16 Search Results for "Jukna, Stasys"


Document
Branching Programs with Bounded Repetitions and Flow Formulas

Authors: Anastasia Sofronova and Dmitry Sokolov

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)


Abstract
Restricted branching programs capture various complexity measures like space in Turing machines or length of proofs in proof systems. In this paper, we focus on the application in the proof complexity that was discovered by Lovasz et al. [László Lovász et al., 1995] who showed the equivalence between regular Resolution and read-once branching programs for "unsatisfied clause search problem" (Search_φ). This connection is widely used, in particular, in the recent breakthrough result about the Clique problem in regular Resolution by Atserias et al. [Albert Atserias et al., 2018]. We study the branching programs with bounded repetitions, so-called (1,+k)-BPs (Sieling [Detlef Sieling, 1996]) in application to the Search_φ problem. On the one hand, it is a natural generalization of read-once branching programs. On the other hand, this model gives a powerful proof system that can efficiently certify the unsatisfiability of a wide class of formulas that is hard for Resolution (Knop [Alexander Knop, 2017]). We deal with Search_φ that is "relatively easy" compared to all known hard examples for the (1,+k)-BPs. We introduce the first technique for proving exponential lower bounds for the (1,+k)-BPs on Search_φ. To do it we combine a well-known technique for proving lower bounds on the size of branching programs [Detlef Sieling, 1996; Detlef Sieling and Ingo Wegener, 1994; Stasys Jukna and Alexander A. Razborov, 1998] with the modification of the "closure" technique [Michael Alekhnovich et al., 2004; Michael Alekhnovich and Alexander A. Razborov, 2003]. In contrast with most Resolution lower bounds, our technique uses not only "local" properties of the formula, but also a "global" structure. Our hard examples are based on the Flow formulas introduced in [Michael Alekhnovich and Alexander A. Razborov, 2003].

Cite as

Anastasia Sofronova and Dmitry Sokolov. Branching Programs with Bounded Repetitions and Flow Formulas. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 17:1-17:25, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{sofronova_et_al:LIPIcs.CCC.2021.17,
  author =	{Sofronova, Anastasia and Sokolov, Dmitry},
  title =	{{Branching Programs with Bounded Repetitions and Flow Formulas}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{17:1--17:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.17},
  URN =		{urn:nbn:de:0030-drops-142915},
  doi =		{10.4230/LIPIcs.CCC.2021.17},
  annote =	{Keywords: proof complexity, branching programs, bounded repetitions, lower bounds}
}
Document
Extended Abstract
Shrinkage Under Random Projections, and Cubic Formula Lower Bounds for AC0 (Extended Abstract)

Authors: Yuval Filmus, Or Meir, and Avishay Tal

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)


Abstract
Håstad showed that any De Morgan formula (composed of AND, OR and NOT gates) shrinks by a factor of O(p²) under a random restriction that leaves each variable alive independently with probability p [SICOMP, 1998]. Using this result, he gave an Ω̃(n³) formula size lower bound for the Andreev function, which, up to lower order improvements, remains the state-of-the-art lower bound for any explicit function. In this work, we extend the shrinkage result of Håstad to hold under a far wider family of random restrictions and their generalization - random projections. Based on our shrinkage results, we obtain an Ω̃(n³) formula size lower bound for an explicit function computed in AC⁰. This improves upon the best known formula size lower bounds for AC⁰, that were only quadratic prior to our work. In addition, we prove that the KRW conjecture [Karchmer et al., Computational Complexity 5(3/4), 1995] holds for inner functions for which the unweighted quantum adversary bound is tight. In particular, this holds for inner functions with a tight Khrapchenko bound. Our random projections are tailor-made to the function’s structure so that the function maintains structure even under projection - using such projections is necessary, as standard random restrictions simplify AC⁰ circuits. In contrast, we show that any De Morgan formula shrinks by a quadratic factor under our random projections, allowing us to prove the cubic lower bound. Our proof techniques build on the proof of Håstad for the simpler case of balanced formulas. This allows for a significantly simpler proof at the cost of slightly worse parameters. As such, when specialized to the case of p-random restrictions, our proof can be used as an exposition of Håstad’s result.

