Volume

Dagstuhl Seminar Proceedings, Volume 6111

Publication Details

  • published at: 2006-10-09
  • Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik

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DOI: 10.4230/DagSemProc.06111.1
06111 Abstracts Collection – Complexity of Boolean Functions

Authors: Matthias Krause, Pavel Pudlák, Rüdiger Reischuk, and Dieter van Melkebeek

Abstract
From 12.03.06 to 17.03.06, the Dagstuhl Seminar 06111 ``Complexity of Boolean Functions'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and 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 paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.
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Matthias Krause, Pavel Pudlák, Rüdiger Reischuk, and Dieter van Melkebeek. 06111 Abstracts Collection – Complexity of Boolean Functions. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{krause_et_al:DagSemProc.06111.1,
  author =	{Krause, Matthias and Pudl\'{a}k, Pavel and Reischuk, R\"{u}diger and van Melkebeek, Dieter},
  title =	{{06111 Abstracts Collection – Complexity of Boolean Functions}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--24},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.1},
  URN =		{urn:nbn:de:0030-drops-7593},
  doi =		{10.4230/DagSemProc.06111.1},
  annote =	{Keywords: Complexity of Boolean functions, Boolean circuits, binary decision diagrams, lower bound proof techniques, combinatorics of Boolean functions, communi algorithmic learning, cryptography, derandomization}
}
Document
DOI: 10.4230/DagSemProc.06111.2
06111 Executive Summary – Complexity of Boolean Functions

Authors: Matthias Krause, Dieter van Melkebeek, Pavel Pudlák, and Rüdiger Reischuk

Abstract
We briefly describe the state of the art concerning the complexity of discrete functions. Computational models and analytical techniques are summarized. After describing the formal organization of the Dagstuhl seminar "Complexity of Boolean Functions" held in March 2006, we introduce the different topics that have been discussed there and mention some of the major achievements. The summary closes with an outlook on the development of discrete computational complexity in the future.
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Matthias Krause, Dieter van Melkebeek, Pavel Pudlák, and Rüdiger Reischuk. 06111 Executive Summary – Complexity of Boolean Functions. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{krause_et_al:DagSemProc.06111.2,
  author =	{Krause, Matthias and van Melkebeek, Dieter and Pudl\'{a}k, Pavel and Reischuk, R\"{u}diger},
  title =	{{06111 Executive Summary – Complexity of Boolean Functions}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--5},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.2},
  URN =		{urn:nbn:de:0030-drops-8409},
  doi =		{10.4230/DagSemProc.06111.2},
  annote =	{Keywords: Boolean and quantum circuits, discrete problems, computational complexity, lower bounds, communication complexity, proof and query complexity, randomization, pseudo-randomness, derandomization, approximation, cryptography, computational learning}
}
Document
DOI: 10.4230/DagSemProc.06111.3
A Generic Time Hierarchy for Semantic Models With One Bit of Advice

Authors: Dieter van Melkebeek and Konstantin Pervyshev

Abstract
We show that for any reasonable semantic model of computation and for any positive integer $a$ and rationals $1 leq c < d$, there exists a language computable in time $n^d$ with $a$ bits of advice but not in time $n^c$ with $a$ bits of advice. A semantic model is one for which there exists a computable enumeration that contains all machines in the model but may also contain others. We call such a model reasonable if it has an efficient universal machine that can be complemented within the model in exponential time and if it is efficiently closed under deterministic transducers. Our result implies the first such hierarchy theorem for randomized machines with zero-sided error, quantum machines with one- or zero-sided error, unambiguous machines, symmetric alternation, Arthur-Merlin games of any signature, interactive proof protocols with one or multiple provers, etc.
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Dieter van Melkebeek and Konstantin Pervyshev. A Generic Time Hierarchy for Semantic Models With One Bit of Advice. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-39, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{vanmelkebeek_et_al:DagSemProc.06111.3,
  author =	{van Melkebeek, Dieter and Pervyshev, Konstantin},
  title =	{{A Generic Time Hierarchy for Semantic Models With One Bit of Advice}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--39},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.3},
  URN =		{urn:nbn:de:0030-drops-6151},
  doi =		{10.4230/DagSemProc.06111.3},
  annote =	{Keywords: Time hierarchy, non-uniformity, one bit of advice, probabilistic algorithms}
}
Document
DOI: 10.4230/DagSemProc.06111.4
Approximability of Minimum AND-Circuits

