Questions tagged [algebraic-complexity]

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Nested dissection for singular matrices

Let $F$ be a field. Define $S_G(F)$ be the set of matrices $A$ in $F^{n\times n}$, such that if we replace all non-zero elements in $A$ with $1$, then we obtain the adjacency matrix of $G$ (the ...
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3 votes
1 answer
159 views

Complexity of matrix diagonalization

I'm probably missing a trivial answer, but somehow I can't find it. Given symmetric matrix $A \in \mathbb R^{n \times n}$, what's the complexity of diagonalizing the matrix, i.e. finding diagonal $\...
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  • 191
8 votes
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166 views

"Addition function" that works for both perm and det simultaneously?

For $f = (f_n)$ a family of polynomials where $f_n$ is a polynomial in $n^2$ variables (which we can think of as the entries of an $n \times n$ matrix), say a function $S(A,B)$ is an addition function ...
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5 votes
1 answer
263 views

Questions about P vs NP and geometric complexity theory

Reading through various papers on geometric complexity theory (GCT), there is one thing, which pops up, while claimed in various places, that it is an approach to P vs NP, all the results seems to ...
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4 votes
1 answer
197 views

Is $GCT$ necessarily a negative result program?

$GCT$ is a candidate program to separate permanent and determinant through symmetries. If indeed permanent and determinant can be handled in similar complexity class would $GCT$ be a program which can ...
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1 vote
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How can I understand the Coppersmith–Winograd algorithm?

I want to do research on matrix multiplication algorithms. I glanced at the Coppersmith-Winograd algorithm paper, but I didn't understand anything. How can I complete the background to read this paper?...
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2 votes
0 answers
131 views

Matrix multiplication when one matrix is fixed

Let $A$ be a fixed positive entried integer matrix of size $a\times n$ with $\ell$ bits per entry One is allowed to pre-process this matrix as appropriate. Given another positive integer entried $B$...
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9 votes
0 answers
151 views

What are some examples of algorithmic applications of noncommutative rational identity testing?

The problem of polynomial identity testing (PIT) is known to be in $\mathsf{RP}$, but not known to be in $\mathsf{P}$. The related problem of noncommutative rational identity testing (NCIT) is known ...
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101 views

Complexity of planted root of a system of quadratic homogeneous polynomials?

Given homogeneous degree $2$ randomly chosen polynomials $f_1,\dots,f_{m}$ in $\mathbb Z[x_1,\dots,x_n,y_1,\dots,y_n]$ each with only monomials $x_iy_j$ with condition that the system $f_1=\dots=f_{m}=...
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7 votes
1 answer
346 views

Can reciprocal inputs speed up monotone computations?

A $(+,\times,1/x_i)$ circuit is a standard monotone arithmetic $(+,\times)$ circuit with the only difference that now besides the input variables $x_1,\ldots,x_n$, also their reciprocals $1/x_1,\...
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Worst case polynomial in elimination theory under rank conditions?

Given $n$ polynomials $h_1(x_1,\dots,x_{2n}),\dots,h_{2n}(x_1,\dots,x_{2n})\in\mathbb Z[x_1,\dots,x_{2n}]$ where each of $h_1(x_1,\dots,x_{2n}),\dots,h_{2n}(x_1,\dots,x_{2n})$ is homogeneous of degree ...
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5 votes
1 answer
170 views

Complexity of counting integer roots of multivariate polynomials in a polyhedron?