Cite as

Yuval Filmus, Or Meir, and Avishay Tal. Shrinkage Under Random Projections, and Cubic Formula Lower Bounds for AC0 (Extended Abstract). In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 89:1-89:7, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{filmus_et_al:LIPIcs.ITCS.2021.89,
  author =	{Filmus, Yuval and Meir, Or and Tal, Avishay},
  title =	{{Shrinkage Under Random Projections, and Cubic Formula Lower Bounds for AC0}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{89:1--89:7},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.89},
  URN =		{urn:nbn:de:0030-drops-136281},
  doi =		{10.4230/LIPIcs.ITCS.2021.89},
  annote =	{Keywords: De Morgan formulas, KRW Conjecture, shrinkage, random restrictions, random projections, bounded depth circuits, constant depth circuits, formula complexity}
}
Document
Hardness Magnification near State-Of-The-Art Lower Bounds

Authors: Igor Carboni Oliveira, Ján Pich, and Rahul Santhanam

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)


Abstract
This work continues the development of hardness magnification. The latter proposes a new strategy for showing strong complexity lower bounds by reducing them to a refined analysis of weaker models, where combinatorial techniques might be successful. We consider gap versions of the meta-computational problems MKtP and MCSP, where one needs to distinguish instances (strings or truth-tables) of complexity <= s_1(N) from instances of complexity >= s_2(N), and N = 2^n denotes the input length. In MCSP, complexity is measured by circuit size, while in MKtP one considers Levin’s notion of time-bounded Kolmogorov complexity. (In our results, the parameters s_1(N) and s_2(N) are asymptotically quite close, and the problems almost coincide with their standard formulations without a gap.) We establish that for Gap-MKtP[s_1,s_2] and Gap-MCSP[s_1,s_2], a marginal improvement over the state-of-the-art in unconditional lower bounds in a variety of computational models would imply explicit super-polynomial lower bounds. Theorem. There exists a universal constant c >= 1 for which the following hold. If there exists epsilon > 0 such that for every small enough beta > 0 (1) Gap-MCSP[2^{beta n}/c n, 2^{beta n}] !in Circuit[N^{1 + epsilon}], then NP !subseteq Circuit[poly]. (2) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in TC^0[N^{1 + epsilon}], then EXP !subseteq TC^0[poly]. (3) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in B_2-Formula[N^{2 + epsilon}], then EXP !subseteq Formula[poly]. (4) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in U_2-Formula[N^{3 + epsilon}], then EXP !subseteq Formula[poly]. (5) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in BP[N^{2 + epsilon}], then EXP !subseteq BP[poly]. (6) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in (AC^0[6])[N^{1 + epsilon}], then EXP !subseteq AC^0[6]. These results are complemented by lower bounds for Gap-MCSP and Gap-MKtP against different models. For instance, the lower bound assumed in (1) holds for U_2-formulas of near-quadratic size, and lower bounds similar to (3)-(5) hold for various regimes of parameters. We also identify a natural computational model under which the hardness magnification threshold for Gap-MKtP lies below existing lower bounds: U_2-formulas that can compute parity functions at the leaves (instead of just literals). As a consequence, if one managed to adapt the existing lower bound techniques against such formulas to work with Gap-MKtP, then EXP !subseteq NC^1 would follow via hardness magnification.

Cite as

Igor Carboni Oliveira, Ján Pich, and Rahul Santhanam. Hardness Magnification near State-Of-The-Art Lower Bounds. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 27:1-27:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{oliveira_et_al:LIPIcs.CCC.2019.27,
  author =	{Oliveira, Igor Carboni and Pich, J\'{a}n and Santhanam, Rahul},
  title =	{{Hardness Magnification near State-Of-The-Art Lower Bounds}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{27:1--27:29},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.27},
  URN =		{urn:nbn:de:0030-drops-108494},
  doi =		{10.4230/LIPIcs.CCC.2019.27},
  annote =	{Keywords: Circuit Complexity, Minimum Circuit Size Problem, Kolmogorov Complexity}
}
Document
Lower Bounds for DeMorgan Circuits of Bounded Negation Width

