Algebraic Complexity Classes

Mahajan, Meena

doi:10.1007/978-3-319-05446-9_4

Cited by 14 publications

(3 citation statements)

References 72 publications

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“…We only give some very minimal notions of arithmetic circuit complexity. For more details we refer the reader to the very accessible recent survey [Mah14].…”

Section: Preliminariesmentioning

confidence: 99%

On the relative power of reduction notions in arithmetic circuit complexity

Ikenmeyer

Mengel

2018

Information Processing Letters

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“…We only give some very minimal notions of arithmetic circuit complexity. For more details we refer the reader to the very accessible recent survey [Mah14].…”

Section: Preliminariesmentioning

confidence: 99%

On the relative power of reduction notions in arithmetic circuit complexity

Ikenmeyer

Mengel

2018

Information Processing Letters

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“…They can be captured via iterated matrix multiplication, weakly-skew circuits, skew circuits, and determinants of a symbolic matrices. See Mahajan (2014) for an overview of these connections.…”

Section: Introductionmentioning

confidence: 99%

Factorization of Polynomials Given by Arithmetic Branching Programs

Sinhababu

Thierauf

2021

comput. complex.

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Given a multivariate polynomial computed by an arithmetic branching program (ABP) of size s, we show that all its factors can be computed by arithmetic branching programs of size poly(s). Kaltofen gave a similar result for polynomials computed by arithmetic circuits. The previously known best upper bound for ABP-factors was poly $$ (s^{ {\rm \log} s}) $$ ( s log s ) .

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“…We call the class of all polynomials computed by an ABP of polynomial size (in the number of variables) VBP. For more interesting connections, the reader is referred to excellent surveys by Mahajan [6] and Shpilka and Yehudayoff [9]. vertices of the graph can be subdivided into sets V 1 , .…”

Section: Introductionmentioning

confidence: 99%

Lower Bounds of Algebraic Branching Programs and Layerization

Engels

2020

Preprint

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In this paper we improve the lower bound of Chatterjee et al. (ECCC 2019) to an Ω(n 2 ) lower bound for unlayered Algebraic Branching Programs. We also study the impact layerization has on Algebraic Branching Programs. We exhibit a polynomial that has an unlayered ABP of size O(n) but any layered ABP has size at least Ω(n √ n). We exhibit a similar dichotomy in the non-commutative setting where the unlayered ABP has size O(n) and any layered ABP has size at least Ω(n log n − log 2 n).

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Algebraic Complexity Classes

Abstract: This survey describes, at an introductory level, the algebraic complexity framework originally proposed by Leslie Valiant in 1979, and some of the insights that have been obtained more recently.

Cited by 14 publications

References 72 publications

On the relative power of reduction notions in arithmetic circuit complexity

On the relative power of reduction notions in arithmetic circuit complexity

Factorization of Polynomials Given by Arithmetic Branching Programs

Lower Bounds of Algebraic Branching Programs and Layerization

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