Deciding integer roots of multivariate polyomials is undecidable. However what is known about counting integer roots of multivariate polynomials in $\mathbb Z[x_1,\dots,x_m]$ with both $m$ and total ...
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7 votes
0 answers
57 views

Complexity of deciding whether subspaces of Z_2^n cover every point 3*x times

When studying the complexity of checking identities in certain finite algebras, I came across the following decision problem: Input: A positive integer $n \in N$ and a set of affine subspaces $H_1,...
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6 votes
1 answer
212 views

VNP is closed under taking coefficients using Valiant's criterion

We consider the family of polynomials $$\{f_n(x_1,\dots,x_n,y_1,\dots,y_n)\}\in\mathsf{VNP}$$ We want to show that the family $$\{h_n(x_1,\dots,x_n)\}$$ is in $\mathsf{VNP}$. Where $h_n(x_1,\dots,...
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1 vote
0 answers
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Relating classes of computational complexity to finding solution to classes of algebraic equations [closed]

Having related classes of computational complexity to finding solution to classes of algebraic equations, we may relate classes of computational complexity to algebraic geometry or complex geometry,...
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2 votes
2 answers
116 views

IPS upper bound for subset sum axiom

I am reading the following paper Michael A. Forbes, Amir Shpilka, Iddo Tzameret, Avi Wigderson ,"Proof Complexity Lower Bounds from Algebraic Circuit Complexity", 2016. IPS is defined as follows: ...
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1 vote
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98 views

What is the relation between computational complexity of algebraic number and computational complexity to find the solution to algebraic equation?

Suppose $\alpha$ is algebraic number, and we have the algorithm with lowest computational complexity to output it, and $f(x)=0$ is algebraic polynomial with $\alpha$ as a root. If an algorithm which ...
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2 votes
0 answers
117 views

On $\Sigma \Pi \Sigma \Pi(2,r)$-circuits

As I understand from the survey "Progress on Polynomial Identity Testing - II" a polynomial-time algorithm for solving PIT for $\Sigma \Pi \Sigma \Pi (2, r)$ is unknown. However, there exists paper ...
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1 vote
1 answer
170 views

What is the computational complexity of solutions over $\mathbb{Q}$ of polynomial equation with coeffiecents over $\mathbb{Z}$

What is the complexity of the following problem? (e.g. best-known running time, space, best upper bound in terms of complexity classes, etc.) Input: A multivariate polynomial $f$ with coefficients ...
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0 votes
1 answer
34 views

Decomposing outer product or general rank factorization over $\Bbb F_q$

Given matrix $M\in\Bbb F_q^{n\times n}$ with the promise that there are two matrices $A\in\Bbb F_q^{n\times 1}$ and $B\in\Bbb F_q^{1\times n}$ such that $AB=M$ is there a deterministic $O((n\log q)^c)$...
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3 votes
0 answers
46 views

Equal degree factoring of homogeneous polynomials over $\Bbb Q[x_1,\dots,x_n]$?

Given $f(x_1,\dots,x_n)\in\Bbb Q[x_1,\dots,x_n]$ of form $\prod_{i=1}^df_i(x_1,\dots,x_n)$ where each of $f,f_i$ are homogeneous and each $f_i$ is irreducible what is the best technique to factor such ...
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9 votes
0 answers
299 views

What does a private coin $\mathsf{IP}$ protocol for Hilbert's Nullstellensatz look like?

$\mathsf{GNI}$ Private Coin In [GMW85], the authors provided the famous interactive proof $\mathsf{IP}$ of Graph Non Isomorphism $\mathsf{GNI}$. The $\mathsf{GNI}$ protocol entails a verifier ...
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  • 822
7 votes
1 answer
542 views

Riemann Hypothesis and Complexity Theory

It is known that "Assuming the generalized Riemann hypothesis (GRH) if VP = VNP then PH collapses to second level". Why would one think of a relation between VP,VNP and the Riemann hypothesis. Where ...
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  • 71
14 votes
1 answer
761 views

VC dimension of polynomials over tropical semirings?