Authors: Jan Arpe and Bodo Manthey

Abstract
Given a set of monomials, the {sc Minimum AND-Circuit} problem asks for a circuit that computes these monomials using AND-gates of fan-in two and being of minimum size. We prove that the problem is not polynomial time approximable within a factor of less than $1.0051$ unless $mathsf{P} = mathsf{NP}$, even if the monomials are restricted to be of degree at most three. For the latter case, we devise several efficient approximation algorithms, yielding an approximation ratio of $1.278$. For the general problem, we achieve an approximation ratio of $d-3/2$, where $d$ is the degree of the largest monomial. In addition, we prove that the problem is fixed parameter tractable with the number of monomials as parameter. Finally, we reveal connections between the {sc Minimum AND-Circuit} problem and several problems from different areas.
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Jan Arpe and Bodo Manthey. Approximability of Minimum AND-Circuits. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{arpe_et_al:DagSemProc.06111.4,
  author =	{Arpe, Jan and Manthey, Bodo},
  title =	{{Approximability of Minimum AND-Circuits}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--21},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.4},
  URN =		{urn:nbn:de:0030-drops-6039},
  doi =		{10.4230/DagSemProc.06111.4},
  annote =	{Keywords: Optimization problems, approximability, automated circuit design}
}
Document
DOI: 10.4230/DagSemProc.06111.5
Bounds on the Fourier Coefficients of the Weighted Sum Function

Authors: Igor E. Shparlinski

Abstract
We estimate Fourier coefficients of a Boolean function which has recently been introduced in the study of read-once branching programs. Our bound implies that this function has an asymptotically ``flat'' Fourier spectrum and thus implies several lower bounds of its various complexity measures.
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Igor E. Shparlinski. Bounds on the Fourier Coefficients of the Weighted Sum Function. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{shparlinski:DagSemProc.06111.5,
  author =	{Shparlinski, Igor E.},
  title =	{{Bounds on the Fourier Coefficients of the Weighted Sum Function}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--9},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.5},
  URN =		{urn:nbn:de:0030-drops-6171},
  doi =		{10.4230/DagSemProc.06111.5},
  annote =	{Keywords: Fourier coefficients, congruences, average sensitivity, decision tree}
}
Document
DOI: 10.4230/DagSemProc.06111.6
Computing Shortest Paths in Series-Parallel Graphs in Logarithmic Space

Authors: Andreas Jakoby and Till Tantau

Abstract
Series-parallel graphs, which are built by repeatedly applying series or parallel composition operations to paths, play an important role in computer science as they model the flow of information in many types of programs. For directed series-parallel graphs, we study the problem of finding a shortest path between two given vertices. Our main result is that we can find such a path in logarithmic space, which shows that the distance problem for series-parallel graphs is L-complete. Previously, it was known that one can compute some path in logarithmic space; but for other graph types, like undirected graphs or tournament graphs, constructing some path between given vertices is possible in logarithmic space while constructing a shortest path is NL-complete.
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Andreas Jakoby and Till Tantau. Computing Shortest Paths in Series-Parallel Graphs in Logarithmic Space. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{jakoby_et_al:DagSemProc.06111.6,
  author =	{Jakoby, Andreas and Tantau, Till},
  title =	{{Computing Shortest Paths in Series-Parallel Graphs in Logarithmic Space}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--9},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.6},
  URN =		{urn:nbn:de:0030-drops-6185},
  doi =		{10.4230/DagSemProc.06111.6},
  annote =	{Keywords: Series-parallel graphs, shortest path, logspace}
}
Document
DOI: 10.4230/DagSemProc.06111.7
Fault Jumping Attacks against Shrinking Generator

