13 Search Results for "Chatterjee, Prerona"

Document
DOI: 10.4230/LIPIcs.ITCS.2026.113
Multi-Quadratic Sum-Of-Squares Lower Bounds Imply VNC ¹ ≠ VNP

Authors: Benjamin Rossman and Davidson Zhu

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
The sum-of-squares (SoS) complexity of a d-multiquadratic polynomial f (quadratic in each of d blocks of n variables) is the minimum s such that f = ∑_{i = 1}^s g_i² with each g_i d-multilinear. In the case d = 2, Hrubeš, Wigderson and Yehudayoff [Hrubeš et al., 2011] showed that an n^{1+Ω(1)} lower bound on the SoS complexity of explicit biquadratic polynomials implies an exponential lower bound for non-commutative arithmetic circuits. In this paper, we establish an analogous connection between general multiquadratic sum-of-squares and commutative arithmetic formulas. Specifically, we show that an n^{d-o(log d)} lower bound on the SoS complexity of explicit d-multiquadratic polynomials, for any d = d(n) with ω(1) ≤ d(n) ≤ O((log n)/(log log n)), would separate the algebraic complexity classes VNC¹ and VNP.
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Benjamin Rossman and Davidson Zhu. Multi-Quadratic Sum-Of-Squares Lower Bounds Imply VNC ¹ ≠ VNP. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 113:1-113:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{rossman_et_al:LIPIcs.ITCS.2026.113,
  author =	{Rossman, Benjamin and Zhu, Davidson},
  title =	{{Multi-Quadratic Sum-Of-Squares Lower Bounds Imply VNC ¹ ≠ VNP}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{113:1--113:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.113},
  URN =		{urn:nbn:de:0030-drops-254006},
  doi =		{10.4230/LIPIcs.ITCS.2026.113},
  annote =	{Keywords: sum-of-squares, arithmetic formulas}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.115
Lower Bounds for Noncommutative Circuits with Low Syntactic Degree

Authors: Pratik Shastri

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
Proving lower bounds on the size of noncommutative arithmetic circuits is an important problem in arithmetic circuit complexity. For explicit n variate polynomials of degree Θ(n), the best known general bound is Ω(n log n) [Strassen, 1973; Walter Baur and Volker Strassen, 1983]. Recent work of Chatterjee and Hrubeš [Chatterjee and Hrubeš, 2023] has provided stronger (Ω(n²)) bounds for the restricted class of homogeneous circuits. The present paper extends these results to a broader class of circuits by using syntactic degree as a complexity measure. The syntactic degree of a circuit is a well known parameter which measures the extent to which high degree computation is used in the circuit. A homogeneous circuit computing a degree d polynomial can be assumed, without loss of generality, to have syntactic degree exactly equal to d [Fournier et al., 2024]. We generalize this by considering circuits that are not necessarily homogeneous but have low syntactic degree. Specifically, for an explicit n variate, degree n polynomial f we show that any circuit with syntactic degree O(n) computing f must have size Ω(n^{1+c}) for some constant c > 0. We also show that any circuit with syntactic degree o(nlog n) computing the same f must have size ω(nlog n). We further analyze the circuit size required to compute f based on the number of distinct syntactic degrees appearing in the circuit. Our analysis yields an ω(nlog n) size lower bound for all but a narrow parameter regime where an improved bound is not obtained. Finally, we observe that low syntactic degree circuits are more powerful than homogeneous circuits in a fine grained sense: there exists an n variate, degree Θ(n) polynomial that has a circuit of size O(nlog ²n) and syntactic degree O(n) but any homogeneous circuit computing it requires size Ω(n²).
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Pratik Shastri. Lower Bounds for Noncommutative Circuits with Low Syntactic Degree. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 115:1-115:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{shastri:LIPIcs.ITCS.2026.115,
  author =	{Shastri, Pratik},
  title =	{{Lower Bounds for Noncommutative Circuits with Low Syntactic Degree}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{115:1--115:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.115},
  URN =		{urn:nbn:de:0030-drops-254028},
  doi =		{10.4230/LIPIcs.ITCS.2026.115},
  annote =	{Keywords: Noncommutative Circuits, Lower Bounds, Circuit Complexity, Algebraic Complexity}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2025.22
IPS Lower Bounds for Formulas and Sum of ROABPs

