38 Search Results for "van Melkebeek, Dieter"

Document
DOI: 10.4230/LIPIcs.FSTTCS.2023.29
Leakage Resilience, Targeted Pseudorandom Generators, and Mild Derandomization of Arthur-Merlin Protocols

Authors: Dieter van Melkebeek and Nicollas Mocelin Sdroievski

Published in: LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

Abstract
Many derandomization results for probabilistic decision processes have been ported to the setting of Arthur-Merlin protocols. Whereas the ultimate goal in the first setting consists of efficient simulations on deterministic machines (BPP vs. P problem), in the second setting it is efficient simulations on nondeterministic machines (AM vs. NP problem). Two notable exceptions that have not yet been ported from the first to the second setting are the equivalence between whitebox derandomization and leakage resilience (Liu and Pass, 2023), and the equivalence between whitebox derandomization and targeted pseudorandom generators (Goldreich, 2011). We develop both equivalences for mild derandomizations of Arthur-Merlin protocols, i.e., simulations on Σ₂-machines. Our techniques also apply to natural simulation models that are intermediate between nondeterministic machines and Σ₂-machines.
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Dieter van Melkebeek and Nicollas Mocelin Sdroievski. Leakage Resilience, Targeted Pseudorandom Generators, and Mild Derandomization of Arthur-Merlin Protocols. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 29:1-29:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{vanmelkebeek_et_al:LIPIcs.FSTTCS.2023.29,
  author =	{van Melkebeek, Dieter and Mocelin Sdroievski, Nicollas},
  title =	{{Leakage Resilience, Targeted Pseudorandom Generators, and Mild Derandomization of Arthur-Merlin Protocols}},
  booktitle =	{43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)},
  pages =	{29:1--29:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-304-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{284},
  editor =	{Bouyer, Patricia and Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.29},
  URN =		{urn:nbn:de:0030-drops-194024},
  doi =		{10.4230/LIPIcs.FSTTCS.2023.29},
  annote =	{Keywords: Hardness versus randomness tradeoff, leakage resilience, Arthur-Merlin protocol, targeted hitting set generator}
}
Document
DOI: 10.4230/LIPIcs.CCC.2023.17
Instance-Wise Hardness Versus Randomness Tradeoffs for Arthur-Merlin Protocols

Authors: Dieter van Melkebeek and Nicollas Mocelin Sdroievski

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Abstract
A fundamental question in computational complexity asks whether probabilistic polynomial-time algorithms can be simulated deterministically with a small overhead in time (the BPP vs. P problem). A corresponding question in the realm of interactive proofs asks whether Arthur-Merlin protocols can be simulated nondeterministically with a small overhead in time (the AM vs. NP problem). Both questions are intricately tied to lower bounds. Prominently, in both settings blackbox derandomization, i.e., derandomization through pseudo-random generators, has been shown equivalent to lower bounds for decision problems against circuits. Recently, Chen and Tell (FOCS'21) established near-equivalences in the BPP setting between whitebox derandomization and lower bounds for multi-bit functions against algorithms on almost-all inputs. The key ingredient is a technique to translate hardness into targeted hitting sets in an instance-wise fashion based on a layered arithmetization of the evaluation of a uniform circuit computing the hard function f on the given instance. In this paper we develop a corresponding technique for Arthur-Merlin protocols and establish similar near-equivalences in the AM setting. As an example of our results in the hardness to derandomization direction, consider a length-preserving function f computable by a nondeterministic algorithm that runs in time n^a. We show that if every Arthur-Merlin protocol that runs in time n^c for c = O(log² a) can only compute f correctly on finitely many inputs, then AM is in NP. Our main technical contribution is the construction of suitable targeted hitting-set generators based on probabilistically checkable proofs for nondeterministic computations. As a byproduct of our constructions, we obtain the first result indicating that whitebox derandomization of AM may be equivalent to the existence of targeted hitting-set generators for AM, an issue raised by Goldreich (LNCS, 2011). Byproducts in the average-case setting include the first uniform hardness vs. randomness tradeoffs for AM, as well as an unconditional mild derandomization result for AM.
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Dieter van Melkebeek and Nicollas Mocelin Sdroievski. Instance-Wise Hardness Versus Randomness Tradeoffs for Arthur-Merlin Protocols. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 17:1-17:36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{vanmelkebeek_et_al:LIPIcs.CCC.2023.17,
  author =	{van Melkebeek, Dieter and Mocelin Sdroievski, Nicollas},
  title =	{{Instance-Wise Hardness Versus Randomness Tradeoffs for Arthur-Merlin Protocols}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{17:1--17:36},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.17},
  URN =		{urn:nbn:de:0030-drops-182870},
  doi =		{10.4230/LIPIcs.CCC.2023.17},
  annote =	{Keywords: Hardness versus randomness tradeoff, Arthur-Merlin protocol, targeted hitting set generator}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.119
Polynomial Identity Testing via Evaluation of Rational Functions

