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Hitting Sets for Orbits of Circuit Classes and Polynomial Families

Authors Chandan Saha, Bhargav Thankey

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Author Details

Chandan Saha
  • Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India
Bhargav Thankey
  • Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India

Acknowledgements

We thank Rohit Gurjar, Ankit Garg, Neeraj Kayal, and Vishwas Bhargava for several stimulating discussions at the onset of this work.

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Chandan Saha and Bhargav Thankey. Hitting Sets for Orbits of Circuit Classes and Polynomial Families. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 50:1-50:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.50

Abstract

The orbit of an n-variate polynomial f(𝐱) over a field 𝔽 is the set {f(A𝐱+𝐛) : A ∈ GL(n,𝔽) and 𝐛 ∈ 𝔽ⁿ}. 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Hitting Sets
  • Orbits
  • ROABPs
  • Rank Concentration

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