Authors: Marcin Gomulkiewicz, Miroslaw Kutylowski, and Pawel Wlaz

Abstract
In this paper we outline two new cryptoanalytic attacks against hardware implementation of the shrinking generator by Coppersmith et al., a classic design in low-cost, simple-design pseudorandom bitstream generator. This is a report on work on progress, since implementation and careful adjusting the attack strategy in order to optimize the atatck is still not completed.
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Marcin Gomulkiewicz, Miroslaw Kutylowski, and Pawel Wlaz. Fault Jumping Attacks against Shrinking Generator. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{gomulkiewicz_et_al:DagSemProc.06111.7,
  author =	{Gomulkiewicz, Marcin and Kutylowski, Miroslaw and Wlaz, Pawel},
  title =	{{Fault Jumping Attacks against Shrinking Generator}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--6},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.7},
  URN =		{urn:nbn:de:0030-drops-6117},
  doi =		{10.4230/DagSemProc.06111.7},
  annote =	{Keywords: Pseudorandom generator, shrinking generator, fault cryptanalysis}
}
Document
DOI: 10.4230/DagSemProc.06111.8
Graphs and Circuits: Some Further Remarks

Authors: Stasys Jukna

Abstract
We consider the power of single level circuits in the context of graph complexity. We first prove that the single level conjecture fails for fanin-$2$ circuits over the basis ${oplus,land,1}$. This shows that the (surpisingly tight) phenomenon, established by Mirwald and Schnorr (1992) for quadratic functions, has no analogon for graphs. We then show that the single level conjecture fails for unbounded fanin circuits over ${lor,land,1}$. This partially answers the question of Pudl'ak, R"odl and Savick'y (1986). We also prove that $Sigma_2 eq Pi_2$ in a restricted version of the hierarhy of communication complexity classes introduced by Babai, Frankl and Simon (1986). Further, we show that even depth-$2$ circuits are surprisingly powerful: every bipartite $n imes n$ graph of maximum degree $Delta$ can be represented by a monotone CNF with $O(Deltalog n)$ clauses. We also discuss a relation between graphs and $ACC$-circuits.
Cite as

Stasys Jukna. Graphs and Circuits: Some Further Remarks. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{jukna:DagSemProc.06111.8,
  author =	{Jukna, Stasys},
  title =	{{Graphs and Circuits: Some Further Remarks}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--16},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.8},
  URN =		{urn:nbn:de:0030-drops-6218},
  doi =		{10.4230/DagSemProc.06111.8},
  annote =	{Keywords: Graph complexity, single level conjecture, Sylvester graphs, communication complexity, ACC-circuits}
}
Document
DOI: 10.4230/DagSemProc.06111.9
Hadamard Tensors and Lower Bounds on Multiparty Communication Complexity

Authors: Jeff Ford and Anna Gál

Abstract
We develop a new method for estimating the discrepancy of tensors associated with multiparty communication problems in the ``Number on the Forehead'' model of Chandra, Furst and Lipton. We define an analogue of the Hadamard property of matrices for tensors in multiple dimensions and show that any $k$-party communication problem represented by a Hadamard tensor must have $Omega(n/2^k)$ multiparty communication complexity. We also exhibit constructions of Hadamard tensors, giving $Omega(n/2^k)$ lower bounds on multiparty communication complexity for a new class of explicitly defined Boolean functions.
Cite as

Jeff Ford and Anna Gál. Hadamard Tensors and Lower Bounds on Multiparty Communication Complexity. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{ford_et_al:DagSemProc.06111.9,
  author =	{Ford, Jeff and G\'{a}l, Anna},
  title =	{{Hadamard Tensors and Lower Bounds on Multiparty Communication Complexity}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--20},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.9},
  URN =		{urn:nbn:de:0030-drops-6076},
  doi =		{10.4230/DagSemProc.06111.9},
  annote =	{Keywords: Multiparty communication complexity, lower bounds}
}
Document
DOI: 10.4230/DagSemProc.06111.10
Incremental branching programs