Authors: Dieter van Melkebeek and Andrew Morgan

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
We introduce a hitting set generator for Polynomial Identity Testing based on evaluations of low-degree univariate rational functions at abscissas associated with the variables. In spite of the univariate nature, we establish an equivalence up to rescaling with a generator introduced by Shpilka and Volkovich, which has a similar structure but uses multivariate polynomials in the abscissas. We study the power of the generator by characterizing its vanishing ideal, i.e., the set of polynomials that it fails to hit. Capitalizing on the univariate nature, we develop a small collection of polynomials that jointly produce the vanishing ideal. As corollaries, we obtain tight bounds on the minimum degree, sparseness, and partition size of set-multi-linearity in the vanishing ideal. Inspired by an alternating algebra representation, we develop a structured deterministic membership test for the vanishing ideal. As a proof of concept we rederive known derandomization results based on the generator by Shpilka and Volkovich, and present a new application for read-once oblivious arithmetic branching programs that provably transcends the usual combinatorial techniques.
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Dieter van Melkebeek and Andrew Morgan. Polynomial Identity Testing via Evaluation of Rational Functions. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 119:1-119:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{vanmelkebeek_et_al:LIPIcs.ITCS.2022.119,
  author =	{van Melkebeek, Dieter and Morgan, Andrew},
  title =	{{Polynomial Identity Testing via Evaluation of Rational Functions}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{119:1--119:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.119},
  URN =		{urn:nbn:de:0030-drops-157158},
  doi =		{10.4230/LIPIcs.ITCS.2022.119},
  annote =	{Keywords: Derandomization, Gr\"{o}bner Basis, Lower Bounds, Polynomial Identity Testing}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2021.54
Cryptographic Hardness Under Projections for Time-Bounded Kolmogorov Complexity

Authors: Eric Allender, John Gouwar, Shuichi Hirahara, and Caleb Robelle

Published in: LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

Abstract
A version of time-bounded Kolmogorov complexity, denoted KT, has received attention in the past several years, due to its close connection to circuit complexity and to the Minimum Circuit Size Problem MCSP. Essentially all results about the complexity of MCSP hold also for MKTP (the problem of computing the KT complexity of a string). Both MKTP and MCSP are hard for SZK (Statistical Zero Knowledge) under BPP-Turing reductions; neither is known to be NP-complete. Recently, some hardness results for MKTP were proved that are not (yet) known to hold for MCSP. In particular, MKTP is hard for DET (a subclass of P) under nonuniform ≤^{NC^0}_m reductions. In this paper, we improve this, to show that the complement of MKTP is hard for the (apparently larger) class NISZK_L under not only ≤^{NC^0}_m reductions but even under projections. Also, the complement of MKTP is hard for NISZK under ≤^{P/poly}_m reductions. Here, NISZK is the class of problems with non-interactive zero-knowledge proofs, and NISZK_L is the non-interactive version of the class SZK_L that was studied by Dvir et al. As an application, we provide several improved worst-case to average-case reductions to problems in NP, and we obtain a new lower bound on MKTP (which is currently not known to hold for MCSP).
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Eric Allender, John Gouwar, Shuichi Hirahara, and Caleb Robelle. Cryptographic Hardness Under Projections for Time-Bounded Kolmogorov Complexity. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 54:1-54:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{allender_et_al:LIPIcs.ISAAC.2021.54,
  author =	{Allender, Eric and Gouwar, John and Hirahara, Shuichi and Robelle, Caleb},
  title =	{{Cryptographic Hardness Under Projections for Time-Bounded Kolmogorov Complexity}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{54:1--54:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.54},
  URN =		{urn:nbn:de:0030-drops-154875},
  doi =		{10.4230/LIPIcs.ISAAC.2021.54},
  annote =	{Keywords: Kolmogorov Complexity, Interactive Proofs, Minimum Circuit Size Problem, Worst-case to Average-case Reductions}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.73
Quantum Logspace Algorithm for Powering Matrices with Bounded Norm

