and Entropy Games by Relative Krasnoselskii-Mann Iteration

Authors: Marianne Akian, Stéphane Gaubert, Ulysse Naepels, and Basile Terver

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Abstract
We analyse an algorithm solving stochastic mean-payoff games, combining the ideas of relative value iteration and of Krasnoselskii-Mann damping. We derive parameterized complexity bounds for several classes of games satisfying irreducibility conditions. We show in particular that an ε-approximation of the value of an irreducible concurrent stochastic game can be computed in a number of iterations in O(|log(ε)|) where the constant in the O(⋅) is explicit, depending on the smallest non-zero transition probabilities. This should be compared with a bound in O(ε^{-1}|log(ε)|) obtained by Chatterjee and Ibsen-Jensen (ICALP 2014) for the same class of games, and to a O(ε^{-1}) bound by Allamigeon, Gaubert, Katz and Skomra (ICALP 2022) for turn-based games. We also establish parameterized complexity bounds for entropy games, a class of matrix multiplication games introduced by Asarin, Cervelle, Degorre, Dima, Horn and Kozyakin. We derive these results by methods of variational analysis, establishing contraction properties of the relative Krasnoselskii-Mann iteration with respect to Hilbert’s semi-norm.
Cite as

Marianne Akian, Stéphane Gaubert, Ulysse Naepels, and Basile Terver. Solving Irreducible Stochastic Mean-Payoff Games and Entropy Games by Relative Krasnoselskii-Mann Iteration. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{akian_et_al:LIPIcs.MFCS.2023.10,
  author =	{Akian, Marianne and Gaubert, St\'{e}phane and Naepels, Ulysse and Terver, Basile},
  title =	{{Solving Irreducible Stochastic Mean-Payoff Games and Entropy Games by Relative Krasnoselskii-Mann Iteration}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{10:1--10:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.10},
  URN =		{urn:nbn:de:0030-drops-185448},
  doi =		{10.4230/LIPIcs.MFCS.2023.10},
  annote =	{Keywords: Stochastic mean-payoff games, concurrent games, entropy games, relative value iteration, Krasnoselskii-Mann fixed point algorithm, Hilbert projective metric}
}
Document
DOI: 10.4230/LITES.8.2.1
Safety Verification of Networked Control Systems by Complex Zonotopes

Authors: Arvind Adimoolam and Thao Dang

Published in: LITES, Volume 8, Issue 2 (2022): Special Issue on Distributed Hybrid Systems. Leibniz Transactions on Embedded Systems, Volume 8, Issue 2

Abstract
Networked control systems (NCS) are widely used in real world applications because of their advantages, such as remote operability and reduced installation costs. However, they are prone to various inaccuracies in execution like delays, packet dropouts, inaccurate sensing and quantization errors. To ensure safety of NCS, their models have to be verified under the consideration of aforementioned uncertainties. In this paper, we tackle the problem of verifying safety of models of NCS under uncertain sampling time, inaccurate output measurement or estimation, and unknown disturbance input. Unbounded-time safety verification requires approximation of reachable sets by invariants, whose computation involves set operations. For uncertain linear dynamics, two important set operations for invariant computation are linear transformation and Minkowski sum operations. Zonotopes have the advantage that linear transformation and Minkowski sum operations can be efficiently approximated. However, they can not encode directions of convergence of trajectories along complex eigenvectors, which is closely related to encoding invariants. Therefore, we extend zonotopes to the complex valued domain by a representation called complex zonotope, which can capture contraction along complex eigenvectors for determining invariants. We prove a related mathematical result that in case of accurate feedback sampling, a complex zonotope will represent an invariant for a stable NCS. In addition, we propose an algorithm to verify the general case based on complex zonotopes, when there is uncertainty in sampling time and in input. We demonstrate the efficiency of our algorithm on benchmark examples and compare it with a state-of-the-art verification tool.
Cite as