Authors: Stasys Jukna and Andrzej Lingas

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


Abstract
We consider Boolean circuits over {or, and, neg} with negations applied only to input variables. To measure the "amount of negation" in such circuits, we introduce the concept of their "negation width". In particular, a circuit computing a monotone Boolean function f(x_1,...,x_n) has negation width w if no nonzero term produced (purely syntactically) by the circuit contains more than w distinct negated variables. Circuits of negation width w=0 are equivalent to monotone Boolean circuits, while those of negation width w=n have no restrictions. Our motivation is that already circuits of moderate negation width w=n^{epsilon} for an arbitrarily small constant epsilon>0 can be even exponentially stronger than monotone circuits. We show that the size of any circuit of negation width w computing f is roughly at least the minimum size of a monotone circuit computing f divided by K=min{w^m,m^w}, where m is the maximum length of a prime implicant of f. We also show that the depth of any circuit of negation width w computing f is roughly at least the minimum depth of a monotone circuit computing f minus log K. Finally, we show that formulas of bounded negation width can be balanced to achieve a logarithmic (in their size) depth without increasing their negation width.

Cite as

Stasys Jukna and Andrzej Lingas. Lower Bounds for DeMorgan Circuits of Bounded Negation Width. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 41:1-41:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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@InProceedings{jukna_et_al:LIPIcs.STACS.2019.41,
  author =	{Jukna, Stasys and Lingas, Andrzej},
  title =	{{Lower Bounds for DeMorgan Circuits of Bounded Negation Width}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{41:1--41:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.41},
  URN =		{urn:nbn:de:0030-drops-102801},
  doi =		{10.4230/LIPIcs.STACS.2019.41},
  annote =	{Keywords: Boolean circuits, monotone circuits, lower bounds}
}
Document
On the Limits of Gate Elimination

Authors: Alexander Golovnev, Edward A. Hirsch, Alexander Knop, and Alexander S. Kulikov

Published in: LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)


Abstract
Although a simple counting argument shows the existence of Boolean functions of exponential circuit complexity, proving superlinear circuit lower bounds for explicit functions seems to be out of reach of the current techniques. There has been a (very slow) progress in proving linear lower bounds with the latest record of 3 1/86*n-o(n). All known lower bounds are based on the so-called gate elimination technique. A typical gate elimination argument shows that it is possible to eliminate several gates from an optimal circuit by making one or several substitutions to the input variables and repeats this inductively. In this note we prove that this method cannot achieve linear bounds of cn beyond a certain constant c, where c depends only on the number of substitutions made at a single step of the induction.

Cite as

Alexander Golovnev, Edward A. Hirsch, Alexander Knop, and Alexander S. Kulikov. On the Limits of Gate Elimination. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 46:1-46:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@InProceedings{golovnev_et_al:LIPIcs.MFCS.2016.46,
  author =	{Golovnev, Alexander and Hirsch, Edward A. and Knop, Alexander and Kulikov, Alexander S.},
  title =	{{On the Limits of Gate Elimination}},
  booktitle =	{41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
  pages =	{46:1--46:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-016-3},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{58},
  editor =	{Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.46},
  URN =		{urn:nbn:de:0030-drops-64593},
  doi =		{10.4230/LIPIcs.MFCS.2016.46},
  annote =	{Keywords: circuit complexity, lower bounds, gate elimination}
}
Document
On Kernelization and Approximation for the Vector Connectivity Problem

Authors: Stefan Kratsch and Manuel Sorge

Published in: LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)


