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Electronic Colloquium on Computational Complexity

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All reports by Author Ramprasad Saptharishi:

TR20-063 | 29th April 2020
Prerona Chatterjee, Mrinal Kumar, C Ramya, Ramprasad Saptharishi, Anamay Tengse

On the Existence of Algebraically Natural Proofs

For every constant c > 0, we show that there is a family {P_{N,c}} of polynomials whose degree and algebraic circuit complexity are polynomially bounded in the number of variables, and that satisfies the following properties:
* For every family {f_n} of polynomials in VP, where f_n is an n ... more >>>

TR19-065 | 1st May 2019
Mrinal Kumar, Ramprasad Saptharishi, Noam Solomon

Derandomization from Algebraic Hardness: Treading the Borders

Revisions: 3

A hitting-set generator (HSG) is a polynomial map $Gen:\mathbb{F}^k \to \mathbb{F}^n$ such that for all $n$-variate polynomials $Q$ of small enough circuit size and degree, if $Q$ is non-zero, then $Q\circ Gen$ is non-zero. In this paper, we give a new construction of such a HSG assuming that we have ... more >>>

TR19-019 | 19th February 2019
Mrinal Kumar, Rafael Mendes de Oliveira, Ramprasad Saptharishi

Towards Optimal Depth Reductions for Syntactically Multilinear Circuits

We show that any $n$-variate polynomial computable by a syntactically multilinear circuit of size $\mathop{poly}(n)$ can be computed by a depth-$4$ syntactically multilinear ($\Sigma\Pi\Sigma\Pi$) circuit of size at most $\exp\left({O\left(\sqrt{n\log n}\right)}\right)$. For degree $d = \omega(n/\log n)$, this improves upon the upper bound of $\exp\left({O(\sqrt{d}\log n)}\right)$ obtained by Tavenas (MFCS ... more >>>

TR18-212 | 26th December 2018
Prerona Chatterjee, Ramprasad Saptharishi

Constructing Faithful Homomorphisms over Fields of Finite Characteristic

We study the question of algebraic rank or transcendence degree preserving homomorphisms over finite fields. This concept was first introduced by Beecken, Mittmann and Saxena (Information and Computing, 2013), and exploited by them, and Agrawal, Saha, Saptharishi and Saxena (Journal of Computing, 2016) to design algebraic independence based identity tests ... more >>>

TR18-132 | 17th July 2018
Mrinal Kumar, Ramprasad Saptharishi, Anamay Tengse

Near-optimal Bootstrapping of Hitting Sets for Algebraic Circuits

Revisions: 2

The classical lemma of Ore-DeMillo-Lipton-Schwartz-Zippel states that any nonzero polynomial $f(x_1,\ldots, x_n)$ of degree at most $s$ will evaluate to a nonzero value at some point on a grid $S^n \subseteq \mathbb{F}^n$ with $|S| > s$. Thus, there is a deterministic polynomial identity test (PIT) for all degree-$s$ size-$s$ ... more >>>

TR17-135 | 10th September 2017
Ramprasad Saptharishi, Anamay Tengse

Quasi-polynomial Hitting Sets for Circuits with Restricted Parse Trees

Revisions: 1

We study the class of non-commutative Unambiguous circuits or Unique-Parse-Tree (UPT) circuits, and a related model of Few-Parse-Trees (FewPT) circuits (which were recently introduced by Lagarde, Malod and Perifel [LMP16] and Lagarde, Limaye and Srinivasan [LLS17]) and give the following constructions:
• An explicit hitting set of quasipolynomial size for ... more >>>

TR16-137 | 3rd September 2016
Mrinal Kumar, Ramprasad Saptharishi

Finer separations between shallow arithmetic circuits

In this paper, we show that there is a family of polynomials $\{P_n\}$, where $P_n$ is a polynomial in $n$ variables of degree at most $d = O(\log^2 n)$, such that

1. $P_n$ can be computed by linear sized homogeneous depth-$5$ circuits.

2. $P_n$ can be computed by ... more >>>

TR16-096 | 14th June 2016
Suryajith Chillara, Mrinal Kumar, Ramprasad Saptharishi, V Vinay

The Chasm at Depth Four, and Tensor Rank : Old results, new insights

Revisions: 2

Agrawal and Vinay [AV08] showed how any polynomial size arithmetic circuit can be thought of as a depth four arithmetic circuit of subexponential size. The resulting circuit size in this simulation was more carefully analyzed by Korian [Koiran] and subsequently by Tavenas [Tav13]. We provide a simple proof of this ... more >>>

TR16-045 | 22nd March 2016
Michael Forbes, Mrinal Kumar, Ramprasad Saptharishi

Functional lower bounds for arithmetic circuits and connections to boolean circuit complexity

We say that a circuit $C$ over a field $F$ functionally computes an $n$-variate polynomial $P \in F[x_1, x_2, \ldots, x_n]$ if for every $x \in \{0,1\}^n$ we have that $C(x) = P(x)$. This is in contrast to {syntactically} computing $P$, when $C \equiv P$ as formal polynomials. In this ... more >>>

TR15-184 | 21st November 2015
Matthew Anderson, Michael Forbes, Ramprasad Saptharishi, Amir Shpilka, Ben Lee Volk

Identity Testing and Lower Bounds for Read-$k$ Oblivious Algebraic Branching Programs

