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**Published in:** LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

In this paper, we introduce a technique we call geometric amortization for enumeration algorithms, which can be used to make the delay of enumeration algorithms more regular with little overhead on the space it uses. More precisely, we consider enumeration algorithms having incremental linear delay, that is, algorithms enumerating, on input x, a set A(x) such that for every t ≤ ♯ A(x), it outputs at least t solutions in time O(t⋅p(|x|)), where p is a polynomial. We call p the incremental delay of the algorithm. While it is folklore that one can transform such an algorithm into an algorithm with maximal delay O(p(|x|)), the naive transformation may use exponential space. We show that, using geometric amortization, such an algorithm can be transformed into an algorithm with delay O(p(|x|)log(♯A(x))) and space O(s log(♯A(x))) where s is the space used by the original algorithm. In terms of complexity, we prove that classes DelayP and IncP₁ with polynomial space coincide.
We apply geometric amortization to show that one can trade the delay of flashlight search algorithms for their average delay up to a factor of O(log(♯A(x))). We illustrate how this tradeoff is advantageous for the enumeration of solutions of DNF formulas.

Florent Capelli and Yann Strozecki. Geometric Amortization of Enumeration Algorithms. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 18:1-18:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{capelli_et_al:LIPIcs.STACS.2023.18, author = {Capelli, Florent and Strozecki, Yann}, title = {{Geometric Amortization of Enumeration Algorithms}}, booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)}, pages = {18:1--18:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-266-2}, ISSN = {1868-8969}, year = {2023}, volume = {254}, editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.18}, URN = {urn:nbn:de:0030-drops-176703}, doi = {10.4230/LIPIcs.STACS.2023.18}, annote = {Keywords: Enumeration, Polynomial Delay, Incremental Delay, Amortization} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

We present a generic strategy improvement algorithm (GSIA) to find an optimal strategy of simple stochastic games (SSG). We prove the correctness of GSIA, and derive a general complexity bound, which implies and improves on the results of several articles. First, we remove the assumption that the SSG is stopping, which is usually obtained by a polynomial blowup of the game. Second, we prove a tight bound on the denominator of the values associated to a strategy, and use it to prove that all strategy improvement algorithms are in fact fixed parameter tractable in the number r of random vertices. All known strategy improvement algorithms can be seen as instances of GSIA, which allows to analyze the complexity of converge from below by Condon [Condon, 1993] and to propose a class of algorithms generalising Gimbert and Horn’s algorithm [Gimbert and Horn, 2008; Gimbert and Horn, 2009]. These algorithms terminate in at most r! iterations, and for binary SSGs, they do less iterations than the current best deterministic algorithm given by Ibsen-Jensen and Miltersen [Ibsen-Jensen and Miltersen, 2012].

David Auger, Xavier Badin de Montjoye, and Yann Strozecki. A Generic Strategy Improvement Method for Simple Stochastic Games. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 12:1-12:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{auger_et_al:LIPIcs.MFCS.2021.12, author = {Auger, David and Badin de Montjoye, Xavier and Strozecki, Yann}, title = {{A Generic Strategy Improvement Method for Simple Stochastic Games}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {12:1--12:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.12}, URN = {urn:nbn:de:0030-drops-144524}, doi = {10.4230/LIPIcs.MFCS.2021.12}, annote = {Keywords: Simple Stochastic Games, Strategy Improvement, Parametrized Complexity, Stopping, Meta Algorithm, f-strategy} }

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**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

The best algorithm so far for solving Simple Stochastic Games is Ludwig’s randomized algorithm [Ludwig, 1995] which works in expected 2^{O(sqrt{n})} time. We first give a simpler iterative variant of this algorithm, using Bland’s rule from the simplex algorithm, which uses exponentially less random bits than Ludwig’s version. Then, we show how to adapt this method to the algorithm of Gimbert and Horn [Gimbert and Horn, 2008] whose worst case complexity is O(k!), where k is the number of random nodes. Our algorithm has an expected running time of 2^{O(k)}, and works for general random nodes with arbitrary outdegree and probability distribution on outgoing arcs.

