All reports by Author Srikanth Srinivasan:

__
TR23-191
| 2nd December 2023
__

Hervé Fournier, Nutan Limaye, Srikanth Srinivasan, Sébastien Tavenas#### On the Power of Homogeneous Algebraic Formulas

__
TR23-009
| 14th February 2023
__

Hervé Fournier, Nutan Limaye, Guillaume Malod, Srikanth Srinivasan, Sébastien Tavenas#### Towards Optimal Depth-Reductions for Algebraic Formulas

__
TR21-094
| 6th July 2021
__

Nutan Limaye, Srikanth Srinivasan, Sébastien Tavenas#### New Non-FPT Lower Bounds for Some Arithmetic Formulas

__
TR21-081
| 14th June 2021
__

Nutan Limaye, Srikanth Srinivasan, Sébastien Tavenas#### Superpolynomial Lower Bounds Against Low-Depth Algebraic Circuits

Revisions: 1

__
TR20-046
| 13th April 2020
__

Srikanth Srinivasan#### A Robust Version of Heged\H{u}s's Lemma, with Applications

__
TR19-133
| 2nd October 2019
__

Nutan Limaye, Srikanth Srinivasan, Utkarsh Tripathi#### More on $AC^0[\oplus]$ and Variants of the Majority Function

Revisions: 1

__
TR19-109
| 21st August 2019
__

Srikanth Srinivasan, Utkarsh Tripathi, S Venkitesh#### Decoding Downset codes over a finite grid

__
TR19-073
| 17th May 2019
__

Igor Carboni Oliveira, Rahul Santhanam, Srikanth Srinivasan#### Parity helps to compute Majority

__
TR19-032
| 4th March 2019
__

Srikanth Srinivasan#### Strongly Exponential Separation Between Monotone VP and Monotone VNP

Revisions: 1

__
TR18-157
| 10th September 2018
__

Nutan Limaye, Karteek Sreenivasiah, Srikanth Srinivasan, Utkarsh Tripathi, S Venkitesh#### The Coin Problem in Constant Depth: Sample Complexity and Parity gates

Revisions: 2

__
TR17-138
| 17th September 2017
__

Srikanth Srinivasan, Madhu Sudan#### Local decoding and testing of polynomials over grids

Revisions: 1

__
TR17-077
| 30th April 2017
__

Guillaume Lagarde, Nutan Limaye, Srikanth Srinivasan#### Lower Bounds and PIT for Non-Commutative Arithmetic circuits with Restricted Parse Trees

__
TR17-022
| 13th February 2017
__

Benjamin Rossman, Srikanth Srinivasan#### Separation of AC$^0[\oplus]$ Formulas and Circuits

__
TR16-204
| 20th December 2016
__

Prahladh Harsha, Srikanth Srinivasan#### Robust Multiplication-based Tests for Reed-Muller Codes

__
TR16-068
| 28th April 2016
__

Prahladh Harsha, Srikanth Srinivasan#### On Polynomial Approximations to $\mathrm{AC}^0$

Revisions: 1

__
TR15-191
| 26th November 2015
__

Ruiwen Chen, Rahul Santhanam, Srikanth Srinivasan#### Average-Case Lower Bounds and Satisfiability Algorithms for Small Threshold Circuits

__
TR15-142
| 28th August 2015
__

Srikanth Srinivasan#### A Compression Algorithm for $AC^0[\oplus]$ circuits using Certifying Polynomials

__
TR13-100
| 15th July 2013
__

Hervé Fournier, Nutan Limaye, Guillaume Malod, Srikanth Srinivasan#### Lower bounds for depth $4$ formulas computing iterated matrix multiplication

__
TR12-102
| 16th August 2012
__

Swastik Kopparty, Srikanth Srinivasan#### Certifying Polynomials for $\mathrm{AC}^0[\oplus]$ circuits, with applications

__
TR12-051
| 25th April 2012
__

Dmytro Gavinsky, Shachar Lovett, Michael Saks, Srikanth Srinivasan#### A Tail Bound for Read-k Families of Functions