Authors: Prerona Chatterjee, Utsab Ghosal, Partha Mukhopadhyay, and Amit Sinhababu

Published in: LIPIcs, Volume 360, 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)

Abstract
We give new lower bounds for the fragments of the Ideal Proof System (IPS) introduced by Grochow and Pitassi [Joshua A. Grochow and Toniann Pitassi, 2018]. The Ideal Proof System is a central topic in algebraic proof complexity developed in the context of Nullstellensatz refutation [Paul Beame et al., 1994] and simulates Extended Frege efficiently. Our main results are as follows. - mult-IPS_{Lin'}: We prove nearly quadratic-size formula lower bound for multilinear refutation (over the Boolean hypercube) of a variant of the subset-sum axiom polynomial. Extending this, we obtain a nearly matching qualitative statement for a constant degree target polynomial. - IPS_{Lin'}: Over the fields of characteristic zero, we prove exponential-size sum-of-ROABPs lower bound for the refutation of a variant of the subset-sum axiom polynomial. The result also extends over the fields of positive characteristics when the target polynomial is suitably modified. The modification is inspired by the recent results [Tuomas Hakoniemi et al., 2024; Amik Raj Behera et al., 2025]. The mult-IPS_{Lin'} lower bound result is obtained by combining the quadratic-size formula lower bound technique of Kalorkoti [Kalorkoti, 1985] with some additional ideas. The proof technique of IPS_{Lin'} lower bound result is inspired by the recent lower bound result of Chatterjee, Kush, Saraf and Shpilka [Prerona Chatterjee et al., 2024].
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Prerona Chatterjee, Utsab Ghosal, Partha Mukhopadhyay, and Amit Sinhababu. IPS Lower Bounds for Formulas and Sum of ROABPs. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 22:1-22:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chatterjee_et_al:LIPIcs.FSTTCS.2025.22,
  author =	{Chatterjee, Prerona and Ghosal, Utsab and Mukhopadhyay, Partha and Sinhababu, Amit},
  title =	{{IPS Lower Bounds for Formulas and Sum of ROABPs}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{22:1--22:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-406-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{360},
  editor =	{Aiswarya, C. and Mehta, Ruta and Roy, Subhajit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2025.22},
  URN =		{urn:nbn:de:0030-drops-251035},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.22},
  annote =	{Keywords: Ideal Proof System, Lower Bound, Algebraic Complexity}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.15
Algebraic Pseudorandomness in VNC⁰

Authors: Robert Andrews

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
We study the arithmetic complexity of hitting set generators, which are pseudorandom objects used for derandomization of the polynomial identity testing problem. We give new explicit constructions of hitting set generators whose outputs are computable in VNC⁰, i.e., can be computed by arithmetic formulas of constant size. Unconditionally, we construct a VNC⁰-computable generator that hits arithmetic circuits of constant depth and polynomial size. We also give conditional constructions, under strong but plausible hardness assumptions, of VNC⁰-computable generators that hit arithmetic formulas and arithmetic branching programs of polynomial size, respectively. As a corollary of our constructions, we derive lower bounds for subsystems of the Geometric Ideal Proof System of Grochow and Pitassi. Constructions of such generators are implicit in prior work of Kayal on lower bounds for the degree of annihilating polynomials. Our main contribution is a construction whose correctness relies on circuit complexity lower bounds rather than degree lower bounds.
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Robert Andrews. Algebraic Pseudorandomness in VNC⁰. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{andrews:LIPIcs.CCC.2025.15,
  author =	{Andrews, Robert},
  title =	{{Algebraic Pseudorandomness in VNC⁰}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{15:1--15:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.15},
  URN =		{urn:nbn:de:0030-drops-237092},
  doi =		{10.4230/LIPIcs.CCC.2025.15},
  annote =	{Keywords: Polynomial identity testing, Algebraic circuits, Ideal Proof System}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2024.10
Explicit Commutative ROABPs from Partial Derivatives

Authors: Vishwas Bhargava and Anamay Tengse

Published in: LIPIcs, Volume 323, 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)