Authors: Uma Girish, Ran Raz, and Wei Zhan

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

Abstract
We give a quantum logspace algorithm for powering contraction matrices, that is, matrices with spectral norm at most 1. The algorithm gets as an input an arbitrary n× n contraction matrix A, and a parameter T ≤ poly(n) and outputs the entries of A^T, up to (arbitrary) polynomially small additive error. The algorithm applies only unitary operators, without intermediate measurements. We show various implications and applications of this result: First, we use this algorithm to show that the class of quantum logspace algorithms with only quantum memory and with intermediate measurements is equivalent to the class of quantum logspace algorithms with only quantum memory without intermediate measurements. This shows that the deferred-measurement principle, a fundamental principle of quantum computing, applies also for quantum logspace algorithms (without classical memory). More generally, we give a quantum algorithm with space O(S + log T) that takes as an input the description of a quantum algorithm with quantum space S and time T, with intermediate measurements (without classical memory), and simulates it unitarily with polynomially small error, without intermediate measurements. Since unitary transformations are reversible (while measurements are irreversible) an interesting aspect of this result is that it shows that any quantum logspace algorithm (without classical memory) can be simulated by a reversible quantum logspace algorithm. This proves a quantum analogue of the result of Lange, McKenzie and Tapp that deterministic logspace is equal to reversible logspace [Lange et al., 2000]. Finally, we use our results to show non-trivial classical simulations of quantum logspace learning algorithms.
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Uma Girish, Ran Raz, and Wei Zhan. Quantum Logspace Algorithm for Powering Matrices with Bounded Norm. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 73:1-73:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{girish_et_al:LIPIcs.ICALP.2021.73,
  author =	{Girish, Uma and Raz, Ran and Zhan, Wei},
  title =	{{Quantum Logspace Algorithm for Powering Matrices with Bounded Norm}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{73:1--73:20},
  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.2021.73},
  URN =		{urn:nbn:de:0030-drops-141426},
  doi =		{10.4230/LIPIcs.ICALP.2021.73},
  annote =	{Keywords: BQL, Matrix Powering, Quantum Circuit, Reversible Computation}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2018.20
Minimum Circuit Size, Graph Isomorphism, and Related Problems

Authors: Eric Allender, Joshua A. Grochow, Dieter van Melkebeek, Cristopher Moore, and Andrew Morgan

Published in: LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

Abstract
We study the computational power of deciding whether a given truth-table can be described by a circuit of a given size (the Minimum Circuit Size Problem, or MCSP for short), and of the variant denoted MKTP where circuit size is replaced by a polynomially-related Kolmogorov measure. All prior reductions from supposedly-intractable problems to MCSP / MKTP hinged on the power of MCSP / MKTP to distinguish random distributions from distributions produced by hardness-based pseudorandom generator constructions. We develop a fundamentally different approach inspired by the well-known interactive proof system for the complement of Graph Isomorphism (GI). It yields a randomized reduction with zero-sided error from GI to MKTP. We generalize the result and show that GI can be replaced by any isomorphism problem for which the underlying group satisfies some elementary properties. Instantiations include Linear Code Equivalence, Permutation Group Conjugacy, and Matrix Subspace Conjugacy. Along the way we develop encodings of isomorphism classes that are efficiently decodable and achieve compression that is at or near the information-theoretic optimum; those encodings may be of independent interest.
Cite as

Eric Allender, Joshua A. Grochow, Dieter van Melkebeek, Cristopher Moore, and Andrew Morgan. Minimum Circuit Size, Graph Isomorphism, and Related Problems. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{allender_et_al:LIPIcs.ITCS.2018.20,
  author =	{Allender, Eric and Grochow, Joshua A. and van Melkebeek, Dieter and Moore, Cristopher and Morgan, Andrew},
  title =	{{Minimum Circuit Size, Graph Isomorphism, and Related Problems}},
  booktitle =	{9th Innovations in Theoretical Computer Science Conference (ITCS 2018)},
  pages =	{20:1--20:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-060-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{94},
  editor =	{Karlin, Anna R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.20},
  URN =		{urn:nbn:de:0030-drops-83455},
  doi =		{10.4230/LIPIcs.ITCS.2018.20},
  annote =	{Keywords: Reductions between NP-intermediate problems, Graph Isomorphism, Minimum Circuit Size Problem, time-bounded Kolmogorov complexity}
}
Document
DOI: 10.4230/LIPIcs.CCC.2017.5
Derandomizing Isolation in Space-Bounded Settings