Abstract
In the Vector Connectivity problem we are given an undirected graph G=(V,E), a demand function phi: V => {0,...,d}, and an integer k. The question is whether there exists a set S of at most k vertices such that every vertex v in V\S has at least phi(v) vertex-disjoint paths to S; this abstractly captures questions about placing servers in a network, or warehouses on a map, relative to demands. The problem is NP-hard already for instances with d=4 (Cicalese et al., Theor. Comput. Sci. 2015), admits a log-factor approximation (Boros et al., Networks 2014), and is fixed-parameter tractable in terms of k (Lokshtanov, unpublished 2014). We prove several results regarding kernelization and approximation for Vector Connectivity and the variant Vector d-Connectivity where the upper bound d on demands is a constant. For Vector d-Connectivity we give a factor d-approximation algorithm and construct a vertex-linear kernelization, i.e., an efficient reduction to an equivalent instance with f(d)k=O(k) vertices. For Vector Connectivity we get a factor opt-approximation and we show that it has no kernelization to size polynomial in k+d unless NP \subseteq coNP/poly, making f(d)\poly(k) optimal for Vector d-Connectivity. Finally, we provide a write-up for fixed-parameter tractability of Vector Connectivity(k) by giving a different algorithm based on matroid intersection.

Cite as

Stefan Kratsch and Manuel Sorge. On Kernelization and Approximation for the Vector Connectivity Problem. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 377-388, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{kratsch_et_al:LIPIcs.IPEC.2015.377,
  author =	{Kratsch, Stefan and Sorge, Manuel},
  title =	{{On Kernelization and Approximation for the Vector Connectivity Problem}},
  booktitle =	{10th International Symposium on Parameterized and Exact Computation (IPEC 2015)},
  pages =	{377--388},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-92-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{43},
  editor =	{Husfeldt, Thore and Kanj, Iyad},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.377},
  URN =		{urn:nbn:de:0030-drops-55985},
  doi =		{10.4230/LIPIcs.IPEC.2015.377},
  annote =	{Keywords: parameterized complexity, kernelization, approximation}
}
Document
Static Analysis for Logic-based Dynamic Programs

Authors: Thomas Schwentick, Nils Vortmeier, and Thomas Zeume

Published in: LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)


Abstract
The goal of dynamic programs as introduced by Patnaik and Immerman (1994) is to maintain the result of a fixed query for an input database which is subject to tuple insertions and deletions. To this end such programs store an auxiliary database whose relations are updated via first-order formulas upon modifications of the input database. One of those auxiliary relations is supposed to store the answer to the query. Several static analysis problems can be associated to such dynamic programs. Is the answer relation of a given dynamic program always empty? Does a program actually maintain a query? That is, is the answer given of the program the same when an input database was reached by two different modification sequences? Even more, is the content of auxiliary relations independent of the modification sequence that lead to an input database? We study the algorithmic properties of those and similar static analysis problems. Since all these problems can easily be seen to be undecidable for full first-order programs, we examine the exact borderline for decidability for restricted programs. Our focus is on restricting the arity of the input databases as well as the auxiliary databases, and to restrict the use of quantifiers.

Cite as

Thomas Schwentick, Nils Vortmeier, and Thomas Zeume. Static Analysis for Logic-based Dynamic Programs. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. 308-324, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{schwentick_et_al:LIPIcs.CSL.2015.308,
  author =	{Schwentick, Thomas and Vortmeier, Nils and Zeume, Thomas},
  title =	{{Static Analysis for Logic-based Dynamic Programs}},
  booktitle =	{24th EACSL Annual Conference on Computer Science Logic (CSL 2015)},
  pages =	{308--324},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-90-3},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{41},
  editor =	{Kreutzer, Stephan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.308},
  URN =		{urn:nbn:de:0030-drops-54221},
  doi =		{10.4230/LIPIcs.CSL.2015.308},
  annote =	{Keywords: Dynamic descriptive complexity, algorithmic problems, emptiness, history independence, consistency}
}
Document
Dependent Random Graphs and Multi-Party Pointer Jumping

Authors: Joshua Brody and Mario Sanchez

Published in: LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)