Read-$k$ oblivious algebraic branching programs are a natural generalization of the well-studied model of read-once oblivious algebraic branching program (ROABPs).
In this work, we give an exponential lower bound of $\exp(n/k^{O(k)})$ on the width of any read-$k$ oblivious ABP computing some explicit multilinear polynomial $f$ that is computed by a ... more >>>

TR15-109 | 1st July 2015
Mrinal Kumar, Ramprasad Saptharishi

An exponential lower bound for homogeneous depth-5 circuits over finite fields

In this paper, we show exponential lower bounds for the class of homogeneous depth-$5$ circuits over all small finite fields. More formally, we show that there is an explicit family $\{P_d : d \in N\}$ of polynomials in $VNP$, where $P_d$ is of degree $d$ in $n = d^{O(1)}$ variables, ... more >>>

TR15-069 | 21st April 2015
Amey Bhangale, Ramprasad Saptharishi, Girish Varma, Rakesh Venkat

On Fortification of General Games

Revisions: 1

A recent result of Moshkovitz~\cite{Moshkovitz14} presented an ingenious method to provide a completely elementary proof of the Parallel Repetition Theorem for certain projection games via a construction called fortification. However, the construction used in \cite{Moshkovitz14} to fortify arbitrary label cover instances using an arbitrary extractor is insufficient to prove parallel ... more >>>

TR13-132 | 23rd September 2013
Michael Forbes, Ramprasad Saptharishi, Amir Shpilka

Pseudorandomness for Multilinear Read-Once Algebraic Branching Programs, in any Order

We give deterministic black-box polynomial identity testing algorithms for multilinear read-once oblivious algebraic branching programs (ROABPs), in n^(lg^2 n) time. Further, our algorithm is oblivious to the order of the variables. This is the first sub-exponential time algorithm for this model. Furthermore, our result has no known analogue in the ... more >>>

TR13-091 | 17th June 2013
Neeraj Kayal, Chandan Saha, Ramprasad Saptharishi

A super-polynomial lower bound for regular arithmetic formulas.

We consider arithmetic formulas consisting of alternating layers of addition $(+)$ and multiplication $(\times)$ gates such that the fanin of all the gates in any fixed layer is the same. Such a formula $\Phi$ which additionally has the property that its formal/syntactic degree is at most twice the (total) degree ... more >>>

TR13-026 | 11th February 2013
Ankit Gupta, Pritish Kamath, Neeraj Kayal, Ramprasad Saptharishi

Arithmetic circuits: A chasm at depth three

Revisions: 1

We show that, over $\mathbb{C}$, if an $n$-variate polynomial of degree $d = n^{O(1)}$ is computable by an arithmetic circuit of size $s$ (respectively by an algebraic branching program of size $s$) then it can also be computed by a depth three circuit (i.e. a $\Sigma \Pi \Sigma$-circuit) of size ... more >>>

TR12-098 | 3rd August 2012
Ankit Gupta, Pritish Kamath, Neeraj Kayal, Ramprasad Saptharishi

An exponential lower bound for homogeneous depth four arithmetic circuits with bounded bottom fanin

Revisions: 3

Agrawal and Vinay (FOCS 2008) have recently shown that an exponential lower bound for depth four homogeneous circuits with bottom layer of $\times$ gates having sublinear fanin translates to an exponential lower bound for a general arithmetic circuit computing the permanent. Motivated by this, we examine the complexity of computing ... more >>>

TR11-143 | 2nd November 2011
Manindra Agrawal, Chandan Saha, Ramprasad Saptharishi, Nitin Saxena

Jacobian hits circuits: Hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k circuits

We present a single, common tool to strictly subsume all known cases of polynomial time blackbox polynomial identity testing (PIT) that have been hitherto solved using diverse tools and techniques. In particular, we show that polynomial time hitting-set generators for identity testing of the two seemingly different and well studied ... more >>>

TR11-021 | 13th February 2011
Chandan Saha, Ramprasad Saptharishi, Nitin Saxena

A Case of Depth-3 Identity Testing, Sparse Factorization and Duality

Finding an efficient solution to the general problem of polynomial identity testing (PIT) is a challenging task. In this work, we study the complexity of two special but natural cases of identity testing - first is a case of depth-$3$ PIT, the other of depth-$4$ PIT.

Our first problem is ... more >>>

TR09-036 | 14th April 2009
Chandan Saha, Ramprasad Saptharishi, Nitin Saxena

The Power of Depth 2 Circuits over Algebras

We study the problem of polynomial identity testing (PIT) for depth
2 arithmetic circuits over matrix algebra. We show that identity
testing of depth 3 (Sigma-Pi-Sigma) arithmetic circuits over a field
F is polynomial time equivalent to identity testing of depth 2
(Pi-Sigma) arithmetic circuits over U_2(F), the ... more >>>

TR08-023 | 10th January 2008
Anindya De, Piyush Kurur, Chandan Saha, Ramprasad Saptharishi

Fast Integer Multiplication using Modular Arithmetic

We give an $O(N\cdot \log N\cdot 2^{O(\log^*N)})$ algorithm for
multiplying two $N$-bit integers that improves the $O(N\cdot \log
N\cdot \log\log N)$ algorithm by
Sch\"{o}nhage-Strassen. Both these algorithms use modular
arithmetic. Recently, F\"{u}rer gave an $O(N\cdot \log
N\cdot 2^{O(\log^*N)})$ algorithm which however uses arithmetic over
complex numbers as opposed to ... more >>>

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