3 Search Results for "Podolskii, Vladimir V."


Document
Polynomial Threshold Functions for Decision Lists

Authors: Vladimir Podolskii and Nikolay V. Proskurin

Published in: LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)


Abstract
For S ⊆ {0,1}ⁿ a Boolean function f : S → {-1,1} is a polynomial threshold function (PTF) of degree d and weight W if there is a polynomial p with integer coefficients of degree d and with sum of absolute coefficients W such that f(x) = sign p(x) for all x ∈ S. We study a representation of decision lists as PTFs over Boolean cubes {0,1}ⁿ and over Hamming balls {0,1}ⁿ_{≤ k}. As our first result, we show that for all d = O((n/(log n))^{1/3}) any decision list over {0,1}ⁿ can be represented by a PTF of degree d and weight 2^O(n/d²). This improves the result by Klivans and Servedio [Adam R. Klivans and Rocco A. Servedio, 2006] by a log² d factor in the exponent of the weight. Our bound is tight for all d = O((n/(log n))^{1/3}) due to the matching lower bound by Beigel [Richard Beigel, 1994]. For decision lists over a Hamming ball {0,1}ⁿ_{≤ k} we show that the upper bound on weight above can be drastically improved to n^O(√k) for d = Θ(√k). We also show that similar improvement is not possible for smaller degrees by proving the lower bound W = 2^Ω(n/d²) for all d = O(√k).

Cite as

Vladimir Podolskii and Nikolay V. Proskurin. Polynomial Threshold Functions for Decision Lists. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 52:1-52:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{podolskii_et_al:LIPIcs.ISAAC.2022.52,
  author =	{Podolskii, Vladimir and Proskurin, Nikolay V.},
  title =	{{Polynomial Threshold Functions for Decision Lists}},
  booktitle =	{33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
  pages =	{52:1--52:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-258-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{248},
  editor =	{Bae, Sang Won and Park, Heejin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.52},
  URN =		{urn:nbn:de:0030-drops-173372},
  doi =		{10.4230/LIPIcs.ISAAC.2022.52},
  annote =	{Keywords: Threshold function, decision list, Hamming ball}
}
Document
Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates

Authors: Alexander S. Kulikov and Vladimir V. Podolskii

Published in: LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)


Abstract
We study the following computational problem: for which values of k, the majority of n bits MAJ_n can be computed with a depth two formula whose each gate computes a majority function of at most k bits? The corresponding computational model is denoted by MAJ_k o MAJ_k. We observe that the minimum value of k for which there exists a MAJ_k o MAJ_k circuit that has high correlation with the majority of n bits is equal to Theta(sqrt(n)). We then show that for a randomized MAJ_k o MAJ_k circuit computing the majority of n input bits with high probability for every input, the minimum value of k is equal to n^(2/3+o(1)). We show a worst case lower bound: if a MAJ_k o MAJ_k circuit computes the majority of n bits correctly on all inputs, then k <= n^(13/19+o(1)). This lower bound exceeds the optimal value for randomized circuits and thus is unreachable for pure randomized techniques. For depth 3 circuits we show that a circuit with k= O(n^(2/3)) can compute MAJ_n correctly on all inputs.

Cite as

Alexander S. Kulikov and Vladimir V. Podolskii. Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 49:1-49:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{kulikov_et_al:LIPIcs.STACS.2017.49,
  author =	{Kulikov, Alexander S. and Podolskii, Vladimir V.},
  title =	{{Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{49:1--49:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Vollmer, Heribert and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.49},
  URN =		{urn:nbn:de:0030-drops-69832},
  doi =		{10.4230/LIPIcs.STACS.2017.49},
  annote =	{Keywords: circuit complexity, computational complexity, threshold, majority, lower bound, upper bound}
}
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-dev.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}
}
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