Abstract
We initiate a study of a relaxed version of the standard Erdos-Renyi random graph model, where each edge may depend on a few other edges. We call such graphs "dependent random graphs". Our main result in this direction is a thorough understanding of the clique number of dependent random graphs. We also obtain bounds for the chromatic number. Surprisingly, many of the standard properties of random graphs also hold in this relaxed setting. We show that with high probability, a dependent random graph will contain a clique of size ((1-o(1))log(n))/log(1/p), and the chromatic number will be at most (nlog(1/(1-p)))/log(n). We expect these results to be of independent interest. As an application and second main result, we give a new communication protocol for the k-player Multi-Party Pointer Jumping problem (MPJk) in the number-on-the-forehead (NOF) model. Multi-Party Pointer Jumping is one of the canonical NOF communication problems, yet even for three players, its communication complexity is not well understood. Our protocol for MPJ3 costs O((n * log(log(n)))/log(n)) communication, improving on a bound from [BrodyChakrabarti08]. We extend our protocol to the non-Boolean pointer jumping problem, achieving an upper bound which is o(n) for any k >= 4 players. This is the first o(n) protocol and improves on a bound of Damm, Jukna, and Sgall, which has stood for almost twenty years.

Cite as

Joshua Brody and Mario Sanchez. Dependent Random Graphs and Multi-Party Pointer Jumping. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 606-624, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{brody_et_al:LIPIcs.APPROX-RANDOM.2015.606,
  author =	{Brody, Joshua and Sanchez, Mario},
  title =	{{Dependent Random Graphs and Multi-Party Pointer Jumping}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{606--624},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.606},
  URN =		{urn:nbn:de:0030-drops-53266},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.606},
  annote =	{Keywords: random graphs, communication complexity, number-on-the-forehead model, pointer jumping}
}
Document
Negation-Limited Formulas

Authors: Siyao Guo and Ilan Komargodski

Published in: LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)


Abstract
We give an efficient structural decomposition theorem for formulas that depends on their negation complexity and demonstrate its power with the following applications. We prove that every formula that contains t negation gates can be shrunk using a random restriction to a formula of size O(t) with the shrinkage exponent of monotone formulas. As a result, the shrinkage exponent of formulas that contain a constant number of negation gates is equal to the shrinkage exponent of monotone formulas. We give an efficient transformation of formulas with t negation gates to circuits with log(t) negation gates. This transformation provides a generic way to cast results for negation-limited circuits to the setting of negation-limited formulas. For example, using a result of Rossman (CCC'15), we obtain an average-case lower bound for formulas of polynomial-size on n variables with n^{1/2-epsilon} negations. In addition, we prove a lower bound on the number of negations required to compute one-way permutations by polynomial-size formulas.

Cite as

Siyao Guo and Ilan Komargodski. Negation-Limited Formulas. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 850-866, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{guo_et_al:LIPIcs.APPROX-RANDOM.2015.850,
  author =	{Guo, Siyao and Komargodski, Ilan},
  title =	{{Negation-Limited Formulas}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{850--866},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.850},
  URN =		{urn:nbn:de:0030-drops-53400},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.850},
  annote =	{Keywords: Negation complexity, De Morgan formulas, Shrinkage}
}
Document
The List-Decoding Size of Fourier-Sparse Boolean Functions

Authors: Ishay Haviv and Oded Regev

Published in: LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)


Abstract
A function defined on the Boolean hypercube is k-Fourier-sparse if it has at most k nonzero Fourier coefficients. For a function f: F_2^n -> R and parameters k and d, we prove a strong upper bound on the number of k-Fourier-sparse Boolean functions that disagree with f on at most d inputs. Our bound implies that the number of uniform and independent random samples needed for learning the class of k-Fourier-sparse Boolean functions on n variables exactly is at most O(n * k * log(k)). As an application, we prove an upper bound on the query complexity of testing Booleanity of Fourier-sparse functions. Our bound is tight up to a logarithmic factor and quadratically improves on a result due to Gur and Tamuz [Chicago J. Theor. Comput. Sci.,2013].