Authors: Dieter van Melkebeek and Gautam Prakriya

Published in: LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

Abstract
We study the possibility of deterministic and randomness-efficient isolation in space-bounded models of computation: Can one efficiently reduce instances of computational problems to equivalent instances that have at most one solution? We present results for the NL-complete problem of reachability on digraphs, and for the LogCFL-complete problem of certifying acceptance on shallow semi-unbounded circuits. A common approach employs small weight assignments that make the solution of minimum weight unique. The Isolation Lemma and other known procedures use Omega(n) random bits to generate weights of individual bitlength O(log(n)). We develop a derandomized version for both settings that uses O(log(n)^{3/2}) random bits and produces weights of bitlength O(log(n)^{3/2}) in logarithmic space. The construction allows us to show that every language in NL can be accepted by a nondeterministic machine that runs in polynomial time and O(log(n)^{3/2}) space, and has at most one accepting computation path on every input. Similarly, every language in LogCFL can be accepted by a nondeterministic machine equipped with a stack that does not count towards the space bound, that runs in polynomial time and O(log(n)^{3/2}) space, and has at most one accepting computation path on every input. We also show that the existence of somewhat more restricted isolations for reachability on digraphs implies that NL can be decided in logspace with polynomial advice. A similar result holds for certifying acceptance on shallow semi-unbounded circuits and LogCFL.
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Dieter van Melkebeek and Gautam Prakriya. Derandomizing Isolation in Space-Bounded Settings. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 5:1-5:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{vanmelkebeek_et_al:LIPIcs.CCC.2017.5,
  author =	{van Melkebeek, Dieter and Prakriya, Gautam},
  title =	{{Derandomizing Isolation in Space-Bounded Settings}},
  booktitle =	{32nd Computational Complexity Conference (CCC 2017)},
  pages =	{5:1--5:32},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-040-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{79},
  editor =	{O'Donnell, Ryan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.5},
  URN =		{urn:nbn:de:0030-drops-75297},
  doi =		{10.4230/LIPIcs.CCC.2017.5},
  annote =	{Keywords: Isolation lemma, derandomization, unambiguous nondeterminism, graph reachability, circuit certification}
}
Document
DOI: 10.4230/DagRep.1.3.42
Computational Complexity of Discrete Problems (Dagstuhl Seminar 11121)

Authors: Martin Grohe, Michal Koucky, Rüdiger Reischik, and Dieter van Melkebeek

Published in: Dagstuhl Reports, Volume 1, Issue 3 (2011)

Abstract
This report documents the program and the outcomes of Dagstuhl Seminar 11121 ``Computational Complexity of Discrete Problems''. The first section gives an overview of the topics covered and the organization of the meeting. Section~2 lists the talks given in chronological order. The last section contains the abstracts of the talks.
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Martin Grohe, Michal Koucky, Rüdiger Reischik, and Dieter van Melkebeek. Computational Complexity of Discrete Problems (Dagstuhl Seminar 11121). In Dagstuhl Reports, Volume 1, Issue 3, pp. 42-66, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@Article{grohe_et_al:DagRep.1.3.42,
  author =	{Grohe, Martin and Koucky, Michal and Reischik, R\"{u}diger and van Melkebeek, Dieter},
  title =	{{Computational Complexity of Discrete Problems (Dagstuhl Seminar 11121)}},
  pages =	{42--66},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2011},
  volume =	{1},
  number =	{3},
  editor =	{Grohe, Martin and Koucky, Michal and Reischik, R\"{u}diger and van Melkebeek, Dieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.1.3.42},
  URN =		{urn:nbn:de:0030-drops-31935},
  doi =		{10.4230/DagRep.1.3.42},
  annote =	{Keywords: Discrete problems, computational complexity, Turing machines, Boolean circuits, quantum computing, communication and query complexity, extractors, pseudorandomness, derandomization, approximation, coding cryptography, algorithmic learning}
}
Document
DOI: 10.4230/DagSemProc.09511.7
Satisfiability Allows No Nontrivial Sparsification Unless The Polynomial-Time Hierarchy Collapses

