Carsten Damm, Markus Holzer, Pierre McKenzie

Tensor calculus over semirings is shown relevant to complexity

theory in unexpected ways. First, evaluating well-formed tensor

formulas with explicit tensor entries is shown complete for $\olpus\P$,

for $\NP$, and for $\#\P$ as the semiring varies. Indeed the

permanent of a matrix is shown expressible as ...
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Mark Jerrum, Eric Vigoda

We present a fully-polynomial randomized approximation scheme

for computing the permanent of an arbitrary matrix

with non-negative entries.

Bruno Codenotti, Igor E. Shparlinski

We show that for several natural classes of ``structured'' matrices, including symmetric, circulant, Hankel and Toeplitz matrices, approximating the permanent modulo a prime $p$ is as hard as computing the exact value. Results of this kind are well known for the class of arbitrary matrices; however the techniques used do ... more >>>

Pascal Koiran

Let $\tau(k)$ be the minimum number of arithmetic operations

required to build the integer $k \in \N$ from the constant 1.

A sequence $x_k$ is said to be ``easy to compute'' if

there exists a polynomial $p$ such that $\tau(x_k) \leq p(\log k)$

for all $k \geq ...
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Oded Goldreich

We highlight a common theme in four relatively recent works

that establish remarkable results by an iterative approach.

Starting from a trivial construct,

each of these works applies an ingeniously designed

sequence of iterations that yields the desired result,

which is highly non-trivial. Furthermore, in each iteration,

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Leonid Gurvits

Let $p(x_1,...,x_n) = p(X) , X \in R^{n}$ be a homogeneous polynomial of degree $n$ in $n$ real variables ,

$e = (1,1,..,1) \in R^n$ be a vector of all ones . Such polynomial $p$ is

called $e$-hyperbolic if for all real vectors $X \in R^{n}$ the univariate polynomial

equation ...
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Peter Buergisser

Let $\tau(n)$ denote the minimum number of arithmetic operations sufficient to build the integer $n$ from the constant~$1$. We prove that if there are arithmetic circuits for computing the permanent of $n$ by $n$ matrices having size polynomial in $n$, then $\tau(n!)$ is polynomially bounded in $\log n$. Under the ... more >>>

Nitin Saxena

In this paper we give the first deterministic polynomial time algorithm for testing whether a {\em diagonal} depth-$3$ circuit $C(\arg{x}{n})$ (i.e. $C$ is a sum of powers of linear functions) is zero. We also prove an exponential lower bound showing that such a circuit will compute determinant or permanent only ... 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 >>>Holger Dell, Thore Husfeldt, Martin Wahlén

The Exponential Time Hypothesis (ETH) says that deciding the satisfiability of $n$-variable 3-CNF formulas requires time $\exp(\Omega(n))$. We relax this hypothesis by introducing its counting version #ETH, namely that every algorithm that counts the satisfying assignments requires time $\exp(\Omega(n))$. We transfer the sparsification lemma for $d$-CNF formulas to the counting ... more >>>

Scott Aaronson, Dieter van Melkebeek

We present an alternate proof of the result by Kabanets and Impagliazzo that derandomizing polynomial identity testing implies circuit lower bounds. Our proof is simpler, scales better, and yields a somewhat stronger result than the original argument.

more >>>Scott Aaronson, Alex Arkhipov

We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a

model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count ...
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Scott Aaronson

One of the crown jewels of complexity theory is Valiant's 1979 theorem that computing the permanent of an n*n matrix is #P-hard. Here we show that, by using the model of linear-optical quantum computing---and in particular, a universality theorem due to Knill, Laflamme, and Milburn---one can give a different and ... more >>>

Neeraj Kayal

An $m$-variate polynomial $f$ is said to be an affine projection of some $n$-variate polynomial $g$ if there exists an $n \times m$ matrix $A$ and an $n$-dimensional vector $b$ such that $f(x) = g(A x + b)$. In other words, if $f$ can be obtained by replacing each variable ... more >>>

Maurice Jansen, Rahul Santhanam

Suppose $f$ is a univariate polynomial of degree $r=r(n)$ that is computed by a size $n$ arithmetic circuit.

