All Questions
1,809 questions
1
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99
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Minimum of the maximum element frequency given the family size and the universe size
[Crossposted at math.stackexchange].
Consider families of sets $\mathcal{F}$ with size $n = |\mathcal{F}|$ and universe $U(\mathcal{F})$ with size $q = |U(\mathcal{F})|$.
I have written and solved ...
- 1,000
1
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0
answers
50
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Computational complexity of deciding if two elements are in the same cycle of a permutation, version 2
This question has relation with this previous one, although the two cases are not likely solved with the same method.
Let us consider a function $P:\{0,1\}^*\to\{0,1\}^*$ that can be calculated in ...
- 469
1
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0
answers
106
views
Quasi polynomial algorithm for NP complete problem [closed]
I know that quasi polynomial algorithm is neither polynomial nor exponential. But I want to know if we find such algorithm for NP complete problem, will it be of any use? Or is there such algorithm ...
- 29
1
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0
answers
68
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Primality testing by reversible computation using the prime number theorem
Suppose we want to build a primality testing algorithm for the numbers limited to the set $A =\{1, ..., 2^n\}$ and $n$ is reasonably large. The prime-number theorem tells us that there are ...
- 283
1
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0
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87
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complexity of membership problem in finite general linear group
Suppose $G$ is a subgroup of $GL(n,q)$ given by a list of generators. What is known about the complexity of the corresponding "membership problem", that is, the problem of deciding whether a ...
- 2,287
1
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0
answers
126
views
Integration in polynomial time
The work of Friedman and Ko and
Müller guarantee the polynomial time computability of the integrals of analytic functions inside the circle of convergence. But do algorithms have practical value? Is ...
- 81
1
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0
answers
171
views
Fast algorithm for computing certain signal transformations
Let $f,g,h:\mathbb Z\to\mathbb C$ supported on $[-n,n]$. For $\tau\in \mathbb Z$, let $\operatorname{sh}_\tau f$ be the shift of $f$ by $\tau$ (i.e. $(\operatorname{sh}_\tau f)(t) = f(t-\tau)$). ...
- 2,639
1
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0
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49
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Computing geodesic length of Euclidean lines in the manifold of positive definite matrices
I am working with the manifold of positive definite matrices $PD(n)$ equipped with the affine-invariant Riemannian metric (AIRM) $g_P(V,W):=tr(P^{-1}VP^{-1}W)$, where $P \in PD(n)$ and $V,W \in T_P PD(...
- 518
1
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0
answers
168
views
Circulant matrix inverse in $GF(p)$
For a polynomial $C(x)=c_0+\dots+c_n x^n$, consider a circulant matrix $C$ such that
$$
C= \begin{pmatrix}
c_0 & c_{n-1} & \cdots & c_2 & c_1 \\
c_1 & c_0 &...
- 1,171
1
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0
answers
52
views
Complexity of the TSP for hypercube graphs
Question:
what is known about the complexity of finding the Hamilton cycle of minimum weight in graphs that resemble hypercubes with weighted edges?
- 13.2k
1
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0
answers
116
views
Sudden drop in complexity class due to the more general correlations
Recently I was asking about the impact of the groundbreaking result MIP*=RE on logic and proof theory (see this discussion). Surprising as it is I got confused with the following: MIP* is a ,,quantum''...
- 9,330
1
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0
answers
51
views
Hardness of an optimization problem when some variables are fixed
Given a general optimization problem, I would like to know what we can say about the hardness of the problem when a subset of its variables are fixed.
With the two (related) examples, it is clear that ...
- 11
1
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0
answers
67
views
Are the lower elementary functions closed under limited recursion?
The lower elementary functions (also called Skolem elementary functions) are functions generated from the successor, modified subtraction, projection functions by the operations of composition and ...
- 1,782
1
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0
answers
145
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Complexity of calculating the expectation of $\operatorname{Tr} h(A)$, $A$ is a random matrix
$A$ is a $d_1\times d_1$ random matrix. Given $\{g_i\}~(1\leq i\leq n)$ iid Gaussian variables, $f_{ij}(g_1,g_2,...,g_n)~(1\leq i,j\leq d_1)$ are degree-$d_2$ polynomials. And $f_{ij}\equiv f_{ji}~(\...
- 91
1
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0
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94
views
Linear Program Optimal Value
If $f(A,b,c)$ is the optimal value of a linear program
$\min c.x$
subject to $A.x \leq b ; x \geq 0.$
Does $f(A,b,c)$ have a piecewise polynomial/rational upper bound in $(A,b,c)$ on the domain of ...
1
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0
answers
114
views
Computing sine of gamma function [closed]
In the sense of bit complexity, how difficult is it to compute $$\sin(a\Gamma(x))$$ where $a$ is a constant and $x>1$? Is it possible to avoid the computation of $\Gamma$ as first step?
Is there a ...
- 67
1
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0
answers
35
views
Are there any known lower complexity bounds on solving positive semidefinite or positive semidefinite feasibility problems?
I've been trying to attack the problem posted here, about quickly checking if a matrix has any positive semidefinite completions. I suspect that the answer to the question is "no", because ...