__
TR11-131
| 29th September 2011
__

Rahul Santhanam, Srikanth Srinivasan#### On the Limits of Sparsification

__
TR09-122
| 23rd November 2009
__

Vikraman Arvind, Srikanth Srinivasan#### Circuit Lower Bounds, Help Functions, and the Remote Point Problem

__
TR09-105
| 27th October 2009
__

Vikraman Arvind, Srikanth Srinivasan#### The Remote Point Problem, Small Bias Spaces, and Expanding Generator Sets

__
TR09-103
| 26th October 2009
__

Vikraman Arvind, Srikanth Srinivasan#### On the Hardness of the Noncommutative Determinant

__
TR09-073
| 6th September 2009
__

Vikraman Arvind, Pushkar Joglekar, Srikanth Srinivasan#### On Lower Bounds for Constant Width Arithmetic Circuits

__
TR08-025
| 3rd January 2008
__

Vikraman Arvind, Partha Mukhopadhyay, Srikanth Srinivasan#### New results on Noncommutative and Commutative Polynomial Identity Testing

Revisions: 2

Hervé Fournier, Nutan Limaye, Srikanth Srinivasan, Sébastien Tavenas

Proving explicit lower bounds on the size of algebraic formulas is a long-standing open problem in the area of algebraic complexity theory. Recent results in the area (e.g. a lower bound against constant-depth algebraic formulas due to Limaye, Srinivasan, and Tavenas (FOCS 2021)) have indicated a way forward for attacking ... more >>>

Hervé Fournier, Nutan Limaye, Guillaume Malod, Srikanth Srinivasan, Sébastien Tavenas

Classical results of Brent, Kuck and Maruyama (IEEE Trans. Computers 1973) and Brent (JACM 1974) show that any algebraic formula of size s can be converted to one of depth O(log s) with only a polynomial blow-up in size. In this paper, we consider a fine-grained version of this result ... more >>>

Nutan Limaye, Srikanth Srinivasan, Sébastien Tavenas

An Algebraic Formula for a polynomial $P\in F[x_1,\ldots,x_N]$ is an algebraic expression for $P(x_1,\ldots,x_N)$ using variables, field constants, additions and multiplications. Such formulas capture an algebraic analog of the Boolean complexity class $\mathrm{NC}^1.$ Proving lower bounds against this model is thus an important problem.

It is known that, to prove ... more >>>

Nutan Limaye, Srikanth Srinivasan, Sébastien Tavenas

An Algebraic Circuit for a polynomial $P\in F[x_1,\ldots,x_N]$ is a computational model for constructing the polynomial $P$ using only additions and multiplications. It is a \emph{syntactic} model of computation, as opposed to the Boolean Circuit model, and hence lower bounds for this model are widely expected to be easier to ... more >>>

Srikanth Srinivasan

Heged\H{u}s's lemma is the following combinatorial statement regarding polynomials over finite fields. Over a field $\mathbb{F}$ of characteristic $p > 0$ and for $q$ a power of $p$, the lemma says that any multilinear polynomial $P\in \mathbb{F}[x_1,\ldots,x_n]$ of degree less than $q$ that vanishes at all points in $\{0,1\}^n$ of ... more >>>

Nutan Limaye, Srikanth Srinivasan, Utkarsh Tripathi

In this paper we prove two results about $AC^0[\oplus]$ circuits.

We show that for $d(N) = o(\sqrt{\log N/\log \log N})$ and $N \leq s(N) \leq 2^{dN^{1/d^2}}$ there is an explicit family of functions $\{f_N:\{0,1\}^N\rightarrow \{0,1\}\}$ such that

$f_N$ has uniform $AC^0$ formulas of depth $d$ and size at ...
more >>>

Srikanth Srinivasan, Utkarsh Tripathi, S Venkitesh

In a recent paper, Kim and Kopparty (Theory of Computing, 2017) gave a deterministic algorithm for the unique decoding problem for polynomials of bounded total degree over a general grid $S_1\times\cdots \times S_m.$ We show that their algorithm can be adapted to solve the unique decoding problem for the general ... more >>>