Authors: Anna Gál, Pierre McKenzie, and Michal Koucký

Abstract
We propose a new model of restricted branching programs which we call {em incremental branching programs}. We show that {em syntactic} incremental branching programs capture previously studied structured models of computation for the problem GEN, namely marking machines [Cook74]. and Poon's extension [Poon93] of jumping automata on graphs [CookRackoff80]. We then prove exponential size lower bounds for our syntactic incremental model, and for some other restricted branching program models as well. We further show that nondeterministic syntactic incremental branching programs are provably stronger than their deterministic counterpart when solving a natural NL-complete GEN subproblem. It remains open if syntactic incremental branching programs are as powerful as unrestricted branching programs for GEN problems. Joint work with Anna Gál and Michal Koucký
Cite as

Anna Gál, Pierre McKenzie, and Michal Koucký. Incremental branching programs. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{gal_et_al:DagSemProc.06111.10,
  author =	{G\'{a}l, Anna and McKenzie, Pierre and Kouck\'{y}, Michal},
  title =	{{Incremental branching programs}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--20},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.10},
  URN =		{urn:nbn:de:0030-drops-6230},
  doi =		{10.4230/DagSemProc.06111.10},
  annote =	{Keywords: Complexity theory, branching programs, logarithmic space, marking machines}
}
Document
DOI: 10.4230/DagSemProc.06111.11
On Probabilistic Time versus Alternating Time

Authors: Emanuele Viola

Abstract
Sipser and Gács, and independently Lautemann, proved in '83 that probabilistic polynomial time is contained in the second level of the polynomial-time hierarchy, i.e. BPP is in Sigma_2 P. This is essentially the only non-trivial upper bound that we have on the power of probabilistic computation. More precisely, the Sipser-Gács-Lautemann simulation shows that probabilistic time can be simulated deterministically, using two quantifiers, **with a quadratic blow-up in the running time**. That is, BPTime(t) is contained in Sigma_2 Time(t^2). In this talk we discuss whether this quadratic blow-up in the running time is necessary. We show that the quadratic blow-up is indeed necessary for black-box simulations that use two quantifiers, such as those of Sipser, Gács, and Lautemann. To obtain this result, we prove a new circuit lower bound for computing **approximate majority**, i.e. computing the majority of a given bit-string whose fraction of 1's is bounded away from 1/2 (by a constant): We show that small depth-3 circuits for approximate majority must have bottom fan-in Omega(log n). On the positive side, we obtain that probabilistic time can be simulated deterministically, using three quantifiers, in quasilinear time. That is, BPTime(t) is contained in Sigma_3 Time(t polylog t). Along the way, we show that approximate majority can be computed by uniform polynomial-size depth-3 circuits. This is a uniform version of a striking result by Ajtai that gives *non-uniform* polynomial-size depth-3 circuits for approximate majority. If time permits, we will discuss some applications of our results to proving lower bounds on randomized Turing machines.
Cite as

Emanuele Viola. On Probabilistic Time versus Alternating Time. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{viola:DagSemProc.06111.11,
  author =	{Viola, Emanuele},
  title =	{{On Probabilistic Time versus Alternating Time}},
  booktitle =	{Complexity of Boolean Functions},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.11},
  URN =		{urn:nbn:de:0030-drops-6194},
  doi =		{10.4230/DagSemProc.06111.11},
  annote =	{Keywords: Probabilistic time, alternating time, polynomial-time hierarchy, approximate majority, constant-depth circuit}
}
Document
DOI: 10.4230/DagSemProc.06111.12
On the Complexity of Numerical Analysis

Authors: Eric Allender, Peter Bürgisser, Johan Kjeldgaard-Pedersen, and Peter Bro Miltersen

Abstract
We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis. We show that both hinge on the question of understanding the complexity of the following problem, which we call PosSlp: Given a division-free straight-line program producing an integer N, decide whether N>0. We show that OrdSlp lies in the counting hierarchy, and combining our results with work of Tiwari, we show that the Euclidean Traveling Salesman Problem lies in the counting hierarchy – the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE.
Cite as