Abstract
The dimension of partial derivatives (Nisan and Wigderson, 1997) is a popular measure for proving lower bounds in algebraic complexity. It is used to give strong lower bounds on the Waring decomposition of polynomials (called Waring rank). This naturally leads to an interesting open question: does this measure essentially characterize the Waring rank of any polynomial? The well-studied model of Read-once Oblivious ABPs (ROABPs for short) lends itself to an interesting hierarchy of "sub-models": Any-Order-ROABPs (ARO), Commutative ROABPs, and Diagonal ROABPs. It follows from previous works that for any polynomial, a bound on its Waring rank implies an analogous bound on its Diagonal ROABP complexity (called the duality trick), and a bound on its dimension of partial derivatives implies an analogous bound on its "ARO complexity": ROABP complexity in any order (Nisan, 1991). Our work strengthens the latter connection by showing that a bound on the dimension of partial derivatives in fact implies a bound on the commutative ROABP complexity. Thus, we improve our understanding of partial derivatives and move a step closer towards answering the above question. Our proof builds on the work of Ramya and Tengse (2022) to show that the commutative-ROABP-width of any homogeneous polynomial is at most the dimension of its partial derivatives. The technique itself is a generalization of the proof of the duality trick due to Saxena (2008).
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Vishwas Bhargava and Anamay Tengse. Explicit Commutative ROABPs from Partial Derivatives. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bhargava_et_al:LIPIcs.FSTTCS.2024.10,
  author =	{Bhargava, Vishwas and Tengse, Anamay},
  title =	{{Explicit Commutative ROABPs from Partial Derivatives}},
  booktitle =	{44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)},
  pages =	{10:1--10:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-355-3},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{323},
  editor =	{Barman, Siddharth and Lasota, S{\l}awomir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2024.10},
  URN =		{urn:nbn:de:0030-drops-221994},
  doi =		{10.4230/LIPIcs.FSTTCS.2024.10},
  annote =	{Keywords: Partial derivatives, Apolar ideals, Commuting matrices, Branching programs}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.20
Lower Bounds for Set-Multilinear Branching Programs

Authors: Prerona Chatterjee, Deepanshu Kush, Shubhangi Saraf, and Amir Shpilka

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
In this paper, we prove super-polynomial lower bounds for the model of sum of ordered set-multilinear algebraic branching programs, each with a possibly different ordering (∑smABP). Specifically, we give an explicit nd-variate polynomial of degree d such that any ∑smABP computing it must have size n^ω(1) for d as low as ω(log n). Notably, this constitutes the first such lower bound in the low degree regime. Moreover, for d = poly(n), we demonstrate an exponential lower bound. This result generalizes the seminal work of Nisan (STOC, 1991), which proved an exponential lower bound for a single ordered set-multilinear ABP. The significance of our lower bounds is underscored by the recent work of Bhargav, Dwivedi, and Saxena (TAMC, 2024), which showed that super-polynomial lower bounds against a sum of ordered set-multilinear branching programs - for a polynomial of sufficiently low degree - would imply super-polynomial lower bounds against general ABPs, thereby resolving Valiant’s longstanding conjecture that the permanent polynomial can not be computed efficiently by ABPs. More precisely, their work shows that if one could obtain such lower bounds when the degree is bounded by O(log n/ log log n), then it would imply super-polynomial lower bounds against general ABPs. Our results strengthen the works of Arvind & Raja (Chic. J. Theor. Comput. Sci., 2016) and Bhargav, Dwivedi & Saxena (TAMC, 2024), as well as the works of Ramya & Rao (Theor. Comput. Sci., 2020) and Ghoshal & Rao (International Computer Science Symposium in Russia, 2021), each of which established lower bounds for related or restricted versions of this model. They also strongly answer a question from the former two, which asked to prove super-polynomial lower bounds for general ∑smABP.
Cite as

Prerona Chatterjee, Deepanshu Kush, Shubhangi Saraf, and Amir Shpilka. Lower Bounds for Set-Multilinear Branching Programs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chatterjee_et_al:LIPIcs.CCC.2024.20,
  author =	{Chatterjee, Prerona and Kush, Deepanshu and Saraf, Shubhangi and Shpilka, Amir},
  title =	{{Lower Bounds for Set-Multilinear Branching Programs}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{20:1--20:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.20},
  URN =		{urn:nbn:de:0030-drops-204167},
  doi =		{10.4230/LIPIcs.CCC.2024.20},
  annote =	{Keywords: Lower Bounds, Algebraic Branching Programs, Set-multilinear polynomials}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2024.27
Determinants vs. Algebraic Branching Programs

