Questions tagged [algebraic-complexity]
The algebraic-complexity tag has no usage guidance.
74
questions
5
votes
1answer
238 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 ...
4
votes
1answer
192 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 ...
1
vote
0answers
987 views
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?...
2
votes
0answers
129 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 $B$...
9
votes
0answers
149 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 ...
1
vote
0answers
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}=...
7
votes
1answer
342 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,\...
1
vote
0answers
22 views
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 ...
5
votes
1answer
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 ...
7
votes
0answers
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,...
6
votes
1answer
208 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,...
1
vote
0answers
68 views
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,...
2
votes
2answers
112 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:
...
1
vote
0answers
93 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 ...
2
votes
0answers
113 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 ...
1
vote
1answer
164 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 ...
3
votes
0answers
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 ...
9
votes
0answers
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 ...
7
votes
1answer
523 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 ...
14
votes
1answer
750 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 ...
9
votes
0answers
263 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 ...
7
votes
0answers
161 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 ...
5
votes
1answer
183 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 ...
5
votes
0answers
235 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 ...
7
votes
1answer
232 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 ...
2
votes
0answers
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 ...
8
votes
0answers
183 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, +, ...
10
votes
2answers
841 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 ...
11
votes
2answers
282 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 ...
1
vote
1answer
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}-...
7
votes
1answer
315 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 ...
4
votes
2answers
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 $...
6
votes
1answer
619 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 ...
14
votes
1answer
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 ...
9
votes
1answer
220 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 ...
6
votes
1answer
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 ...
12
votes
1answer
764 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 ...
5
votes
1answer
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 ...
7
votes
1answer
391 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 ...
6
votes
2answers
640 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(...
6
votes
1answer
380 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?
14
votes
0answers
297 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 ...
3
votes
1answer
241 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?
20
votes
0answers
610 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 $\...
1
vote
1answer
149 views
Morgenstern's Theorem
Morgenstern proves a $\Omega(n\log n)$ lower bound for Fourier transform in the bounded coefficient model.
Let $x=[x_1,x_2,\ldots,x_n]'$ be given vector and $F$ be Fourier transform matrix.
It is ...
5
votes
2answers
290 views
Commutative matrix multiplication algorithms
What is known about commutative algorithms like Winograd algorithm and its variants for Matrix Multiplication? Why is there not much study on them? Can they be asymptotically as efficient as Non-...
11
votes
2answers
1k views
Determinants and Matrix Multiplication - Similarity and differences in algorithmic complexity and arithmetic circuit size
I am trying to understand the relation between algorithmic complexity and circuit complexity of Determinants and Matrix Multiplication.
It is known that the determinant of an $n\times n$ matrix can ...
10
votes
2answers
735 views
Cancellation and determinant
Berkowitz algorithm provides a polynomial size circuit with logarithmic depth for determinant of a square matrix using matrix powers. The algorithm implicitly uses cancellation. Is cancellation ...
9
votes
3answers
599 views
Find the remainder of a large fixed polynomial when divided by a small unknown polynomial
Assume we operate in a finite field. We are given a large fixed polynomial p(x) (of, say, degree 1000) over this field. This polynomial is known beforehand and we are allowed to do computation using a ...
9
votes
1answer
440 views
(Cryptographic) problems solvable in a polynomial number of arithmetic steps
In the paper from Adi Shamir [1] from 1979 he shows, that factoring can be done in a polynomial number of arithmetic steps. This fact was restated, and thus came to my attention, in the recent paper ...