Eric Allender, Peter Bürgisser, Johan Kjeldgaard-Pedersen, and Peter Bro Miltersen. On the Complexity of Numerical Analysis. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{allender_et_al:DagSemProc.06111.12,
  author =	{Allender, Eric and B\"{u}rgisser, Peter and Kjeldgaard-Pedersen, Johan and Miltersen, Peter Bro},
  title =	{{On the Complexity of Numerical Analysis}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--9},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.12},
  URN =		{urn:nbn:de:0030-drops-6130},
  doi =		{10.4230/DagSemProc.06111.12},
  annote =	{Keywords: Blum-Shub-Smale Model, Euclidean Traveling Salesman Problem, Counting Hierarchy}
}
Document
DOI: 10.4230/DagSemProc.06111.13
On the Teachability of Randomized Learners

Authors: Frank J. Balbach and Thomas Zeugmann

Abstract
The present paper introduces a new model for teaching {em randomized learners}. Our new model, though based on the classical teaching dimension model, allows to study the influence of various parameters such as the learner's memory size, its ability to provide or to not provide feedback, and the influence of the order in which examples are presented. Furthermore, within the new model it is possible to investigate new aspects of teaching like teaching from positive data only or teaching with inconsistent teachers. Furthermore, we provide characterization theorems for teachability from positive data for both ordinary teachers and inconsistent teachers with and without feedback.
Cite as

Frank J. Balbach and Thomas Zeugmann. On the Teachability of Randomized Learners. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{balbach_et_al:DagSemProc.06111.13,
  author =	{Balbach, Frank J. and Zeugmann, Thomas},
  title =	{{On the Teachability of Randomized Learners}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--20},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.13},
  URN =		{urn:nbn:de:0030-drops-6203},
  doi =		{10.4230/DagSemProc.06111.13},
  annote =	{Keywords: Algorithmic Teaching, Complexity of teaching}
}
Document
DOI: 10.4230/DagSemProc.06111.14
Quantum Network Coding

Authors: Masahito Hayashi, Kazuo Iwama, Harumichi Nishimura, Rudy Raymond, and Shigeru Yamashita

Abstract
Since quantum information is continuous, its handling is sometimes surprisingly harder than the classical counterpart. A typical example is cloning; making a copy of digital information is straightforward but it is not possible exactly for quantum information. The question in this paper is whether or not {em quantum} network coding is possible. Its classical counterpart is another good example to show that digital information flow can be done much more efficiently than conventional (say, liquid) flow. Our answer to the question is similar to the case of cloning, namely, it is shown that quantum network coding is possible if approximation is allowed, by using a simple network model called Butterfly. In this network, there are two flow paths, $s_1$ to $t_1$ and $s_2$ to $t_2$, which shares a single bottleneck channel of capacity one. In the classical case, we can send two bits simultaneously, one for each path, in spite of the bottleneck. Our results for quantum network coding include: (i) We can send any quantum state $|psi_1 angle$ from $s_1$ to $t_1$ and $|psi_2 angle$ from $s_2$ to $t_2$ simultaneously with a fidelity strictly greater than $1/2$. (ii) If one of $|psi_1 angle$ and $|psi_2 angle$ is classical, then the fidelity can be improved to $2/3$. (iii) Similar improvement is also possible if $|psi_1 angle$ and $|psi_2 angle$ are restricted to only a finite number of (previously known) states. (iv) Several impossibility results including the general upper bound of the fidelity are also given.
Cite as

Masahito Hayashi, Kazuo Iwama, Harumichi Nishimura, Rudy Raymond, and Shigeru Yamashita. Quantum Network Coding. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{hayashi_et_al:DagSemProc.06111.14,
  author =	{Hayashi, Masahito and Iwama, Kazuo and Nishimura, Harumichi and Raymond, Rudy and Yamashita, Shigeru},
  title =	{{Quantum Network Coding}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--17},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.14},
  URN =		{urn:nbn:de:0030-drops-6080},
  doi =		{10.4230/DagSemProc.06111.14},
  annote =	{Keywords: Network coding, quantum computation, quantum information}
}
Document
DOI: 10.4230/DagSemProc.06111.15
Quantum vs. Classical Read-Once Branching Programs

