2 Search Results for "Medini, Dori"

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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2024.16

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

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

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

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.

Cite as

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)

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@InProceedings{baraskar_et_al:LIPIcs.ICALP.2024.16,
  author =	{Baraskar, Omkar and Dewan, Agrim and Saha, Chandan and Sinha, Pulkit},
  title =	{{NP-Hardness of Testing Equivalence to Sparse Polynomials and to Constant-Support Polynomials}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{16:1--16:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.16},
  URN =		{urn:nbn:de:0030-drops-201598},
  doi =		{10.4230/LIPIcs.ICALP.2024.16},
  annote =	{Keywords: Equivalence testing, MCSP, sparse polynomials, 3SAT}
}

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DOI: 10.4230/LIPIcs.CCC.2021.19

Hitting Sets and Reconstruction for Dense Orbits in VP_{e} and ΣΠΣ Circuits

Authors: Dori Medini and Amir Shpilka

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Abstract

In this paper we study polynomials in VP_{e} (polynomial-sized formulas) and in ΣΠΣ (polynomial-size depth-3 circuits) whose orbits, under the action of the affine group GL^{aff}_n(𝔽) (the action of (A,b) ∈ GL^{aff}_n(𝔽) on a polynomial f ∈ 𝔽[x] is defined as (A,b)∘f = f(A^Tx+b)), are dense in their ambient class. We construct hitting sets and interpolating sets for these orbits as well as give reconstruction algorithms. Specifically, we obtain the following results: 1) For C_n(ℓ_1(x),…,ℓ_n(x)) ≜ Trace(\begin{pmatrix} 𝓁₁(x) & 1 \\ 1 & 0 \end{pmatrix} ⋅ … ⋅ \begin{pmatrix} 𝓁_n(x) & 1 \\ 1 & 0 \end{pmatrix}), where the 𝓁_is are linearly independent linear functions, we construct a polynomial-sized interpolating set, and give a polynomial-time reconstruction algorithm. By a result of Bringmann, Ikenmeyer and Zuiddam, the set of all such polynomials is dense in VP_e [Karl Bringmann et al., 2018], thus our construction gives the first polynomial-size interpolating set for a dense subclass of VP_e. 2) For polynomials of the form ANF_Δ(𝓁₁(x),…,𝓁_{4^Δ}(x)), where ANF_Δ(x) is the canonical read-once formula in alternating normal form, of depth 2Δ, and the 𝓁_is are linearly independent linear functions, we provide a quasipolynomial-size interpolating set. We also observe that the reconstruction algorithm of [Ankit Gupta et al., 2014] works for all polynomials in this class. This class is also dense in VP_e. 3) Similarly, we give a quasipolynomial-sized hitting set for read-once formulas (not necessarily in alternating normal form) composed with a set of linearly independent linear functions. This gives another dense class in VP_e. 4) We give a quasipolynomial-sized hitting set for polynomials of the form f(𝓁₁(x),…,𝓁_{m}(x)), where f is an m-variate s-sparse polynomial. and the 𝓁_is are linearly independent linear functions in n ≥ m variables. This class is dense in ΣΠΣ. 5) For polynomials of the form ∑_{i=1}^{s}∏_{j=1}^{d}𝓁_{i,j}(x), where the 𝓁_{i,j}s are linearly independent linear functions, we construct a polynomial-sized interpolating set. We also observe that the reconstruction algorithm of [Neeraj Kayal and Chandan Saha, 2019] works for every polynomial in the class. This class is dense in ΣΠΣ. As VP = VNC², our results for VP_{e} translate immediately to VP with a quasipolynomial blow up in parameters. If any of our hitting or interpolating sets could be made robust then this would immediately yield a hitting set for the superclass in which the relevant class is dense, and as a consequence also a lower bound for the superclass. Unfortunately, we also prove that the kind of constructions that we have found (which are defined in terms of k-independent polynomial maps) do not necessarily yield robust hitting sets.

Cite as

Dori Medini and Amir Shpilka. Hitting Sets and Reconstruction for Dense Orbits in VP_{e} and ΣΠΣ Circuits. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 19:1-19:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{medini_et_al:LIPIcs.CCC.2021.19,
  author =	{Medini, Dori and Shpilka, Amir},
  title =	{{Hitting Sets and Reconstruction for Dense Orbits in VP\underline\{e\} and \Sigma\Pi\Sigma Circuits}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{19:1--19:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.19},
  URN =		{urn:nbn:de:0030-drops-142930},
  doi =		{10.4230/LIPIcs.CCC.2021.19},
  annote =	{Keywords: Algebraic complexity, VP, VNP, algebraic circuits, algebraic formula}
}

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1 Baraskar, Omkar
1 Dewan, Agrim
1 Medini, Dori
1 Saha, Chandan
1 Shpilka, Amir
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2 Theory of computation → Algebraic complexity theory

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1 3SAT
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