Igor Carboni Oliveira, Rahul Santhanam, Srikanth Srinivasan

We study the complexity of computing symmetric and threshold functions by constant-depth circuits with Parity gates, also known as AC$^0[\oplus]$ circuits. Razborov (1987) and Smolensky (1987, 1993) showed that Majority requires depth-$d$ AC$^0[\oplus]$ circuits of size $2^{\Omega(n^{1/2(d-1)})}$. By using a divide-and-conquer approach, it is easy to show that Majority can ... more >>>

Srikanth Srinivasan

We show that there is a sequence of explicit multilinear polynomials $P_n(x_1,\ldots,x_n)\in \mathbb{R}[x_1,\ldots,x_n]$ with non-negative coefficients that lies in monotone VNP such that any monotone algebraic circuit for $P_n$ must have size $\exp(\Omega(n)).$ This builds on (and strengthens) a result of Yehudayoff (2018) who showed a lower bound of $\exp(\tilde{\Omega}(\sqrt{n})).$

more >>>Nutan Limaye, Karteek Sreenivasiah, Srikanth Srinivasan, Utkarsh Tripathi, S Venkitesh

The $\delta$-Coin Problem is the computational problem of distinguishing between coins that are heads with probability $(1+\delta)/2$ or $(1-\delta)/2,$ where $\delta$ is a parameter that is going to $0$. We study the complexity of this problem in the model of constant-depth Boolean circuits and prove the following results.

1. Upper ... more >>>

Srikanth Srinivasan, Madhu Sudan

The well-known DeMillo-Lipton-Schwartz-Zippel lemma says that $n$-variate

polynomials of total degree at most $d$ over

grids, i.e. sets of the form $A_1 \times A_2 \times \cdots \times A_n$, form

error-correcting codes (of distance at least $2^{-d}$ provided $\min_i\{|A_i|\}\geq 2$).

In this work we explore their local

decodability and local testability. ...
more >>>

Guillaume Lagarde, Nutan Limaye, Srikanth Srinivasan

We investigate the power of Non-commutative Arithmetic Circuits, which compute polynomials over the free non-commutative polynomial ring $\mathbb{F}\langle x_1,\dots,x_N \rangle$, where variables do not commute. We consider circuits that are restricted in the ways in which they can compute monomials: this can be seen as restricting the families of parse ... more >>>

Benjamin Rossman, Srikanth Srinivasan

This paper gives the first separation between the power of {\em formulas} and {\em circuits} of equal depth in the $\mathrm{AC}^0[\oplus]$ basis (unbounded fan-in AND, OR, NOT and MOD$_2$ gates). We show, for all $d(n) \le O(\frac{\log n}{\log\log n})$, that there exist {\em polynomial-size depth-$d$ circuits} that are not equivalent ... more >>>

Prahladh Harsha, Srikanth Srinivasan

We consider the following multiplication-based tests to check if a given function $f: \mathbb{F}_q^n\to \mathbb{F}_q$ is the evaluation of a degree-$d$ polynomial over $\mathbb{F}_q$ for $q$ prime.

* $\mathrm{Test}_{e,k}$: Pick $P_1,\ldots,P_k$ independent random degree-$e$ polynomials and accept iff the function $fP_1\cdots P_k$ is the evaluation of a degree-$(d+ek)$ polynomial.

... more >>>Prahladh Harsha, Srikanth Srinivasan

We make progress on some questions related to polynomial approximations of $\mathrm{AC}^0$. It is known, by works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. $6$th CCC 1991), that any $\mathrm{AC}^0$ circuit of size $s$ and depth $d$ has an $\varepsilon$-error probabilistic polynomial over the reals ... more >>>