Authors: Abhranil Chatterjee, Mrinal Kumar, and Ben Lee Volk

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
We show that for every homogeneous polynomial of degree d, if it has determinantal complexity at most s, then it can be computed by a homogeneous algebraic branching program (ABP) of size at most O(d⁵s). Moreover, we show that for most homogeneous polynomials, the width of the resulting homogeneous ABP is just s-1 and the size is at most O(ds). Thus, for constant degree homogeneous polynomials, their determinantal complexity and ABP complexity are within a constant factor of each other and hence, a super-linear lower bound for ABPs for any constant degree polynomial implies a super-linear lower bound on determinantal complexity; this relates two open problems of great interest in algebraic complexity. As of now, super-linear lower bounds for ABPs are known only for polynomials of growing degree [Mrinal Kumar, 2019; Prerona Chatterjee et al., 2022], and for determinantal complexity the best lower bounds are larger than the number of variables only by a constant factor [Mrinal Kumar and Ben Lee Volk, 2022]. While determinantal complexity and ABP complexity are classically known to be polynomially equivalent [Meena Mahajan and V. Vinay, 1997], the standard transformation from the former to the latter incurs a polynomial blow up in size in the process, and thus, it was unclear if a super-linear lower bound for ABPs implies a super-linear lower bound on determinantal complexity. In particular, a size preserving transformation from determinantal complexity to ABPs does not appear to have been known prior to this work, even for constant degree polynomials.
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Abhranil Chatterjee, Mrinal Kumar, and Ben Lee Volk. Determinants vs. Algebraic Branching Programs. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 27:1-27:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chatterjee_et_al:LIPIcs.ITCS.2024.27,
  author =	{Chatterjee, Abhranil and Kumar, Mrinal and Volk, Ben Lee},
  title =	{{Determinants vs. Algebraic Branching Programs}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{27:1--27:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.27},
  URN =		{urn:nbn:de:0030-drops-195550},
  doi =		{10.4230/LIPIcs.ITCS.2024.27},
  annote =	{Keywords: Determinant, Algebraic Branching Program, Lower Bounds, Singular Variety}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2023.11
Monotone Classes Beyond VNP

Authors: Prerona Chatterjee, Kshitij Gajjar, and Anamay Tengse

Published in: LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

Abstract
In this work, we study the natural monotone analogues of various equivalent definitions of VPSPACE: a well studied class (Poizat 2008, Koiran & Perifel 2009, Malod 2011, Mahajan & Rao 2013) that is believed to be larger than VNP. We observe that these monotone analogues are not equivalent unlike their non-monotone counterparts, and propose monotone VPSPACE (mVPSPACE) to be defined as the monotone analogue of Poizat’s definition. With this definition, mVPSPACE turns out to be exponentially stronger than mVNP and also satisfies several desirable closure properties that the other analogues may not. Our initial goal was to understand the monotone complexity of transparent polynomials, a concept that was recently introduced by Hrubeš & Yehudayoff (2021). In that context, we show that transparent polynomials of large sparsity are hard for the monotone analogues of all the known definitions of VPSPACE, except for the one due to Poizat.
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Prerona Chatterjee, Kshitij Gajjar, and Anamay Tengse. Monotone Classes Beyond VNP. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 11:1-11:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chatterjee_et_al:LIPIcs.FSTTCS.2023.11,
  author =	{Chatterjee, Prerona and Gajjar, Kshitij and Tengse, Anamay},
  title =	{{Monotone Classes Beyond VNP}},
  booktitle =	{43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)},
  pages =	{11:1--11:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-304-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{284},
  editor =	{Bouyer, Patricia and Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.11},
  URN =		{urn:nbn:de:0030-drops-193846},
  doi =		{10.4230/LIPIcs.FSTTCS.2023.11},
  annote =	{Keywords: Algebraic Complexity, Monotone Computation, VPSPACE, Transparent Polynomials}
}
Document
DOI: 10.4230/LIPIcs.CCC.2023.13
New Lower Bounds Against Homogeneous Non-Commutative Circuits

