License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.50
URN: urn:nbn:de:0030-drops-147433
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14743/
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Saha, Chandan ; Thankey, Bhargav

Hitting Sets for Orbits of Circuit Classes and Polynomial Families

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LIPIcs-APPROX50.pdf (0.9 MB)

Abstract

The orbit of an n-variate polynomial f(?) over a field ? is the set {f(A?+?) : A ∈ GL(n,?) and ? ∈ ?ⁿ}. In this paper, we initiate the study of explicit hitting sets for the orbits of polynomials computable by several natural and well-studied circuit classes and polynomial families. In particular, we give quasi-polynomial time hitting sets for the orbits of:
1) Low-individual-degree polynomials computable by commutative ROABPs. This implies quasi-polynomial time hitting sets for the orbits of the elementary symmetric polynomials.
2) Multilinear polynomials computable by constant-width ROABPs. This implies a quasi-polynomial time hitting set for the orbits of the family {IMM_{3,d}}_{d ∈ ℕ}, which is complete for arithmetic formulas.
3) Polynomials computable by constant-depth, constant-occur formulas. This implies quasi-polynomial time hitting sets for the orbits of multilinear depth-4 circuits with constant top fan-in, and also polynomial-time hitting sets for the orbits of the power symmetric and the sum-product polynomials.
4) Polynomials computable by occur-once formulas.

BibTeX - Entry

@InProceedings{saha_et_al:LIPIcs.APPROX/RANDOM.2021.50,
  author =	{Saha, Chandan and Thankey, Bhargav},
  title =	{{Hitting Sets for Orbits of Circuit Classes and Polynomial Families}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{50:1--50:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14743},
  URN =		{urn:nbn:de:0030-drops-147433},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.50},
  annote =	{Keywords: Hitting Sets, Orbits, ROABPs, Rank Concentration}
}

Keywords: Hitting Sets, Orbits, ROABPs, Rank Concentration
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)
Issue Date: 2021
Date of publication: 15.09.2021

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