- 111
1
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0
answers
34
views
Does $2$ variable linear Diophantine equation in $NC$ imply $2$ dimensional shortest vector is in $NC$?
Consider the Linear Diophantine in known $a,b,c\in\mathbb Z$
$$ax+by=c.$$
Above can be solve by Extended Euclidean which is not in $NC$ as far as we know. It is clear if Extended Euclidean is in $NC$ ...
- 13.9k
1
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0
answers
101
views
On determinant and permanent of certain homotopy defined simple matrices
Let $A_1,A_2,B_1,B_2$ be four $n\times n$ $0/1$ square matrices where $$\det(A_1)=\det(A_2)=per(A_1)=per(A_2)=1$$
$$\det(B_1)=\det(B_2)=per(B_1)=per(B_2)=0$$
hold ($per$ refers to permanent).
I. What ...
- 13.9k
1
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0
answers
161
views
On an optimization question
Suppose we have a square matrix $M=(1-z)A+zB$ where $A,B$ have integer entries from $\{0,1\}$ with $\det(A)+\det(B)=1$ and $\det(A),\det(B),per(A),per(B)\in\{0,1\}$ and we want to find $z\in[0,1]$ ...
- 13.9k
1
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0
answers
223
views
Fastest algorithm for finding the closest semi-definite matrix?
Given a real-valued, symmetric matrix $A \in \mathbb{R}^{n \times n}$, I'm interested in finding the closest positive semi-definite matrix $X^*\in \mathbb{R}^{n \times n}$:
$$
X^* = \mathop{\text{...
- 723
1
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0
answers
61
views
Is there any lower bound for basis computation in finite Abelian groups?
Victor Shoup in this paper has given a lower bound for discrete logarithm. The algorithms that I have come across use discrete logarithms (extended discrete logarithms) to compute a basis for a finite ...
- 11
1
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0
answers
56
views
Over a given finite field, how many couples of matrices there are, for which their minimal polynomials are co-prime?
Let ${\mathbb F}_{q}$ be a given finite field. How many couples of $n\times n$ matrices $\left(A,B\right)$ over ${\mathbb F}_{q}$, such that $\gcd\left(\mu_{A}\left(\lambda\right),\mu_{B}\left(\lambda\...
- 21
1
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0
answers
36
views
Does Hoffman constant keep the same after a very tiny perturbation on the polyhedron such that the bases are even unchanegd?
Suppose that $P$ is a polyhedron represented by
$$P:=\{x \in \mathbb{R}^n: A x \le b \} \text{ for }A \in \mathbb{R}^{m\times n},\ b \in \mathbb{R}^m,$$
and $P$ contains interior points. Moreover, the ...
- 33
1
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0
answers
96
views
On optimizing a multivariate quadratic function subject to certain conditions
The problem is to maximize $f(x_1,x_2,\cdots,x_n)=\sum\limits_{i=1}^{n}\Big(x_i-k_i\Big)^2$ for $n\ge 3$ subject to the conditions (1) $\sum\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}k_i\le n(n-1)$ ...
- 61
1
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0
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336
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Closed-form solution of a particular linear program
(Note: I asked a similar question at math.stackexchange but the present one is more precise.)
I have a linear program of the form:
$$\text{minimize} \space\space x_1 \space\space \text{subject to:}$$
$...
- 1,000
1
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0
answers
70
views
Is this factorization problem in EXP?
Factorization is not known to have a polynomial time algorithm. Traditionally the input length is number of bits in representation of the integer to be factored.
However now consider integers of form $...
- 13.9k
1
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0
answers
46
views
Computational hardness of a discrete generalized rectangle packing problem
I have a decision problem that is clearly in NP, but I cannot seem to prove that it is in P, nor can I prove its NP-hardness. I attribute this more to my inexperience than to the problem's difficulty (...
1
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0
answers
47
views
Nash Equilibria change linearly in (some) game parameters. Already known / follows from a more general result?
EDIT: The key thing that I am wondering about is the linearity of the P2 strategy, not the constancy of P1. (The latter is straightforward.)
Question: Is the following result already known? Or is it a ...
1
vote
0
answers
121
views
Frog game on tree graphs is in NP but not in P (NP-complete)?
Problem
We can restrict ourselves to tree graphs. What is the complexity of the following problem?
Let $G$ be simple connected graph with vertices in $V$, edges in $E$, and a vertex weighted function $...
- 611
1
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0
answers
123
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Finding the optimal arithmetic circuit for evaluating a given polynomial
The Horner's algorithm takes as input a univariate polynomial $f(X)$ and an evaluation point $x$ and computes $f(x)$ using $O(\deg(f))$ field operations.
Suppose now that the polynomial $f(X)$ is ...
- 297
1
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0
answers
83
views
What is the complexity of elgamal cryptosystem? [closed]
Its clear generation of keys based
On cyclic group and its generator for z_p
So my question
Does finding the generator efect on complexity
Moreove does the size of message M effect on the complexity?
- 11
1
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0
answers
59
views
How do I incorporate Ito's lemma into the solution for a finite-horizon stochastic cake-eating problem?