Authors: Holger Dell and Dieter van Melkebeek

Published in: Dagstuhl Seminar Proceedings, Volume 9511, Parameterized complexity and approximation algorithms (2010)

Abstract
Consider the following two-player communication process to decide a language $L$: The first player holds the entire input $x$ but is polynomially bounded; the second player is computationally unbounded but does not know any part of $x$; their goal is to cooperatively decide whether $x$ belongs to $L$ at small cost, where the cost measure is the number of bits of communication from the first player to the second player. For any integer $d geq 3$ and positive real $epsilon$ we show that if satisfiability for $n$-variable $d$-CNF formulas has a protocol of cost $O(n^{d-epsilon})$ then coNP is in NP/poly, which implies that the polynomial-time hierarchy collapses to its third level. The result even holds when the first player is conondeterministic, and is tight as there exists a trivial protocol for $epsilon = 0$. Under the hypothesis that coNP is not in NP/poly, our result implies tight lower bounds for parameters of interest in several areas, namely sparsification, kernelization in parameterized complexity, lossy compression, and probabilistically checkable proofs. By reduction, similar results hold for other NP-complete problems. For the vertex cover problem on $n$-vertex $d$-uniform hypergraphs, the above statement holds for any integer $d geq 2$. The case $d=2$ implies that no NP-hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of $O(k^{2-epsilon})$ edges unless coNP is in NP/poly, where $k$ denotes the size of the deletion set. Kernels consisting of $O(k^2)$ edges are known for several problems in the class, including vertex cover, feedback vertex set, and bounded-degree deletion.
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Holger Dell and Dieter van Melkebeek. Satisfiability Allows No Nontrivial Sparsification Unless The Polynomial-Time Hierarchy Collapses. In Parameterized complexity and approximation algorithms. Dagstuhl Seminar Proceedings, Volume 9511, pp. 1-29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{dell_et_al:DagSemProc.09511.7,
  author =	{Dell, Holger and van Melkebeek, Dieter},
  title =	{{Satisfiability Allows No Nontrivial Sparsification Unless The Polynomial-Time Hierarchy Collapses}},
  booktitle =	{Parameterized complexity and approximation algorithms},
  pages =	{1--29},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2010},
  volume =	{9511},
  editor =	{Erik D. Demaine and MohammadTaghi Hajiaghayi and D\'{a}niel Marx},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.09511.7},
  URN =		{urn:nbn:de:0030-drops-25043},
  doi =		{10.4230/DagSemProc.09511.7},
  annote =	{Keywords: Sparsification, Kernelization, Parameterized Complexity, Probabilistically Checkable Proofs, Satisfiability, Vertex Cover}
}
Document
DOI: 10.4230/DagSemProc.08381.4
Depth Reduction for Circuits with a Single Layer of Modular Counting Gates

Authors: Kristoffer Arnsfelt Hansen

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/DagSemProc.08381.6
Understanding space in resolution: optimal lower bounds and exponential trade-offs

Authors: Eli Ben-Sasson and Jakob Nordström

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

Abstract
We continue the study of tradeoffs between space and length of resolution proofs and focus on two new results: begin{enumerate} item We show that length and space in resolution are uncorrelated. This is proved by exhibiting families of CNF formulas of size $O(n)$ that have proofs of length $O(n)$ but require space $Omega(n / log n)$. Our separation is the strongest possible since any proof of length $O(n)$ can always be transformed into a proof in space $O(n / log n)$, and improves previous work reported in [Nordstr"{o}m 2006, Nordstr"{o}m and H{aa}stad 2008]. item We prove a number of trade-off results for space in the range from constant to $O(n / log n)$, most of them superpolynomial or even exponential. This is a dramatic improvement over previous results in [Ben-Sasson 2002, Hertel and Pitassi 2007, Nordstr"{o}m 2007]. end{enumerate} The key to our results is the following, somewhat surprising, theorem: Any CNF formula $F$ can be transformed by simple substitution transformation into a new formula $F'$ such that if $F$ has the right properties, $F'$ can be proven in resolution in essentially the same length as $F$ but the minimal space needed for $F'$ is lower-bounded by the number of variables that have to be mentioned simultaneously in any proof for $F$. Applying this theorem to so-called pebbling formulas defined in terms of pebble games over directed acyclic graphs and analyzing black-white pebbling on these graphs yields our results.
Cite as