Authors: Prerona Chatterjee and Pavel Hrubeš

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Abstract
We give several new lower bounds on size of homogeneous non-commutative circuits. We present an explicit homogeneous bivariate polynomial of degree d which requires homogeneous non-commutative circuit of size Ω(d/log d). For an n-variate polynomial with n > 1, the result can be improved to Ω(nd), if d ≤ n, or Ω(nd (log n)/(log d)), if d ≥ n. Under the same assumptions, we also give a quadratic lower bound for the ordered version of the central symmetric polynomial.
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Prerona Chatterjee and Pavel Hrubeš. New Lower Bounds Against Homogeneous Non-Commutative Circuits. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 13:1-13:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chatterjee_et_al:LIPIcs.CCC.2023.13,
  author =	{Chatterjee, Prerona and Hrube\v{s}, Pavel},
  title =	{{New Lower Bounds Against Homogeneous Non-Commutative Circuits}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{13:1--13:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.13},
  URN =		{urn:nbn:de:0030-drops-182835},
  doi =		{10.4230/LIPIcs.CCC.2023.13},
  annote =	{Keywords: Algebraic circuit complexity, Non-Commutative Circuits, Homogeneous Computation, Lower bounds against algebraic circuits}
}
Document
DOI: 10.4230/LIPIcs.STACS.2022.44
If VNP Is Hard, Then so Are Equations for It

Authors: Mrinal Kumar, C. Ramya, Ramprasad Saptharishi, and Anamay Tengse

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
Assuming that the Permanent polynomial requires algebraic circuits of exponential size, we show that the class VNP does not have efficiently computable equations. In other words, any nonzero polynomial that vanishes on the coefficient vectors of all polynomials in the class VNP requires algebraic circuits of super-polynomial size. In a recent work of Chatterjee, Kumar, Ramya, Saptharishi and Tengse (FOCS 2020), it was shown that the subclasses of VP and VNP consisting of polynomials with bounded integer coefficients do have equations with small algebraic circuits. Their work left open the possibility that these results could perhaps be extended to all of VP or VNP. The results in this paper show that assuming the hardness of Permanent, at least for VNP, allowing polynomials with large coefficients does indeed incur a significant blow up in the circuit complexity of equations.
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Mrinal Kumar, C. Ramya, Ramprasad Saptharishi, and Anamay Tengse. If VNP Is Hard, Then so Are Equations for It. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 44:1-44:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kumar_et_al:LIPIcs.STACS.2022.44,
  author =	{Kumar, Mrinal and Ramya, C. and Saptharishi, Ramprasad and Tengse, Anamay},
  title =	{{If VNP Is Hard, Then so Are Equations for It}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{44:1--44:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.44},
  URN =		{urn:nbn:de:0030-drops-158547},
  doi =		{10.4230/LIPIcs.STACS.2022.44},
  annote =	{Keywords: Computational Complexity, Algebraic Circuits, Algebraic Natural Proofs}
}
Document
DOI: 10.4230/LIPIcs.CCC.2021.7
Separating ABPs and Some Structured Formulas in the Non-Commutative Setting

Authors: Prerona Chatterjee

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

Abstract
The motivating question for this work is a long standing open problem, posed by Nisan [Noam Nisan, 1991], regarding the relative powers of algebraic branching programs (ABPs) and formulas in the non-commutative setting. Even though the general question remains open, we make some progress towards its resolution. To that effect, we generalise the notion of ordered polynomials in the non-commutative setting (defined by Hrubeš, Wigderson and Yehudayoff [Hrubeš et al., 2011]) to define abecedarian polynomials and models that naturally compute them. Our main contribution is a possible new approach towards resolving the VF_{nc} vs VBP_{nc} question, via lower bounds against abecedarian formulas. In particular, we show the following. There is an explicit n²-variate degree d abecedarian polynomial f_{n,d}(𝐱) such that - f_{n, d}(𝐱) can be computed by an abecedarian ABP of size O(nd); - any abecedarian formula computing f_{n, log n}(𝐱) must have size at least n^{Ω(log log n)}. We also show that a super-polynomial lower bound against abecedarian formulas for f_{log n, n}(𝐱) would separate the powers of formulas and ABPs in the non-commutative setting.
Cite as

Prerona Chatterjee. Separating ABPs and Some Structured Formulas in the Non-Commutative Setting. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 7:1-7:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chatterjee:LIPIcs.CCC.2021.7,
  author =	{Chatterjee, Prerona},
  title =	{{Separating ABPs and Some Structured Formulas in the Non-Commutative Setting}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{7:1--7:24},
  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.2021.7},
  URN =		{urn:nbn:de:0030-drops-142812},
  doi =		{10.4230/LIPIcs.CCC.2021.7},
  annote =	{Keywords: Non-Commutative Formulas, Lower Bound, Separating ABPs and Formulas}
}
Document
DOI: 10.4230/LIPIcs.CCC.2020.2
A Quadratic Lower Bound for Algebraic Branching Programs

