Progress on Polynomial Identity Testing-II

Saxena, Nitin

doi:10.1007/978-3-319-05446-9_7

Cited by 57 publications

(57 citation statements)

References 51 publications

(42 reference statements)

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“…One of the main reasons for interest in this problem is the connection between PIT algorithms and circuit lower bounds (Heintz & Schnorr [HS80], Kabanets & Impagliazzo [KI03] and Agrawal [Agr05,Agr06]). Refer to surveys for a detailed treatment of PIT [Sax09,AS09].…”

Section: Introductionmentioning

confidence: 99%

Blackbox Identity Testing for Bounded Top-Fanin Depth-3 Circuits: The Field Doesn't Matter

Saxena¹,

Seshadhri²

2012

SIAM J. Comput.

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Section: Introductionmentioning

confidence: 99%

Blackbox Identity Testing for Bounded Top-Fanin Depth-3 Circuits: The Field Doesn't Matter

Saxena¹,

Seshadhri²

2012

SIAM J. Comput.

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“…It is also known that any sub-exponential time algorithm for PIT implies a lower bound [KI03,Agr05]. See also the surveys [Sax09,Sax14,SY10].…”

Section: Introductionmentioning

confidence: 99%

Deterministic Identity Testing for Sum of Read-Once Oblivious Arithmetic Branching Programs

et al. 2016

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A read-once oblivious arithmetic branching program (ROABP) is an arithmetic branching program (ABP) where each variable occurs in at most one layer. We give the first polynomial time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding blackbox algorithm with quasi-polynomial time complexity n O(log n) . In both the cases, our time complexity is double exponential in the number of ROABPs.ROABPs are a generalization of set-multilinear depth-3 circuits. The prior results for the sum of constantly many set-multilinear depth-3 circuits were only slightly better than brute-force, i.e. exponential-time.Our techniques are a new interplay of three concepts for ROABP: low evaluation dimension, basis isolating weight assignment and low-support rank concentration. We relate basis isolation to rank concentration and extend it to a sum of two ROABPs using evaluation dimension (or partial derivatives).

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“…In 2004, Impagliazzo and Kabanets [KI04] showed that a deterministic polynomial time algorithm for PIT implies lower bounds (either NEXP ⊂ P/poly or permanent does not have polynomial size arithmetic circuits), thus making it one of the central problems in algebraic complexity. Following [KI04], intense efforts over the last decade have been directed towards de-randomizing PIT (see for instance [SY10,Sax14]). The attempts fall into two categories: considering special cases ( [Sax14]), and optimizing the random bits used in the Schwartz-Zippel test [BHS08,BE11].…”

Section: Introductionmentioning

confidence: 99%

“…Following [KI04], intense efforts over the last decade have been directed towards de-randomizing PIT (see for instance [SY10,Sax14]). The attempts fall into two categories: considering special cases ( [Sax14]), and optimizing the random bits used in the Schwartz-Zippel test [BHS08,BE11].…”

Section: Introductionmentioning

confidence: 99%

“…In 2008, Agrawal and Vinay [AV08] explained this difficulty, showing that deterministic polynomial time algorithms for PIT on depth four circuits implies sub-exponential time deterministic algorithms for general circuits. There have been several interesting approaches towards obtaining black-box algorithms for PIT on restricted classes of depth three and four circuits, see [Sax14,SY10] for further details. Recently, Kamath, Kayal and Saptharishi [GKKS13] showed that, over infinite fields, deterministic polynomial time algorithms for PIT on depth three circuits would also imply lower bounds for the permanent.…”

Section: Introductionmentioning

confidence: 99%

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Building above Read-once Polynomials: Identity Testing and Hardness of Representation

Mahajan¹,

Rao

Sreenivasaiah³

2014

Lecture Notes in Computer Science

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Abstract. Polynomial Identity Testing (PIT) algorithms have focussed on polynomials computed either by small alternation-depth arithmetic circuits, or by read-restricted formulas. Read-once polynomials (ROPs) are computed by read-once formulas (ROFs) and are the simplest of read-restricted polynomials. Building structures above these, we show:1. A deterministic polynomial-time non-black-box PIT algorithm for (2) · ·ROF. 2. Weak hardness of representation theorems for sums of powers of constant-free ROPs and for 0-justified alternation-depth-3 ROPs.

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Progress on Polynomial Identity Testing-II

Cited by 57 publications

References 51 publications

Blackbox Identity Testing for Bounded Top-Fanin Depth-3 Circuits: The Field Doesn't Matter

Blackbox Identity Testing for Bounded Top-Fanin Depth-3 Circuits: The Field Doesn't Matter

Deterministic Identity Testing for Sum of Read-Once Oblivious Arithmetic Branching Programs

Building above Read-once Polynomials: Identity Testing and Hardness of Representation

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