Authors: Martin Sauerhoff

Abstract
A simple, explicit boolean function on 2n input bits is presented that is computable by errorfree quantum read-once branching programs of size O(n^3), while each classical randomized read-once branching program and each quantum OBDD for this function with bounded two-sided error requires size 2^{omega(n)}.
Cite as

Martin Sauerhoff. Quantum vs. Classical Read-Once Branching Programs. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{sauerhoff:DagSemProc.06111.15,
  author =	{Sauerhoff, Martin},
  title =	{{Quantum vs. Classical Read-Once Branching Programs}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--13},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.15},
  URN =		{urn:nbn:de:0030-drops-6161},
  doi =		{10.4230/DagSemProc.06111.15},
  annote =	{Keywords: Quantum branching program, randomized branching program, read-once}
}
Document
DOI: 10.4230/DagSemProc.06111.16
Secure Linear Algebra Using Linearly Recurrent Sequences

Authors: Eike Kiltz and Enav Weinreb

Abstract
In this work we present secure two-party protocols for various core problems in linear algebra. Our main building block is a protocol to obliviously decide singularity of an encrypted matrix: Bob holds an $n imes n$ matrix $M$, encrypted with Alice's secret key, and wants to learn whether the matrix is singular or not (and nothing beyond that). We give an interactive protocol between Alice and Bob that solves the above problem with optimal communication complexity while at the same time achieving low round complexity. More precisely, the number of communication rounds in our protocol is $polylog(n)$ and the overall communication is roughly $O(n^2)$ (note that the input size is $n^2$). At the core of our protocol we exploit some nice mathematical properties of linearly recurrent sequences and their relation to the characteristic polynomial of the matrix $M$, following [Wiedemann, 1986]. With our new techniques we are able to improve the round complexity of the communication efficient solution of [Nissim and Weinreb, 2006] from $n^{0.275}$ to $polylog(n)$. Based on our singularity protocol we further extend our result to the problems of securely computing the rank of an encrypted matrix and solving systems of linear equations.
Cite as

Eike Kiltz and Enav Weinreb. Secure Linear Algebra Using Linearly Recurrent Sequences. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{kiltz_et_al:DagSemProc.06111.16,
  author =	{Kiltz, Eike and Weinreb, Enav},
  title =	{{Secure Linear Algebra Using Linearly Recurrent Sequences}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--19},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.16},
  URN =		{urn:nbn:de:0030-drops-6101},
  doi =		{10.4230/DagSemProc.06111.16},
  annote =	{Keywords: Secure Linear Algebra, Linearly Recurrent Sequences, Wiedemann's Algorithm}
}
Document
DOI: 10.4230/DagSemProc.06111.17
The Cell Probe Complexity of Succinct Data Structures

Authors: Anna Gál and Peter Bro Miltersen

Abstract
In the cell probe model with word size 1 (the bit probe model), a static data structure problem is given by a map $f: {0,1}^n imes {0,1}^m ightarrow {0,1}$, where ${0,1}^n$ is a set of possible data to be stored, ${0,1}^m$ is a set of possible queries (for natural problems, we have $m ll n$) and $f(x,y)$ is the answer to question $y$ about data $x$. A solution is given by a representation $phi: {0,1}^n ightarrow {0,1}^s$ and a query algorithm $q$ so that $q(phi(x), y) = f(x,y)$. The time $t$ of the query algorithm is the number of bits it reads in $phi(x)$. In this paper, we consider the case of {em succinct} representations where $s = n + r$ for some {em redundancy} $r ll n$. For a boolean version of the problem of polynomial evaluation with preprocessing of coefficients, we show a lower bound on the redundancy-query time tradeoff of the form [ (r+1) t geq Omega(n/log n).] In particular, for very small redundancies $r$, we get an almost optimal lower bound stating that the query algorithm has to inspect almost the entire data structure (up to a logarithmic factor). We show similar lower bounds for problems satisfying a certain combinatorial property of a coding theoretic flavor. Previously, no $omega(m)$ lower bounds were known on $t$ in the general model for explicit functions, even for very small redundancies. By restricting our attention to {em systematic} or {em index} structures $phi$ satisfying $phi(x) = x cdot phi^*(x)$ for some map $phi^*$ (where $cdot$ denotes concatenation) we show similar lower bounds on the redundancy-query time tradeoff for the natural data structuring problems of Prefix Sum and Substring Search.
Cite as

