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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

In the consensus halving problem we are given n agents with valuations over the interval [0,1]. The goal is to divide the interval into at most n+1 pieces (by placing at most n cuts), which may be combined to give a partition of [0,1] into two sets valued equally by all agents. The existence of a solution may be established by the Borsuk-Ulam theorem. We consider the task of computing an approximation of an exact solution of the consensus halving problem, where the valuations are given by distribution functions computed by algebraic circuits. Here approximation refers to computing a point that is ε-close to an exact solution, also called strong approximation. We show that this task is polynomial time equivalent to computing an approximation to an exact solution of the Borsuk-Ulam search problem defined by a continuous function that is computed by an algebraic circuit.
The Borsuk-Ulam search problem is the defining problem of the complexity class BU. We introduce a new complexity class BBU to also capture an alternative formulation of the Borsuk-Ulam theorem from a computational point of view. We investigate their relationship and prove several structural results for these classes as well as for the complexity class FIXP.

Eleni Batziou, Kristoffer Arnsfelt Hansen, and Kasper Høgh. Strong Approximate Consensus Halving and the Borsuk-Ulam Theorem. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{batziou_et_al:LIPIcs.ICALP.2021.24, author = {Batziou, Eleni and Hansen, Kristoffer Arnsfelt and H{\o}gh, Kasper}, title = {{Strong Approximate Consensus Halving and the Borsuk-Ulam Theorem}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {24:1--24:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.24}, URN = {urn:nbn:de:0030-drops-140939}, doi = {10.4230/LIPIcs.ICALP.2021.24}, annote = {Keywords: Consensus halving, Computational Complexity, Borsuk-Ulam} }

Document

**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

We show that the problem of deciding whether in a multi-player perfect information recursive game (i.e. a stochastic game with terminal rewards) there exists a stationary Nash equilibrium ensuring each player a certain payoff is ∃ℝ-complete. Our result holds for acyclic games, where a Nash equilibrium may be computed efficiently by backward induction, and even for deterministic acyclic games with non-negative terminal rewards. We further extend our results to the existence of Nash equilibria where a single player is surely winning. Combining our result with known gadget games without any stationary Nash equilibrium, we obtain that for cyclic games, just deciding existence of any stationary Nash equilibrium is ∃ℝ-complete. This holds for reach-a-set games, stay-in-a-set games, and for deterministic recursive games.

Kristoffer Arnsfelt Hansen and Steffan Christ Sølvsten. ∃ℝ-Completeness of Stationary Nash Equilibria in Perfect Information Stochastic Games. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 45:1-45:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{hansen_et_al:LIPIcs.MFCS.2020.45, author = {Hansen, Kristoffer Arnsfelt and S{\o}lvsten, Steffan Christ}, title = {{\exists\mathbb{R}-Completeness of Stationary Nash Equilibria in Perfect Information Stochastic Games}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {45:1--45:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.45}, URN = {urn:nbn:de:0030-drops-127113}, doi = {10.4230/LIPIcs.MFCS.2020.45}, annote = {Keywords: Existential Theory of the Reals, Stationary Nash Equilibrium, Perfect Information Stochastic Games} }

Document

**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Low rank approximation of matrices is an important tool in machine learning. Given a data matrix, low rank approximation helps to find factors, patterns, and provides concise representations for the data. Research on low rank approximation usually focuses on real matrices. However, in many applications data are binary (categorical) rather than continuous. This leads to the problem of low rank approximation of binary matrices. Here we are given a d x n binary matrix A and a small integer k < d. The goal is to find two binary matrices U and V of sizes d x k and k x n respectively, so that the Frobenius norm of A - U V is minimized. There are two models of this problem, depending on the definition of the dot product of binary vectors: The GF(2) model and the Boolean semiring model. Unlike low rank approximation of a real matrix which can be efficiently solved by Singular Value Decomposition, we show that approximation of a binary matrix is NP-hard, even for k=1.
In this paper, our main concern is the problem of Column Subset Selection (CSS), in which the low rank matrix U must be formed by k columns of the data matrix, and we are interested in the approximation ratio achievable by CSS for binary matrices. For the GF(2) model, we show that CSS has approximation ratio bounded by k/2+1+k/(2(2^k-1)) and this is asymptotically tight. For the Boolean model, it turns out that CSS is no longer sufficient to obtain a bound. We then develop a Generalized CSS (GCSS) procedure in which the columns of U are generated from Boolean formulas operating bitwise on selected columns of the data matrix. We show that the approximation ratio achieved by GCSS is bounded by 2^(k-1)+1, and argue that an exponential dependency on k is seems inherent.