Ruiwen Chen, Rahul Santhanam, Srikanth Srinivasan

We show average-case lower bounds for explicit Boolean functions against bounded-depth threshold circuits with a superlinear number of wires. We show that for each integer d > 1, there is \epsilon_d > 0 such that Parity has correlation at most 1/n^{\Omega(1)} with depth-d threshold circuits which have at most

n^{1+\epsilon_d} ...
more >>>

Srikanth Srinivasan

A recent work of Chen, Kabanets, Kolokolova, Shaltiel and Zuckerman (CCC 2014, Computational Complexity 2015) introduced the Compression problem for a class $\mathcal{C}$ of circuits, defined as follows. Given as input the truth table of a Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$ that has a small (say size $s$) circuit from ... more >>>

Hervé Fournier, Nutan Limaye, Guillaume Malod, Srikanth Srinivasan

We study the arithmetic complexity of iterated matrix multiplication. We show that any multilinear homogeneous depth $4$ arithmetic formula computing the product of $d$ generic matrices of size $n \times n$, IMM$_{n,d}$, has size $n^{\Omega(\sqrt{d})}$ as long as $d \leq n^{1/10}$. This improves the result of Nisan and Wigderson (Computational ... more >>>

Swastik Kopparty, Srikanth Srinivasan

In this paper, we introduce and develop the method of certifying polynomials for proving $\mathrm{AC}^0[\oplus]$ circuit lower bounds.

We use this method to show that Approximate Majority cannot be computed by $\mathrm{AC}^0[\oplus]$ circuits of size $n^{1+o(1)}$. This implies a separation between the power of $\mathrm{AC}^0[\oplus]$ circuits of near-linear size and ... more >>>

Dmytro Gavinsky, Shachar Lovett, Michael Saks, Srikanth Srinivasan

We prove a Chernoff-like large deviation bound on the sum of non-independent random variables that have the following dependence structure. The variables $Y_1,\ldots,Y_r$ are arbitrary Boolean functions of independent random variables $X_1,\ldots,X_m$, modulo a restriction that every $X_i$ influences at most $k$ of the variables $Y_1,\ldots,Y_r$.

more >>>Rahul Santhanam, Srikanth Srinivasan

Impagliazzo, Paturi and Zane (JCSS 2001) proved a sparsification lemma for $k$-CNFs:

every k-CNF is a sub-exponential size disjunction of $k$-CNFs with a linear

number of clauses. This lemma has subsequently played a key role in the study

of the exact complexity of the satisfiability problem. A natural question is

more >>>

Vikraman Arvind, Srikanth Srinivasan

We investigate the power of Algebraic Branching Programs (ABPs) augmented with help polynomials, and constant-depth Boolean circuits augmented with help functions. We relate the problem of proving explicit lower bounds in both these models to the Remote Point Problem (introduced by Alon, Panigrahy, and Yekhanin (RANDOM '09)). More precisely, proving ... more >>>

Vikraman Arvind, Srikanth Srinivasan

Using $\epsilon$-bias spaces over F_2 , we show that the Remote Point Problem (RPP), introduced by Alon et al [APY09], has an $NC^2$ algorithm (achieving the same parameters as [APY09]). We study a generalization of the Remote Point Problem to groups: we replace F_n by G^n for an arbitrary fixed ... more >>>

Vikraman Arvind, Srikanth Srinivasan

In this paper we study the computational complexity of computing the noncommutative determinant. We first consider the arithmetic circuit complexity of computing the noncommutative determinant polynomial. Then, more generally, we also examine the complexity of computing the determinant (as a function) over noncommutative domains. Our hardness results are summarized below:

... more >>>Vikraman Arvind, Pushkar Joglekar, Srikanth Srinivasan

The motivation for this paper is to study the complexity of constant-width arithmetic circuits. Our main results are the following.

1. For every k > 1, we provide an explicit polynomial that can be computed by a linear-sized monotone circuit of width 2k but has no subexponential-sized monotone circuit ...
more >>>

Vikraman Arvind, Partha Mukhopadhyay, Srikanth Srinivasan

Using ideas from automata theory we design a new efficient

(deterministic) identity test for the \emph{noncommutative}

polynomial identity testing problem (first introduced and studied by

Raz-Shpilka in 2005 and Bogdanov-Wee in 2005). More precisely,

given as input a noncommutative

circuit $C(x_1,\cdots,x_n)$ computing a ...
more >>>