David Auger, Pierre Coucheney, and Yann Strozecki. Solving Simple Stochastic Games with Few Random Nodes Faster Using Bland’s Rule. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 9:1-9:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{auger_et_al:LIPIcs.STACS.2019.9, author = {Auger, David and Coucheney, Pierre and Strozecki, Yann}, title = {{Solving Simple Stochastic Games with Few Random Nodes Faster Using Bland’s Rule}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {9:1--9:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.9}, URN = {urn:nbn:de:0030-drops-102488}, doi = {10.4230/LIPIcs.STACS.2019.9}, annote = {Keywords: simple stochastic games, randomized algorithm, parametrized complexity, strategy improvement, Bland’s rule} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

In this paper we address the problem of generating all elements obtained by the saturation of an initial set by some operations. More precisely, we prove that we can generate the closure by polymorphisms of a boolean relation with a polynomial delay. Therefore we can compute with polynomial delay the closure of a family of sets by any set of "set operations" (e.g. by union, intersection, difference, symmetric difference ...). To do so, we prove that for any set of operations F, one can decide in polynomial time whether an element belongs to the closure by F of a family of sets. When the relation is over a domain larger than two elements, we prove that our generic enumeration method fails, since the associated decision problem is NP-hard.

Arnaud Mary and Yann Strozecki. Efficient Enumeration of Solutions Produced by Closure Operations. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 52:1-52:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{mary_et_al:LIPIcs.STACS.2016.52, author = {Mary, Arnaud and Strozecki, Yann}, title = {{Efficient Enumeration of Solutions Produced by Closure Operations}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {52:1--52:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.52}, URN = {urn:nbn:de:0030-drops-57538}, doi = {10.4230/LIPIcs.STACS.2016.52}, annote = {Keywords: enumeration, set saturation, polynomial delay, Post’s lattice} }

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**Published in:** LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

Polynomial identity testing and arithmetic circuit lower bounds are two central questions in algebraic complexity theory. It is an intriguing fact that these questions are actually related.
One of the authors of the present paper has recently proposed
a "real tau-conjecture" which is inspired by this connection.
The real tau-conjecture states that the number of real roots of
a sum of products of sparse univariate polynomials should be
polynomially bounded. It implies a superpolynomial lower bound on the
size of arithmetic circuits computing the permanent polynomial.
In this paper we show that the real tau-conjecture holds true for a restricted class of sums of products of sparse polynomials.
This result yields lower bounds for a restricted class of depth-4 circuits: we show that polynomial size circuits from this class cannot compute the permanent, and we also give a deterministic polynomial identity testing algorithm for the same class of circuits.

Bruno Grenet, Pascal Koiran, Natacha Portier, and Yann Strozecki. The Limited Power of Powering: Polynomial Identity Testing and a Depth-four Lower Bound for the Permanent. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 127-139, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)

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@InProceedings{grenet_et_al:LIPIcs.FSTTCS.2011.127, author = {Grenet, Bruno and Koiran, Pascal and Portier, Natacha and Strozecki, Yann}, title = {{The Limited Power of Powering: Polynomial Identity Testing and a Depth-four Lower Bound for the Permanent}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)}, pages = {127--139}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-34-7}, ISSN = {1868-8969}, year = {2011}, volume = {13}, editor = {Chakraborty, Supratik and Kumar, Amit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.127}, URN = {urn:nbn:de:0030-drops-33501}, doi = {10.4230/LIPIcs.FSTTCS.2011.127}, annote = {Keywords: Algebraic Complexity, Sparse Polynomials, Descartes' Rule of Signs, Lower Bound for the Permanent, Polynomial Identity Testing} }

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**Published in:** LIPIcs, Volume 12, Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL (2011)

We consider query problems defined by first order formulas of the form F(x,T) with free first order and second order variables and study the data complexity of enumerating results of such queries. By considering the number of alternations in the quantifier prefixes of formulas, we show that such query problems either admit a constant delay or a polynomial delay enumeration algorithm or are hard to enumerate.
We also exhibit syntactically defined fragments inside the hard cases that still admit good enumeration algorithms and discuss the case of some restricted classes of database structures as inputs.

Arnaud Durand and Yann Strozecki. Enumeration Complexity of Logical Query Problems with Second-order Variables. In Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 12, pp. 189-202, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)

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@InProceedings{durand_et_al:LIPIcs.CSL.2011.189, author = {Durand, Arnaud and Strozecki, Yann}, title = {{Enumeration Complexity of Logical Query Problems with Second-order Variables}}, booktitle = {Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL}, pages = {189--202}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-32-3}, ISSN = {1868-8969}, year = {2011}, volume = {12}, editor = {Bezem, Marc}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2011.189}, URN = {urn:nbn:de:0030-drops-32313}, doi = {10.4230/LIPIcs.CSL.2011.189}, annote = {Keywords: descriptive complexity, enumeration, query problem} }

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