Chen Dan, Kristoffer Arnsfelt Hansen, He Jiang, Liwei Wang, and Yuchen Zhou. Low Rank Approximation of Binary Matrices: Column Subset Selection and Generalizations. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 41:1-41:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dan_et_al:LIPIcs.MFCS.2018.41, author = {Dan, Chen and Hansen, Kristoffer Arnsfelt and Jiang, He and Wang, Liwei and Zhou, Yuchen}, title = {{Low Rank Approximation of Binary Matrices: Column Subset Selection and Generalizations}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {41:1--41:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.41}, URN = {urn:nbn:de:0030-drops-96239}, doi = {10.4230/LIPIcs.MFCS.2018.41}, annote = {Keywords: Approximation Algorithms, Low Rank Approximation, Binary Matrices} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

We consider two player, zero-sum, finite-state concurrent reachability games, played for an infinite number of rounds, where in every round, each player simultaneously and independently of the other players chooses an action, whereafter the successor state is determined by a probability distribution given by the current state and the chosen actions. Player 1 wins iff a designated goal state is eventually visited. We are interested in the complexity of stationary strategies measured by their patience, which is defined as the inverse of the smallest non-zero probability employed. Our main results are as follows: We show that: (i) the optimal bound on the patience of optimal and epsilon-optimal strategies, for both players is doubly exponential; and (ii) even in games with a single non-absorbing state exponential (in the number of actions) patience is necessary.

Krishnendu Chatterjee, Kristoffer Arnsfelt Hansen, and Rasmus Ibsen-Jensen. Strategy Complexity of Concurrent Safety Games. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 55:1-55:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chatterjee_et_al:LIPIcs.MFCS.2017.55, author = {Chatterjee, Krishnendu and Hansen, Kristoffer Arnsfelt and Ibsen-Jensen, Rasmus}, title = {{Strategy Complexity of Concurrent Safety Games}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {55:1--55:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.55}, URN = {urn:nbn:de:0030-drops-81203}, doi = {10.4230/LIPIcs.MFCS.2017.55}, annote = {Keywords: Concurrent games, Reachability and safety, Patience of strategies} }

Document

**Published in:** Dagstuhl Seminar Proceedings, Volume 8381, Computational Complexity of Discrete Problems (2008)

We consider the class of constant depth AND/OR circuits augmented with
a layer of modular counting gates at the bottom layer, i.e ${AC}^0 circ {MOD}_m$ circuits. We show that the following
holds for several types of gates $G$: by adding a gate of type $G$ at
the output, it is possible to obtain an equivalent randomized depth 2
circuit of quasipolynomial size consisting of a gate of type $G$ at
the output and a layer of modular counting gates, i.e $G circ {MOD}_m$ circuits. The types of gates $G$ we consider are modular
counting gates and threshold-style gates. For all of these, strong
lower bounds are known for (deterministic) $G circ {MOD}_m$
circuits.

Kristoffer Arnsfelt Hansen. Depth Reduction for Circuits with a Single Layer of Modular Counting Gates. In Computational Complexity of Discrete Problems. Dagstuhl Seminar Proceedings, Volume 8381, pp. 1-11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{hansen:DagSemProc.08381.4, author = {Hansen, Kristoffer Arnsfelt}, title = {{Depth Reduction for Circuits with a Single Layer of Modular Counting Gates}}, booktitle = {Computational Complexity of Discrete Problems}, pages = {1--11}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8381}, editor = {Peter Bro Miltersen and R\"{u}diger Reischuk and Georg Schnitger and Dieter van Melkebeek}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.08381.4}, URN = {urn:nbn:de:0030-drops-17824}, doi = {10.4230/DagSemProc.08381.4}, annote = {Keywords: Boolean Circuits, Randomized Polynomials, Fourier sums} }

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