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DOI: 10.4230/LIPIcs.ITCS.2020.72

Trade-Offs Between Size and Degree in Polynomial Calculus

Authors: Guillaume Lagarde, Jakob Nordström, Dmitry Sokolov, and Joseph Swernofsky

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract

Building on [Clegg et al. '96], [Impagliazzo et al. '99] established that if an unsatisfiable k-CNF formula over n variables has a refutation of size S in the polynomial calculus resolution proof system, then this formula also has a refutation of degree k + O(√(n log S)). The proof of this works by converting a small-size refutation into a small-degree one, but at the expense of increasing the proof size exponentially. This raises the question of whether it is possible to achieve both small size and small degree in the same refutation, or whether the exponential blow-up is inherent. Using and extending ideas from [Thapen '16], who studied the analogous question for the resolution proof system, we prove that a strong size-degree trade-off is necessary.

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Guillaume Lagarde, Jakob Nordström, Dmitry Sokolov, and Joseph Swernofsky. Trade-Offs Between Size and Degree in Polynomial Calculus. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 72:1-72:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{lagarde_et_al:LIPIcs.ITCS.2020.72,
  author =	{Lagarde, Guillaume and Nordstr\"{o}m, Jakob and Sokolov, Dmitry and Swernofsky, Joseph},
  title =	{{Trade-Offs Between Size and Degree in Polynomial Calculus}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{72:1--72:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.72},
  URN =		{urn:nbn:de:0030-drops-117573},
  doi =		{10.4230/LIPIcs.ITCS.2020.72},
  annote =	{Keywords: proof complexity, polynomial calculus, polynomial calculus resolution, PCR, size-degree trade-off, resolution, colored polynomial local search}
}

Document

DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.26

Tensor Rank is Hard to Approximate

Authors: Joseph Swernofsky

Published in: LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

Abstract

We prove that approximating the rank of a 3-tensor to within a factor of 1 + 1/1852 - delta, for any delta > 0, is NP-hard over any field. We do this via reduction from bounded occurrence 2-SAT.

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Joseph Swernofsky. Tensor Rank is Hard to Approximate. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 26:1-26:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{swernofsky:LIPIcs.APPROX-RANDOM.2018.26,
  author =	{Swernofsky, Joseph},
  title =	{{Tensor Rank is Hard to Approximate}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{26:1--26:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-085-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{116},
  editor =	{Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.26},
  URN =		{urn:nbn:de:0030-drops-94309},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.26},
  annote =	{Keywords: tensor rank, high rank tensor, slice elimination, approximation algorithm, hardness of approximation}
}

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Trade-Offs Between Size and Degree in Polynomial Calculus

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Tensor Rank is Hard to Approximate

Abstract

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