Eli Ben-Sasson and Jakob Nordström. Understanding space in resolution: optimal lower bounds and exponential trade-offs. In Computational Complexity of Discrete Problems. Dagstuhl Seminar Proceedings, Volume 8381, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{bensasson_et_al:DagSemProc.08381.6,
  author =	{Ben-Sasson, Eli and Nordstr\"{o}m, Jakob},
  title =	{{Understanding space in resolution: optimal lower bounds and exponential trade-offs}},
  booktitle =	{Computational Complexity of Discrete Problems},
  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.08381.6},
  URN =		{urn:nbn:de:0030-drops-17815},
  doi =		{10.4230/DagSemProc.08381.6},
  annote =	{Keywords: Proof complexity, Resolution, Pebbling.}
}
Document
DOI: 10.4230/DagSemProc.08381.1
08381 Abstracts Collection – Computational Complexity of Discrete Problems

Authors: Peter Bro Miltersen, Rüdiger Reischuk, Georg Schnitger, and Dieter van Melkebeek

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

Abstract
From the 14th of September to the 19th of September, the Dagstuhl Seminar 08381 ``Computational Complexity of Discrete Problems'' was held in Schloss Dagstuhl - Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work as well as open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this report. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.
Cite as

Peter Bro Miltersen, Rüdiger Reischuk, Georg Schnitger, and Dieter van Melkebeek. 08381 Abstracts Collection – Computational Complexity of Discrete Problems. In Computational Complexity of Discrete Problems. Dagstuhl Seminar Proceedings, Volume 8381, pp. 1-18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{miltersen_et_al:DagSemProc.08381.1,
  author =	{Miltersen, Peter Bro and Reischuk, R\"{u}diger and Schnitger, Georg and van Melkebeek, Dieter},
  title =	{{08381 Abstracts Collection – Computational Complexity of Discrete Problems}},
  booktitle =	{Computational Complexity of Discrete Problems},
  pages =	{1--18},
  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.08381.1},
  URN =		{none},
  doi =		{10.4230/DagSemProc.08381.1},
  annote =	{Keywords: Computational complexity, discrete problems, Turing machines, circuits, proof complexity, pseudorandomness, derandomization, cryptography, computational learning, communication complexity, query complexity, hardness of approximation}
}
Document
DOI: 10.4230/DagSemProc.08381.2
08381 Executive Summary – Computational Complexity of Discrete Problems

Authors: Peter Bro Miltersen, Rüdiger Reischuk, Georg Schnitger, and Dieter van Melkebeek

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

Abstract
Estimating the computational complexity of discrete problems constitutes one of the central and classical topics in the theory of computation. Mathematicians and computer scientists have long tried to classify natural families of Boolean relations according to fundamental complexity measures like time and space, both in the uniform and in the nonuniform setting. Several models of computation have been developed in order to capture the various capabilities of digital computing devices, including parallelism, randomness, and quantum interference.
Cite as

Peter Bro Miltersen, Rüdiger Reischuk, Georg Schnitger, and Dieter van Melkebeek. 08381 Executive Summary – Computational Complexity of Discrete Problems. In Computational Complexity of Discrete Problems. Dagstuhl Seminar Proceedings, Volume 8381, pp. 1-7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{miltersen_et_al:DagSemProc.08381.2,
  author =	{Miltersen, Peter Bro and Reischuk, R\"{u}diger and Schnitger, Georg and van Melkebeek, Dieter},
  title =	{{08381 Executive Summary – Computational Complexity of Discrete Problems}},
  booktitle =	{Computational Complexity of Discrete Problems},
  pages =	{1--7},
  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.08381.2},
  URN =		{urn:nbn:de:0030-drops-17789},
  doi =		{10.4230/DagSemProc.08381.2},
  annote =	{Keywords: Computational complexity, discrete problems, Turing machines, circuits, proof complexity, pseudorandomness, derandomization, cryptography, computational learning, communication complexity, query complexity, hardness of approximation}
}
Document
DOI: 10.4230/DagSemProc.08381.3
Approximation norms and duality for communication complexity lower bounds