Arvind Adimoolam and Thao Dang. Safety Verification of Networked Control Systems by Complex Zonotopes. In LITES, Volume 8, Issue 2 (2022): Special Issue on Distributed Hybrid Systems. Leibniz Transactions on Embedded Systems, Volume 8, Issue 2, pp. 01:1-01:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Article{adimoolam_et_al:LITES.8.2.1,
  author =	{Adimoolam, Arvind and Dang, Thao},
  title =	{{Safety Verification of Networked Control Systems by Complex Zonotopes}},
  journal =	{Leibniz Transactions on Embedded Systems},
  pages =	{01:1--01:22},
  ISSN =	{2199-2002},
  year =	{2022},
  volume =	{8},
  number =	{2},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LITES.8.2.1},
  URN =		{urn:nbn:de:0030-drops-192934},
  doi =		{10.4230/LITES.8.2.1},
  annote =	{Keywords: Safety Verification, Networked Control System, Reachability Analysis, Complex Zonotope}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2022.110
Universal Complexity Bounds Based on Value Iteration and Application to Entropy Games

Authors: Xavier Allamigeon, Stéphane Gaubert, Ricardo D. Katz, and Mateusz Skomra

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
We develop value iteration-based algorithms to solve in a unified manner different classes of combinatorial zero-sum games with mean-payoff type rewards. These algorithms rely on an oracle, evaluating the dynamic programming operator up to a given precision. We show that the number of calls to the oracle needed to determine exact optimal (positional) strategies is, up to a factor polynomial in the dimension, of order R/sep, where the "separation" sep is defined as the minimal difference between distinct values arising from strategies, and R is a metric estimate, involving the norm of approximate sub and super-eigenvectors of the dynamic programming operator. We illustrate this method by two applications. The first one is a new proof, leading to improved complexity estimates, of a theorem of Boros, Elbassioni, Gurvich and Makino, showing that turn-based mean payoff games with a fixed number of random positions can be solved in pseudo-polynomial time. The second one concerns entropy games, a model introduced by Asarin, Cervelle, Degorre, Dima, Horn and Kozyakin. The rank of an entropy game is defined as the maximal rank among all the ambiguity matrices determined by strategies of the two players. We show that entropy games with a fixed rank, in their original formulation, can be solved in polynomial time, and that an extension of entropy games incorporating weights can be solved in pseudo-polynomial time under the same fixed rank condition.
Cite as

Xavier Allamigeon, Stéphane Gaubert, Ricardo D. Katz, and Mateusz Skomra. Universal Complexity Bounds Based on Value Iteration and Application to Entropy Games. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 110:1-110:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{allamigeon_et_al:LIPIcs.ICALP.2022.110,
  author =	{Allamigeon, Xavier and Gaubert, St\'{e}phane and Katz, Ricardo D. and Skomra, Mateusz},
  title =	{{Universal Complexity Bounds Based on Value Iteration and Application to Entropy Games}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{110:1--110:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.110},
  URN =		{urn:nbn:de:0030-drops-164511},
  doi =		{10.4230/LIPIcs.ICALP.2022.110},
  annote =	{Keywords: Mean-payoff games, entropy games, value iteration, Perron root, separation bounds, parameterized complexity}
}
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
DOI: 10.4230/DagRep.6.11.168
Algorithms and Effectivity in Tropical Mathematics and Beyond (Dagstuhl Seminar 16482)

Authors: Stéphane Gaubert, Dima Grigoriev, Michael Joswig, and Thorsten Theobald

Published in: Dagstuhl Reports, Volume 6, Issue 11 (2017)

Abstract
This report documents the Dagstuhl Seminar on Algorithms and Effectivity in Tropical Mathematics and Beyond, which took place from November 27 -- December 02, 2016. The report contains an executive summary as well as abstracts of the talks which reflect recent progress in the topic of the meeting.
Cite as