I'm interested in finite-horizon, continuous-time cake-eating problems in which the agent has a time-horizon $W$ over which to eat the cake, and then chooses an optimal consumption path $\{h_t\}_0^W$, ...
- 11
1
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0
answers
42
views
If statement in the algebraic group model (AGM)
In the algebraic group model (https://eprint.iacr.org/2017/620.pdf), can one use "if" statement? For example, can one do the following in AGM?
input: x, y, z
if (x = y) then z = x else z = ...
- 651
1
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0
answers
92
views
Formalizing intuition of search hardness
Basically, this is a search problem of an object that is promised to exist. Suppose we have an object that can be described completely and uniquely by $m$ properties (each n bits). Suppose a search ...
- 2,993
1
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0
answers
42
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What is the complexity of the matrix multiplication closure for a given generating system?
Given a generating set of $k$ matrices $X = \{M_1, M_2, \ldots, M_k\}$, with $M_i\in \mathrm{Mat}(\mathbb{C},n)$, what is the worst case complexity for computing the algebraic closure w.r.t. matrix ...
- 21
1
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0
answers
101
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Fast algorithm to compute nimber product
It is known that nimbers (Grundy numbers) below $2^{2^n}$ form a field with the nim addition $\oplus$ and the nim product $\cdot$.
Generally, one can develop an algorithm to compute the product of two ...
- 1,171
1
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1
answer
331
views
Finding a special solution in a solution set over F2
Given a solution set of a linear system of the following form
$$
\{ \begin{bmatrix}
x_{1} \\
\vdots \\
x_{n}
\end{bmatrix} = \vec{v_1} * x_1 + \dots + \vec{...
- 51
1
vote
0
answers
61
views
Linear programming robustness to input perturbations
I'm running a linear program whose parametrization depends on the output of a neural network. I was wondering if there exist results on how robust linear programs are towards perturbations in their ...
- 11
1
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0
answers
98
views
Solution of a simple optimization problem
Let $\mathbf{U}_1$ and $\mathbf{U}_2$ be two arbitrary unitary matrices and $\mathbf{D}$ be a diagonal matrix. What is the solution of the following optimization problem?
\begin{align}
\min_{\mathbf{...
- 287
1
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0
answers
35
views
How to chose the start vector for the MTZ variables
In the context of LP-formulations for the Traveling Salesman Problem the MTZ constraints prevent subtours via $n$ (i.e. effectively $n-1$) additional variables $$u_1=1\\2\le u_2,\,\dots ,\,u_n\le n\\ ...
- 13.2k
1
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0
answers
200
views
Drawing a 3D object in a 3D environment, and converting to math [closed]
So I have been granted a free time and I want to work on a project but first I had to research.
As we know, lines have infinite points, and with lines, we can create infinite shapes. I want to let ...
- 11
1
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0
answers
69
views
On perfect matchings on planar graphs - is there a linear time deterministic algorithm?
The slides here provide a way to get a pfaffian orientation from Minimum Spanning Tree.
MST can be found in linear time if graph is planar and weights are $1$ and the slides give a linear time ...
- 13.9k
1
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0
answers
181
views
Does fraction-free Gaussian elimination use fractional row operations?
I would like to understand whether Gaussian elimination of an integer matrix, which uses only row operations of the form
Addition (or subtraction) of row $i$ to row $j$
can be performed in ...
- 111
1
vote
0
answers
58
views
Second-order envelope theorem for linear programming
Consider parameterized linear programming $V(\theta) = \max_x \langle c(\theta),x\rangle$ s.t. $A(\theta)x\leq b(\theta)$, $x\geq 0$. Let's also assume $c,A,b$ are infinitely differentiable with ...
- 373
1
vote
0
answers
37
views
Sum of all integer binary solutions of a TUM linear system
I have the following problem: $A x = b$ where $A$ is a $m \times n$ total unimodular matrix (TUM) with entries in $\{0,1\}$ and $b$ is a $m$-vector of strictly positive integers. Let $\mathcal X$ be ...
- 11
1
vote
0
answers
83
views
Literature about graph isomorphism and incidence matrix [closed]
I would like to read some paper, if any, for some classes of graphs, regarding inverting (right/left inverting) the incidence matrix to solve the graph isomorphism problem. Or anyway some known facts ...
- 1,000
1
vote
0
answers
185
views
Maximum independent set in dense graphs
Let $0 < A < 1$ and $G$ be connected d-regular graph
with degree $d=[A n]$. The density of $G$ is about $A$.
Q1 Are there constraints on $A$ such that finding maximum
independent set of $G$ is ...
- 25.4k
1
vote
0
answers
583
views
Langlands program and complexity theory
Back when I was an undergraduate, I spent some time reading the about the modularity conjecture, but the details are fuzzy now.
One of the motivations I imagined for the Langlands program was for ...
- 59
1
vote
0
answers
75
views
Subgraph isomorphism problem with linear map
I am working on proving the NP-hardness of a problem by reducing it from the subgraph isomorphism problem. Currently, I can reduce it from the following problem:
Problem 1: Given two graphs $G=(V, E)$ ...
- 101