Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Garcia-Marco, Ignacio; Koiran, Pascal; Pecatte, Timothée; Thomassé, Stéphan https://www.dagstuhl.de/lipics License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
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URN: urn:nbn:de:0030-drops-69964
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On the Complexity of Partial Derivatives

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Abstract

The method of partial derivatives is one of the most successful lower bound methods for arithmetic circuits. It uses as a complexity measure the dimension of the span of the partial derivatives of a polynomial. In this paper, we consider this complexity measure as a computational problem: for an input polynomial given as the sum of its nonzero monomials, what is the complexity of computing the dimension of its space of partial derivatives?

We show that this problem is #P-hard and we ask whether it belongs to #P. We analyze the "trace method", recently used in combinatorics and in algebraic complexity to lower bound the rank of certain matrices. We show that this method provides a polynomial-time computable lower bound on the dimension of the span of partial derivatives, and from this method we derive closed-form lower bounds. We leave as an open problem the existence of an approximation algorithm with reasonable performance guarantees.

BibTeX - Entry

@InProceedings{garciamarco_et_al:LIPIcs:2017:6996,
  author =	{Ignacio Garcia-Marco and Pascal Koiran and Timoth{\'e}e Pecatte and St{\'e}phan Thomass{\'e}},
  title =	{{On the Complexity of Partial Derivatives}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{37:1--37:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Heribert Vollmer and Brigitte Vallée},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/6996},
  URN =		{urn:nbn:de:0030-drops-69964},
  doi =		{10.4230/LIPIcs.STACS.2017.37},
  annote =	{Keywords: counting complexity, simplicial complex, lower bounds, arithmetic circuits}
}

Keywords: counting complexity, simplicial complex, lower bounds, arithmetic circuits
Seminar: 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)
Issue date: 2017
Date of publication: 06.03.2017


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