Anna Gál and Peter Bro Miltersen. The Cell Probe Complexity of Succinct Data Structures. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{gal_et_al:DagSemProc.06111.17,
  author =	{G\'{a}l, Anna and Miltersen, Peter Bro},
  title =	{{The Cell Probe Complexity of Succinct Data Structures}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--13},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.17},
  URN =		{urn:nbn:de:0030-drops-6065},
  doi =		{10.4230/DagSemProc.06111.17},
  annote =	{Keywords: Cell probe model, data structures, lower bounds, time-space tradeoffs}
}
Document
DOI: 10.4230/DagSemProc.06111.18
The complexity of Boolean functions from cryptographic viewpoint

Authors: Claude Carlet

Abstract
Cryptographic Boolean functions must be complex to satisfy Shannon's principle of confusion. But the cryptographic viewpoint on complexity is not the same as in circuit complexity. The two main criteria evaluating the cryptographic complexity of Boolean functions on $F_2^n$ are the nonlinearity (and more generally the $r$-th order nonlinearity, for every positive $r< n$) and the algebraic degree. Two other criteria have also been considered: the algebraic thickness and the non-normality. After recalling the definitions of these criteria and why, asymptotically, almost all Boolean functions are deeply non-normal and have high algebraic degrees, high ($r$-th order) nonlinearities and high algebraic thicknesses, we study the relationship between the $r$-th order nonlinearity and a recent cryptographic criterion called the algebraic immunity. This relationship strengthens the reasons why the algebraic immunity can be considered as a further cryptographic complexity criterion.
Cite as

Claude Carlet. The complexity of Boolean functions from cryptographic viewpoint. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{carlet:DagSemProc.06111.18,
  author =	{Carlet, Claude},
  title =	{{The complexity of Boolean functions from cryptographic viewpoint}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--15},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.18},
  URN =		{urn:nbn:de:0030-drops-6044},
  doi =		{10.4230/DagSemProc.06111.18},
  annote =	{Keywords: Boolean function, nonlinearity, Reed-Muller code}
}
Document
DOI: 10.4230/DagSemProc.06111.19
The optimal sequence compression

Authors: Alexander E. Andreev

Abstract
This paper presents the optimal compression for sequences with undefined values. Let we have $(N-m)$ undefined and $m$ defined positions in the boolean sequence $vv V$ of length $N$. The sequence code length can't be less then $m$ in general case, otherwise at least two sequences will have the same code. We present the coding algorithm which generates codes of almost $m$ length, i.e. almost equal to the lower bound. The paper presents the decoding circuit too. The circuit has low complexity which depends from the inverse density of defined values $D(vv V) = frac{N}{m}$. The decoding circuit includes RAM and random logic. It performs sequential decoding. The total RAM size is proportional to the $$logleft(D(vv V) ight) ,$$ the number of random logic cells is proportional to $$log logleft(D(vv V) ight) * left(log log logleft(D(vv V) ight) ight)^2 .$$ So the decoding circuit will be small enough even for the very low density sequences. The decoder complexity doesn't depend of the sequence length at all.
Cite as

Alexander E. Andreev. The optimal sequence compression. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{andreev:DagSemProc.06111.19,
  author =	{Andreev, Alexander E.},
  title =	{{The optimal sequence compression}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--11},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.19},
  URN =		{urn:nbn:de:0030-drops-6025},
  doi =		{10.4230/DagSemProc.06111.19},
  annote =	{Keywords: Compression, partial boolean function}
}
Document
DOI: 10.4230/DagSemProc.06111.20
Time-Space Lower Bounds for the Polynomial-Time Hierarchy on Randomized Machines