It is a basic fact of algebra that a nonzero univariate polynomial of degree $r$ can vanish on at most $r$ points. This implies that for checking whether $f$ is identically zero, ...
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Ruiwen Chen, Valentine Kabanets

A family of Boolean circuits $\{C_n\}_{n\geq 0}$ is called \emph{$\gamma(n)$-weakly uniform} if

there is a polynomial-time algorithm for deciding the direct-connection language of every $C_n$,

given \emph{advice} of size $\gamma(n)$. This is a relaxation of the usual notion of uniformity, which allows one

to interpolate between complete uniformity (when $\gamma(n)=0$) ...
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Sanjeev Arora, Arnab Bhattacharyya, Rajsekar Manokaran, Sushant Sachdeva

Suppose we are given an oracle that claims to approximate the permanent for most matrices $X$, where $X$ is chosen from the Gaussian ensemble (the matrix entries are i.i.d. univariate complex Gaussians). Can we test that the oracle satisfies this claim? This paper gives a polynomial-time algorithm for the task.

... more >>>Ankit Gupta, Pritish Kamath, Neeraj Kayal, Ramprasad Saptharishi

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 >>>

Markus Bläser

We consider the complexity of computing the determinant over arbitrary finite-dimensional algebras. We first consider the case that $A$ is fixed. We obtain the following dichotomy: If $A/rad(A)$ is noncommutative, then computing the determinant over $A$ is hard. ``Hard'' here means $\#P$-hard over fields of characteristic $0$ and $ModP_p$-hard over ... more >>>

Scott Aaronson, Travis Hance

Around 2002, Leonid Gurvits gave a striking randomized algorithm to approximate the permanent of an n×n matrix A. The algorithm runs in O(n^2/?^2) time, and approximates Per(A) to within ±?||A||^n additive error. A major advantage of Gurvits's algorithm is that it works for arbitrary matrices, not just for nonnegative matrices. ... more >>>

Scott Aaronson

BosonSampling, which we proposed three years ago, is a scheme for using linear-optical networks to solve sampling problems that appear to be intractable for a classical computer. In a recent manuscript, Gogolin et al. claimed that even an ideal BosonSampling device's output would be "operationally indistinguishable" from a uniform random ... more >>>

Leonid Gurvits

We prove a new efficiently computable lower bound on the coefficients of stable homogeneous polynomials and present its algorthmic and combinatorial applications. Our main application is the first poly-time deterministic algorithm which approximates the partition functions associated with

boolean matrices with prescribed row and (uniformly bounded) column sums within simply ...
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Craig Gentry

We show that, for many noncommutative algebras A, the hardness of computing the determinant of matrices over A follows almost immediately from Barrington’s Theorem. Barrington showed how to express a NC1 circuit as a product program over any non-solvable group. We construct a simple matrix directly from Barrington’s product program ... more >>>

Pushkar Joglekar, Aravind N.R.

We introduce and study the notion of read-$k$ projections of the determinant: a polynomial $f \in \mathbb{F}[x_1, \ldots, x_n]$ is called a {\it read-$k$ projection of determinant} if $f=det(M)$, where entries of matrix $M$ are either field elements or variables such that each variable appears at most $k$ times in ... more >>>

Joshua Grochow

In this short note, we show that the permanent is not complete for non-negative polynomials in $VNP_{\mathbb{R}}$ under monotone p-projections. In particular, we show that Hamilton Cycle polynomial and the cut polynomials are not monotone p-projections of the permanent. To prove this we introduce a new connection between monotone projections ... more >>>

Ryan Williams

We present an efficient proof system for Multipoint Arithmetic Circuit Evaluation: for every arithmetic circuit $C(x_1,\ldots,x_n)$ of size $s$ and degree $d$ over a field ${\mathbb F}$, and any inputs $a_1,\ldots,a_K \in {\mathbb F}^n$,

$\bullet$ the Prover sends the Verifier the values $C(a_1), \ldots, C(a_K) \in {\mathbb F}$ and ...
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Daniel Grier, Luke Schaeffer

In 2011, Aaronson gave a striking proof, based on quantum linear optics, showing that the problem of computing the permanent of a matrix is #P-hard. Aaronson's proof led naturally to hardness of approximation results for the permanent, and it was arguably simpler than Valiant's seminal proof of the same fact ... more >>>

Radu Curticapean, Nutan Limaye, Srikanth Srinivasan

A polynomial $P\in F[x_1,\ldots,x_n]$ is said to be symmetric if it is invariant under any permutation of its input variables. The study of symmetric polynomials is a classical topic in mathematics, specifically in algebraic combinatorics and representation theory. More recently, they have been studied in several works in computer science, ... more >>>