As in this question, I am interested the $\mathbf{BPP}$ vs. $\mathbf{P}$/$\mathrm{poly}$ problem for tropical $(\max,+)$ and $(\min,+)$ circuits. This question reduces to showing upper bounds for the ...
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1 vote
0 answers
24 views

A question about a claim in "No occurrence obstructions in geometric complexity theory"

It the new preprint Peter Bürgisser, Christian Ikenmeyer, Greta Panova, "No occurrence obstructions in geometric complexity theory", 2016 it is stated that 1.3. Conjecture (Mulmuley and Sohoni ...
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  • 12.5k
9 votes
0 answers
264 views

Checking whether two quadratic equations have a common zero

Given two quadratic equations (with integer coefficients): $x^T A_1 x+ b^T_1 x + c_1=0$ and $x^T A_2 x+ b^T_2 x + c_2=0$ The problem is to decide whether they have a common zero. Here $x$ is a ...
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7 votes
0 answers
162 views

Recognition of a primitive root

Adleman and McCurley published a paper in 1994 called "Open problems in number theoretic complexity, II" (http://ww.cstheory.com/papers/open.ps.gz) Problem 18 of this list of open problems is about ...
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  • 521
5 votes
1 answer
186 views

Implications of a recent negative result to geometric complexity

A paper was posted in arxiv http://arxiv.org/pdf/1512.03798.pdf titled 'Rectangular Kronecker coefficients and plethysms in geometric complexity theory' by Christian Ikenmeyer and Greta Panova with ...
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5 votes
0 answers
239 views

What is the status of Determinantal Complexity of Permanent

Recently Landsberg and Ressayre announced an exponential lower bound for permanent's determinantal complexity assuming certain symmetry conditions. What is the status of the problem of Permanent's ...
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7 votes
1 answer
234 views

Sorting using ring operations

Sorting is in $\mathsf{NP}$. Given a sorted list, it is trivial to check sortedness in linear time. Is there any evidence sorting of elements from an ordered gcd domain(eg: $\Bbb Z$) cannot be done ...
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2 votes
0 answers
125 views

Complexity of a particular determinant

Suppose we have an $n\times n$ matrix $A$ with non-negative integer entries such that $\mathsf{Tr}(A^i)=0$ at every $i\in\{1,2,\dots,n-2,n-1\}$ and $\mathsf{Tr}(A^n)\neq0$, then from Trace-Determinant ...
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6 votes
0 answers
106 views

Polynomial decomposition complexity

Given $N(x)\in\Bbb Z_{\geq0}[x]$ with non-negative coefficients with promise that there exists $a(x),b(x),c(x),d(x)\in\Bbb Z_{\geq0}[x]$ with non-negative coefficients such that $N(x)=a(x)b(x)+c(x)d(x)...
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8 votes
0 answers
184 views

Speed-up of Boolean over Algebraic computation

I would like to know what is the maximum speed-up of algebraic computation when we work in the word RAM model. This question is motivated by this theorem from Ryan's paper: Theorem 1.2 Let $(R, +, ...
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  • 1,120
10 votes
2 answers
849 views

Implications of Riemann Hypothesis variants in TCS

The over ~1½ century old Riemann Hypothesis has deep implications in mathematics and a large edifice of math theory is now proved conditionally on it and numerous variants. I recently came across a ...
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11 votes
2 answers
285 views

Straight line complexity of monomials

Let $k$ be some field. As usual, for an $f\in k[x_{1},x_{2},\ldots,x_{n}]$ we define $L(f)$ to be the straight-line complexity of $f$ over $k$. Let $F$ be the set of monomials of $f$, namely the ...
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1 vote
1 answer
113 views

Hitting set of very restricted linear forms

We say that $f\in\mathbb{Z}[x_{1},\dots,x_{n}]$ is a {-1,0,1}-linear form if $f=\sum_{i\in S}x_{i}-\sum_{i\in T}x_{i}$ where $S,T\subseteq[n]$. A hitting set $H\subseteq\mathbb{Z}^{n}$ for {-1,0,1}-...
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7 votes
1 answer
320 views

$NP \not\subseteq BPP \implies NP_{\mathbb{C}} \not\subseteq P_{\mathbb{C}}$

Stephen Smale claims in Mathematical Problems for the Next Century that $$NP \not\subseteq BPP \implies NP_{\mathbb{C}} \not\subseteq P_{\mathbb{C}}.$$ Can someone sketch the argument or provide a ...
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  • 3,993
4 votes
2 answers
1k views