Cite as

Ishay Haviv and Oded Regev. The List-Decoding Size of Fourier-Sparse Boolean Functions. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 58-71, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{haviv_et_al:LIPIcs.CCC.2015.58,
  author =	{Haviv, Ishay and Regev, Oded},
  title =	{{The List-Decoding Size of Fourier-Sparse Boolean Functions}},
  booktitle =	{30th Conference on Computational Complexity (CCC 2015)},
  pages =	{58--71},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-81-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{33},
  editor =	{Zuckerman, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.58},
  URN =		{urn:nbn:de:0030-drops-50600},
  doi =		{10.4230/LIPIcs.CCC.2015.58},
  annote =	{Keywords: Fourier-sparse functions, list-decoding, learning theory, property testing}
}
Document
Kolmogorov Width of Discrete Linear Spaces: an Approach to Matrix Rigidity

Authors: Alex Samorodnitsky, Ilya Shkredov, and Sergey Yekhanin

Published in: LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)


Abstract
A square matrix V is called rigid if every matrix V' obtained by altering a small number of entries of $V$ has sufficiently high rank. While random matrices are rigid with high probability, no explicit constructions of rigid matrices are known to date. Obtaining such explicit matrices would have major implications in computational complexity theory. One approach to establishing rigidity of a matrix V is to come up with a property that is satisfied by any collection of vectors arising from a low-dimensional space, but is not satisfied by the rows of V even after alterations. In this paper we propose such a candidate property that has the potential of establishing rigidity of combinatorial design matrices over the field F_2. Stated informally, we conjecture that under a suitable embedding of F_2^n into R^n, vectors arising from a low dimensional F_2-linear space always have somewhat small Kolmogorov width, i.e., admit a non-trivial simultaneous approximation by a low dimensional Euclidean space. This implies rigidity of combinatorial designs, as their rows do not admit such an approximation even after alterations. Our main technical contribution is a collection of results establishing weaker forms and special cases of the conjecture above.

Cite as

Alex Samorodnitsky, Ilya Shkredov, and Sergey Yekhanin. Kolmogorov Width of Discrete Linear Spaces: an Approach to Matrix Rigidity. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 347-364, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{samorodnitsky_et_al:LIPIcs.CCC.2015.347,
  author =	{Samorodnitsky, Alex and Shkredov, Ilya and Yekhanin, Sergey},
  title =	{{Kolmogorov Width of Discrete Linear Spaces: an Approach to Matrix Rigidity}},
  booktitle =	{30th Conference on Computational Complexity (CCC 2015)},
  pages =	{347--364},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-81-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{33},
  editor =	{Zuckerman, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.347},
  URN =		{urn:nbn:de:0030-drops-50703},
  doi =		{10.4230/LIPIcs.CCC.2015.347},
  annote =	{Keywords: Matrix rigidity, linear codes, Kolmogorov width}
}
Document
Tropical Effective Primary and Dual Nullstellens"atze

Authors: Dima Grigoriev and Vladimir V. Podolskii

Published in: LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)


Abstract
Tropical algebra is an emerging field with a number of applications in various areas of mathematics. In many of these applications appeal to tropical polynomials allows to study properties of mathematical objects such as algebraic varieties and algebraic curves from the computational point of view. This makes it important to study both mathematical and computational aspects of tropical polynomials. In this paper we prove tropical Nullstellensatz and moreover we show effective formulation of this theorem. Nullstellensatz is a next natural step in building algebraic theory of tropical polynomials and effective version is relevant for computational aspects of this field.

Cite as

Dima Grigoriev and Vladimir V. Podolskii. Tropical Effective Primary and Dual Nullstellens"atze. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 379-391, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{grigoriev_et_al:LIPIcs.STACS.2015.379,
  author =	{Grigoriev, Dima and Podolskii, Vladimir V.},
  title =	{{Tropical Effective Primary and Dual Nullstellens"atze}},
  booktitle =	{32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
  pages =	{379--391},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-78-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{30},
  editor =	{Mayr, Ernst W. and Ollinger, Nicolas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.379},
  URN =		{urn:nbn:de:0030-drops-49286},
  doi =		{10.4230/LIPIcs.STACS.2015.379},
  annote =	{Keywords: tropical algebra, tropical geometry, Nullstellensatz}
}
Document
Space Pseudorandom Generators by Communication Complexity Lower Bounds

Authors: Anat Ganor and Ran Raz

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)