Authors: Troy Lee and Adi Shraibman

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

Abstract
Abstract: We will discuss a general norm based framework for showing lower bounds on communication complexity. An advantage of this approach is that one can use duality theory to obtain a lower bound quantity phrased as a maximization problem, which can be more convenient to work with in showing lower bounds. We discuss two applications of this approach. 1. The approximation rank of a matrix A is the minimum rank of a matrix close to A in ell_infty norm. The logarithm of approximation rank lower bounds quantum communication complexity and is one of the most powerful techniques available, albeit difficult to compute in practice. We show that an approximation norm known as gamma_2 is polynomially related to approximation rank. This results in a polynomial time algorithm to approximate approximation rank, and also shows that the logarithm of approximation rank lower bounds quantum communication complexity even with entanglement which was previously not known. 2. By means of an approximation norm which lower bounds multiparty number-on-the-forehead complexity, we show non-trivial lower bounds on the complexity of the disjointness function for up to c log log n players, c <1.
Cite as

Troy Lee and Adi Shraibman. Approximation norms and duality for communication complexity lower bounds. In Computational Complexity of Discrete Problems. Dagstuhl Seminar Proceedings, Volume 8381, pp. 1-9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{lee_et_al:DagSemProc.08381.3,
  author =	{Lee, Troy and Shraibman, Adi},
  title =	{{Approximation norms and duality for communication complexity lower bounds}},
  booktitle =	{Computational Complexity of Discrete Problems},
  pages =	{1--9},
  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.08381.3},
  URN =		{urn:nbn:de:0030-drops-17768},
  doi =		{10.4230/DagSemProc.08381.3},
  annote =	{Keywords: Communication complexity, lower bounds}
}
Document
DOI: 10.4230/DagSemProc.08381.5
Fast polynomial factorization and modular composition

Authors: Kiran Kedlaya and Christopher Umans

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

Abstract
We obtain randomized algorithms for factoring degree $n$ univariate polynomials over $F_q$ requiring $O(n^{1.5 + o(1)} log^{1+o(1)} q+ n^{1 + o(1)}log^{2+o(1)} q)$ bit operations. When $log q < n$, this is asymptotically faster than the best previous algorithms (von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998)); for $log q ge n$, it matches the asymptotic running time of the best known algorithms. The improvements come from new algorithms for modular composition of degree $n$ univariate polynomials, which is the asymptotic bottleneck in fast algorithms for factoring polynomials over finite fields. The best previous algorithms for modular composition use $O(n^{(omega + 1)/2})$ field operations, where $omega$ is the exponent of matrix multiplication (Brent & Kung (1978)), with a slight improvement in the exponent achieved by employing fast rectangular matrix multiplication (Huang & Pan (1997)). We show that modular composition and multipoint evaluation of multivariate polynomials are essentially equivalent, in the sense that an algorithm for one achieving exponent $alpha$ implies an algorithm for the other with exponent $alpha + o(1)$, and vice versa. We then give two new algorithms that solve the problem optimally (up to lower order terms): an algebraic algorithm for fields of characteristic at most $n^{o(1)}$, and a nonalgebraic algorithm that works in arbitrary characteristic. The latter algorithm works by lifting to characteristic 0, applying a small number of rounds of {em multimodular reduction}, and finishing with a small number of multidimensional FFTs. The final evaluations are reconstructed using the Chinese Remainder Theorem. As a bonus, this algorithm produces a very efficient data structure supporting polynomial evaluation queries, which is of independent interest. Our algorithms use techniques which are commonly employed in practice, so they may be competitive for real problem sizes. This contrasts with all previous subquadratic algorithsm for these problems, which rely on fast matrix multiplication. This is joint work with Kiran Kedlaya.
Cite as

Kiran Kedlaya and Christopher Umans. Fast polynomial factorization and modular composition. In Computational Complexity of Discrete Problems. Dagstuhl Seminar Proceedings, Volume 8381, pp. 1-33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{kedlaya_et_al:DagSemProc.08381.5,
  author =	{Kedlaya, Kiran and Umans, Christopher},
  title =	{{Fast polynomial factorization and modular composition}},
  booktitle =	{Computational Complexity of Discrete Problems},
  pages =	{1--33},
  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.08381.5},
  URN =		{urn:nbn:de:0030-drops-17771},
  doi =		{10.4230/DagSemProc.08381.5},
  annote =	{Keywords: Modular composition; polynomial factorization; multipoint evaluation; Chinese Remaindering}
}
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