Authors: Scott Diehl and Dieter van Melkebeek

Abstract
In this talk, we establish lower bounds for the running time of randomized machines with two-sided error which use a small amount of workspace to solve complete problems in the polynomial-time hierarchy. In particular, we show that for integers $l > 1$, a randomized machine with two-sided error using subpolynomial space requires time $n^{l - o(1)}$ to solve QSATl, where QSATl denotes the problem of deciding the validity of a Boolean first-order formula with at most $l-1$ quantifier alternations. This represents the first time-space lower bounds for complete problems in the polynomial-time hierarchy on randomized machines with two-sided error. Corresponding to $l = 1$, we show that a randomized machine with one-sided error using subpolynomial space requires time $n^{1.759}$ to decide the set of Boolean tautologies. As a corollary, this gives the same lower bound for satisfiability on deterministic machines, improving on the previously best known such result.
Cite as

Scott Diehl and Dieter van Melkebeek. Time-Space Lower Bounds for the Polynomial-Time Hierarchy on Randomized Machines. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{diehl_et_al:DagSemProc.06111.20,
  author =	{Diehl, Scott and van Melkebeek, Dieter},
  title =	{{Time-Space Lower Bounds for the Polynomial-Time Hierarchy on Randomized Machines}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--33},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.20},
  URN =		{urn:nbn:de:0030-drops-6054},
  doi =		{10.4230/DagSemProc.06111.20},
  annote =	{Keywords: Time-space lower bounds, lower bounds, randomness, polynomial-time hierarchy}
}
Document
DOI: 10.4230/DagSemProc.06111.21
Using Quantum Oblivious Transfer to Cheat Sensitive Quantum Bit Commitment

Authors: Andreas Jakoby, Maciej Liskiewicz, and Aleksander Madry

Abstract
We define $(varepsilon,delta)$-secure quantum computations between two parties that can play dishonestly to maximise advantage $delta$, however keeping small the probability $varepsilon$ that the computation fails in evaluating correct value. We present a simple quantum protocol for computing one-out-of-two oblivious transfer that is $(O(sqrt{varepsilon}),varepsilon)$-secure. Using the protocol as a black box we construct a scheme for cheat sensitive quantum bit commitment which guarantee that a mistrustful party has a nonzero probability of detecting a cheating.
Cite as

Andreas Jakoby, Maciej Liskiewicz, and Aleksander Madry. Using Quantum Oblivious Transfer to Cheat Sensitive Quantum Bit Commitment. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{jakoby_et_al:DagSemProc.06111.21,
  author =	{Jakoby, Andreas and Liskiewicz, Maciej and Madry, Aleksander},
  title =	{{Using Quantum Oblivious Transfer to Cheat Sensitive Quantum Bit Commitment}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--12},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.21},
  URN =		{urn:nbn:de:0030-drops-6223},
  doi =		{10.4230/DagSemProc.06111.21},
  annote =	{Keywords: Two-Party Computations, Quantum Protocols, Bit Commitment, Oblivious Transfer.}
}
Document
DOI: 10.4230/DagSemProc.06111.22
Very Large Cliques are Easy to Detect

Authors: Alexander E. Andreev and Stasys Jukna

Abstract
It is known that, for every constant $kgeq 3$, the presence of a $k$-clique (a complete subgraph on $k$ vertices) in an $n$-vertex graph cannot be detected by a monotone boolean circuit using fewer than $Omega((n/log n)^k)$ gates. We show that, for every constant $k$, the presence of an $(n-k)$-clique in an $n$-vertex graph can be detected by a monotone circuit using only $O(n^2log n)$ gates. Moreover, if we allow unbounded fanin, then $O(log n)$ gates are enough.
Cite as

Alexander E. Andreev and Stasys Jukna. Very Large Cliques are Easy to Detect. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{andreev_et_al:DagSemProc.06111.22,
  author =	{Andreev, Alexander E. and Jukna, Stasys},
  title =	{{Very Large Cliques are Easy to Detect}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--7},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.22},
  URN =		{urn:nbn:de:0030-drops-6092},
  doi =		{10.4230/DagSemProc.06111.22},
  annote =	{Keywords: Clique function, monotone circuits, perfect hashing}
}

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