Stéphane Gaubert, Dima Grigoriev, Michael Joswig, and Thorsten Theobald. Algorithms and Effectivity in Tropical Mathematics and Beyond (Dagstuhl Seminar 16482). In Dagstuhl Reports, Volume 6, Issue 11, pp. 168-184, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@Article{gaubert_et_al:DagRep.6.11.168,
  author =	{Gaubert, St\'{e}phane and Grigoriev, Dima and Joswig, Michael and Theobald, Thorsten},
  title =	{{Algorithms and Effectivity in Tropical Mathematics and Beyond (Dagstuhl Seminar 16482)}},
  pages =	{168--184},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2017},
  volume =	{6},
  number =	{11},
  editor =	{Gaubert, St\'{e}phane and Grigoriev, Dima and Joswig, Michael and Theobald, Thorsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.6.11.168},
  URN =		{urn:nbn:de:0030-drops-71073},
  doi =		{10.4230/DagRep.6.11.168},
  annote =	{Keywords: Algorithms in tropical mathematics, complexity, effective bounds, optimization, zero-sum games}
}
Document
DOI: 10.4230/LIPIcs.STACS.2017.6
The Operator Approach to Entropy Games

Authors: Marianne Akian, Stéphane Gaubert, Julien Grand-Clément, and Jérémie Guillaud

Published in: LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Abstract
Entropy games and matrix multiplication games have been recently introduced by Asarin et al. They model the situation in which one player (Despot) wishes to minimize the growth rate of a matrix product, whereas the other player (Tribune) wishes to maximize it. We develop an operator approach to entropy games. This allows us to show that entropy games can be cast as stochastic mean payoff games in which some action spaces are simplices and payments are given by a relative entropy (Kullback-Leibler divergence). In this way, we show that entropy games with a fixed number of states belonging to Despot can be solved in polynomial time. This approach also allows us to solve these games by a policy iteration algorithm, which we compare with the spectral simplex algorithm developed by Protasov.
Cite as

Marianne Akian, Stéphane Gaubert, Julien Grand-Clément, and Jérémie Guillaud. The Operator Approach to Entropy Games. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{akian_et_al:LIPIcs.STACS.2017.6,
  author =	{Akian, Marianne and Gaubert, St\'{e}phane and Grand-Cl\'{e}ment, Julien and Guillaud, J\'{e}r\'{e}mie},
  title =	{{The Operator Approach to Entropy Games}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{6:1--6:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Vollmer, Heribert and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.6},
  URN =		{urn:nbn:de:0030-drops-70260},
  doi =		{10.4230/LIPIcs.STACS.2017.6},
  annote =	{Keywords: Stochastic games, Shapley operators, policy iteration, Perron eigenvalues, Risk sensitive control}
}
Document
DOI: 10.4230/LIPIcs.STACS.2010.2443
The Tropical Double Description Method

Authors: Xavier Allamigeon, Stéphane Gaubert, and Éric Goubault

Published in: LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

Abstract
We develop a tropical analogue of the classical double description method allowing one to compute an internal representation (in terms of vertices) of a polyhedron defined externally (by inequalities). The heart of the tropical algorithm is a characterization of the extreme points of a polyhedron in terms of a system of constraints which define it. We show that checking the extremality of a point reduces to checking whether there is only one minimal strongly connected component in an hypergraph. The latter problem can be solved in almost linear time, which allows us to eliminate quickly redundant generators. We report extensive tests (including benchmarks from an application to static analysis) showing that the method outperforms experimentally the previous ones by orders of magnitude. The present tools also lead to worst case bounds which improve the ones provided by previous methods.
Cite as

Xavier Allamigeon, Stéphane Gaubert, and Éric Goubault. The Tropical Double Description Method. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 47-58, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

Copy BibTex To Clipboard

@InProceedings{allamigeon_et_al:LIPIcs.STACS.2010.2443,
  author =	{Allamigeon, Xavier and Gaubert, St\'{e}phane and Goubault, \'{E}ric},
  title =	{{The Tropical Double Description Method}},
  booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{47--58},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-16-3},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{5},
  editor =	{Marion, Jean-Yves and Schwentick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2443},
  URN =		{urn:nbn:de:0030-drops-24435},
  doi =		{10.4230/LIPIcs.STACS.2010.2443},
  annote =	{Keywords: Convexity in tropical algebra, algorithmics and combinatorics of tropical polyhedra, computational geometry, discrete event systems, static analysis}
}
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