Authors: Prerona Chatterjee, Mrinal Kumar, Adrian She, and Ben Lee Volk

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Abstract
We show that any Algebraic Branching Program (ABP) computing the polynomial ∑_{i=1}^n xⁿ_i has at least Ω(n²) vertices. This improves upon the lower bound of Ω(nlog n), which follows from the classical result of Baur and Strassen [Volker Strassen, 1973; Walter Baur and Volker Strassen, 1983], and extends the results of Kumar [Mrinal Kumar, 2019], which showed a quadratic lower bound for homogeneous ABPs computing the same polynomial. Our proof relies on a notion of depth reduction which is reminiscent of similar statements in the context of matrix rigidity, and shows that any small enough ABP computing the polynomial ∑_{i=1}^n xⁿ_i can be depth reduced to essentially a homogeneous ABP of the same size which computes the polynomial ∑_{i=1}^n xⁿ_i + ε(𝐱), for a structured "error polynomial" ε(𝐱). To complete the proof, we then observe that the lower bound in [Mrinal Kumar, 2019] is robust enough and continues to hold for all polynomials ∑_{i=1}^n xⁿ_i + ε(𝐱), where ε(𝐱) has the appropriate structure.
Cite as

Prerona Chatterjee, Mrinal Kumar, Adrian She, and Ben Lee Volk. A Quadratic Lower Bound for Algebraic Branching Programs. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 2:1-2:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chatterjee_et_al:LIPIcs.CCC.2020.2,
  author =	{Chatterjee, Prerona and Kumar, Mrinal and She, Adrian and Volk, Ben Lee},
  title =	{{A Quadratic Lower Bound for Algebraic Branching Programs}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{2:1--2:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.2},
  URN =		{urn:nbn:de:0030-drops-125546},
  doi =		{10.4230/LIPIcs.CCC.2020.2},
  annote =	{Keywords: Algebraic Branching Programs, Lower Bound}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2019.11
Constructing Faithful Homomorphisms over Fields of Finite Characteristic

Authors: Prerona Chatterjee and Ramprasad Saptharishi

Published in: LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

Abstract
We study the question of algebraic rank or transcendence degree preserving homomorphisms over finite fields. This concept was first introduced by Beecken et al. [Malte Beecken et al., 2013] and exploited by them and Agrawal et al. [Manindra Agrawal et al., 2016] to design algebraic independence based identity tests using the Jacobian criterion over characteristic zero fields. An analogue of such constructions over finite characteristic fields were unknown due to the failure of the Jacobian criterion over finite characteristic fields. Building on a recent criterion of Pandey, Saxena and Sinhababu [Anurag Pandey et al., 2018], we construct explicit faithful maps for some natural classes of polynomials in fields of positive characteristic, when a certain parameter called the inseparable degree of the underlying polynomials is bounded (this parameter is always 1 in fields of characteristic zero). This presents the first generalisation of some of the results of Beecken, Mittmann and Saxena [Malte Beecken et al., 2013] and Agrawal, Saha, Saptharishi, Saxena [Manindra Agrawal et al., 2016] in the positive characteristic setting.
Cite as

Prerona Chatterjee and Ramprasad Saptharishi. Constructing Faithful Homomorphisms over Fields of Finite Characteristic. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chatterjee_et_al:LIPIcs.FSTTCS.2019.11,
  author =	{Chatterjee, Prerona and Saptharishi, Ramprasad},
  title =	{{Constructing Faithful Homomorphisms over Fields of Finite Characteristic}},
  booktitle =	{39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)},
  pages =	{11:1--11:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-131-3},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{150},
  editor =	{Chattopadhyay, Arkadev and Gastin, Paul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.11},
  URN =		{urn:nbn:de:0030-drops-115733},
  doi =		{10.4230/LIPIcs.FSTTCS.2019.11},
  annote =	{Keywords: Faithful Homomorphisms, Identity Testing, Algebraic Independence, Finite characteristic fields}
}
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