Abstract
In 1989, Babai, Nisan and Szegedy gave a construction of a pseudorandom generator for logspace, based on lower bounds for multiparty communication complexity. The seed length of their pseudorandom generator was relatively large, because the best lower bounds for multiparty communication complexity are relatively weak. Subsequently, pseudorandom generators for logspace with seed length O(log^2 n) were given by Nisan, and Impagliazzo, Nisan and Wigderson. In this paper, we show how to use the pseudorandom generator construction of Babai, Nisan and Szegedy to obtain a third construction of a pseudorandom generator with seed length O(log^2 n), achieving the same parameters as Nisan, and Impagliazzo, Nisan and Wigderson. We achieve this by concentrating on protocols in a restricted model of multiparty communication complexity that we call the conservative one-way unicast model and is based on the conservative one-way model of Damm, Jukna and Sgall. We observe that bounds in the conservative one-way unicast model (rather than the standard Number On the Forehead model) are sufficient for the pseudorandom generator construction of Babai, Nisan and Szegedy to work. Roughly speaking, in a conservative one-way unicast communication protocol, the players speak in turns, one after the other in a fixed order, and every message is visible only to the next player. Moreover, before the beginning of the protocol, each player only knows the inputs of the players that speak after she does and a certain function of the inputs of the players that speak before she does. We prove a lower bound for the communication complexity of conservative one-way unicast communication protocols that compute a family of functions obtained by compositions of strong extractors. Our final pseudorandom generator construction is related to, but different from the constructions of Nisan, and Impagliazzo, Nisan and Wigderson.

Cite as

Anat Ganor and Ran Raz. Space Pseudorandom Generators by Communication Complexity Lower Bounds. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 692-703, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


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@InProceedings{ganor_et_al:LIPIcs.APPROX-RANDOM.2014.692,
  author =	{Ganor, Anat and Raz, Ran},
  title =	{{Space Pseudorandom Generators by Communication Complexity Lower Bounds}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{692--703},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.692},
  URN =		{urn:nbn:de:0030-drops-47324},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.692},
  annote =	{Keywords: Communication complexity, Logspace, Pseudorandom generator}
}
Document
Communication Complexity of Set-Disjointness for All Probabilities

Authors: Mika Göös and Thomas Watson

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)


Abstract
We study set-disjointness in a generalized model of randomized two-party communication where the probability of acceptance must be at least alpha(n) on yes-inputs and at most beta(n) on no-inputs, for some functions alpha(n)>beta(n). Our main result is a complete characterization of the private-coin communication complexity of set-disjointness for all functions alpha and beta, and a near-complete characterization for public-coin protocols. In particular, we obtain a simple proof of a theorem of Braverman and Moitra (STOC 2013), who studied the case where alpha=1/2+epsilon(n) and beta=1/2-epsilon(n). The following contributions play a crucial role in our characterization and are interesting in their own right. (1) We introduce two communication analogues of the classical complexity class that captures small bounded-error computations: we define a "restricted" class SBP (which lies between MA and AM) and an "unrestricted" class USBP. The distinction between them is analogous to the distinction between the well-known communication classes PP and UPP. (2) We show that the SBP communication complexity is precisely captured by the classical corruption lower bound method. This sharpens a theorem of Klauck (CCC 2003). (3) We use information complexity arguments to prove a linear lower bound on the USBP complexity of set-disjointness.

Cite as

Mika Göös and Thomas Watson. Communication Complexity of Set-Disjointness for All Probabilities. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 721-736, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


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@InProceedings{goos_et_al:LIPIcs.APPROX-RANDOM.2014.721,
  author =	{G\"{o}\"{o}s, Mika and Watson, Thomas},
  title =	{{Communication Complexity of Set-Disjointness for All Probabilities}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{721--736},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.721},
  URN =		{urn:nbn:de:0030-drops-47342},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.721},
  annote =	{Keywords: Communication Complexity, Set-Disjointness, All Probabilities}
}
Document
Graphs and Circuits: Some Further Remarks

Authors: Stasys Jukna

Published in: Dagstuhl Seminar Proceedings, Volume 6111, Complexity of Boolean Functions (2006)


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}
}
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