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NP-Hardness of Testing Equivalence to Sparse Polynomials and to Constant-Support Polynomials

Authors Omkar Baraskar, Agrim Dewan, Chandan Saha, Pulkit Sinha

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Author Details

Omkar Baraskar
  • University of Waterloo, Canada
Agrim Dewan
  • Indian Institute of Science, Bengaluru, India
Chandan Saha
  • Indian Institute of Science, Bengaluru, India
Pulkit Sinha
  • University of Waterloo, Canada

Funding

  • Saha, Chandan: Partially supported by a MATRICS grant of the Science and Engineering Research Board, DST, India.
  • Sinha, Pulkit: Partially supported by a Mike and Ophelia Lazardis Fellowship, and partially supported by NSERC Canada.

Acknowledgements

We thank the anonymous reviewers for their detailed and constructive feedback, which has helped us improve the presentation of this work. In particular, we thank one of the reviewers for pointing out some inaccuracies in the original proofs of Lemmas 46 and 47; simpler proofs for both lemmas came up in the process of fixing these inaccuracies.

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Omkar Baraskar, Agrim Dewan, Chandan Saha, and Pulkit Sinha. NP-Hardness of Testing Equivalence to Sparse Polynomials and to Constant-Support Polynomials. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 16:1-16:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.16

Abstract

An s-sparse polynomial has at most s monomials with nonzero coefficients. The Equivalence Testing problem for sparse polynomials (ETsparse) asks to decide if a given polynomial f is equivalent to (i.e., in the orbit of) some s-sparse polynomial. In other words, given f ∈ 𝔽[𝐱] and s ∈ ℕ, ETsparse asks to check if there exist A ∈ GL(|𝐱|, 𝔽) and 𝐛 ∈ 𝔽^|𝐱| such that f(A𝐱 + 𝐛) is s-sparse. We show that ETsparse is NP-hard over any field 𝔽, if f is given in the sparse representation, i.e., as a list of nonzero coefficients and exponent vectors. This answers a question posed by Gupta, Saha and Thankey (SODA 2023) and also, more explicitly, by Baraskar, Dewan and Saha (STACS 2024). The result implies that the Minimum Circuit Size Problem (MCSP) is NP-hard for a dense subclass of depth-3 arithmetic circuits if the input is given in sparse representation. We also show that approximating the smallest s₀ such that a given s-sparse polynomial f is in the orbit of some s₀-sparse polynomial to within a factor of s^{1/3 - ε} is NP-hard for any ε > 0; observe that s-factor approximation is trivial as the input is s-sparse. Finally, we show that for any constant σ ≥ 6, checking if a polynomial (given in sparse representation) is in the orbit of some support-σ polynomial is NP-hard. Support of a polynomial f is the maximum number of variables present in any monomial of f. These results are obtained via direct reductions from the 3-SAT problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Equivalence testing
  • MCSP
  • sparse polynomials
  • 3SAT

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