Complexity of the inverse modulo a composite number

Supposing $M$ is a composite number and supposing $a$ is an integer such that $a^{-1}\mod M$ exists, can we compute $a^{-1} \bmod M$ by using $O(\log^{b}(M))$ ring operations in the RAM model, where $...
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6 votes
1 answer
631 views

Lower bounds for Polynomials computing the boolean functions

Expressing a boolean function $f$ $:\{ 0,1 \}^{n} \rightarrow \{0,1 \}$ using a polynomial $P(x_{1},...,x_{n})$, where $x_{1},...,x_{n}$ may be integer, finite fields, or other fields. One of the most ...
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  • 201
14 votes
1 answer
2k views

A course for learning algebraic complexity

I want to learn about algebraic algorithms and complexity thoery. In particular, I am interested in PIT. Is there a set of lecture notes, books, papers and surveys for students who have read standard ...
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  • 201
9 votes
1 answer
221 views

Checking if a polynomial factors into linear factors

Let $f\in\mathbb{Q}[x_{1},x_{2},\ldots,x_{n}]$ be a polynomial given by an arithmetic circuit $C$ of size $s$. Given $C$ as the input, is there a deterministic algorithm to check whether all the ...
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6 votes
1 answer
157 views

Where does randomness help when deciding algebraic geometry over $\mathbb{C}$?

If we have a single straight line program expressing a multivariate polynomial equation with integer coefficients, the Schwartz-Zippel lemma gives a simple randomized algorithm for deciding whether ...
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12 votes
1 answer
769 views

Expressing Determinant as Permanent

One major problem in TCS is the problem of expressing a permanent as a determinant. I was reading Agrawal's paper Determinant Versus Permanent and in one paragraph he claims the reverse problem is ...
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  • 131
5 votes
1 answer
175 views

Size of Formulas with no negative sign for Matrix Permanent

What is the best lower bound for algebraic formulas for Permanent of a matrix given that the formulas have no negative sign? Is there an exponential lower bound known for such formulas and what would ...
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  • 12.5k
7 votes
1 answer
394 views

Uniformity vs. nonuniformity in algebraic complexity theory

I understand that the study of Boolean circuits and nonuniform complexity classes was introduced to (hopefully) prove separation of uniform complexity classes. In this sense, nonuniform computational ...
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  • 875
7 votes
1 answer
700 views

Factoring with LLL when the form of the factors is given

Given a degree $2k$ reducible polynomial $$f(x)=\sum_{i=0}^{2k}a_ix^i\in\Bbb Z[x]$$ with $$\text{gcd}(a_{2k},\dots,a_0)=1$$ that is known to be of the form $f_1(x)f_2(x)$ with $\text{deg}\big(...
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  • 12.5k
6 votes
1 answer
393 views

Degree restriction for polynomials in $\mathsf{VP}$

why do we need the restriction on the degree for a polynomial to be in VP due to which $(\prod_{i=0}^{n} x_i)^{2^n}$ $\notin$ VP even though it has a poly-size circuit?
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  • 193
14 votes
0 answers
299 views

Exponential-time factorization of polynomials

Let an explicit field be a field for which equality is decidable (in some standard model of computation). I am interested in the factorization of univariate polynomials over an explicit field. It is ...
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  • 4,400
3 votes
1 answer
242 views

Convolution without FFT

What is the best upper and lower bound known for convolution without FFT? Is FFT proven to be essential for time complexity reduction? Is cancellation essential as well?
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  • 12.5k
20 votes
0 answers
616 views

Identifying Reducible/Irreducible polynomials over $Z[x]$

It is well known LLL algorithm provides a fully polynomial algorithm to factor a reducible primitive polynomial over $\mathbb{Z}[x]$. Say one only seeks to identify whether a given polynomial over $\...
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