All Questions
1,809 questions
1
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1
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137
views
Complexity of edge coloring graphs with $\Delta(G) \ge n/3$ assuming the overfull conjecture
Closely related to this on cstheory.
Let $G$ be graph of order $n$ with $\Delta(G) \ge n/3$.
Assume the overfull conjecture.
Can we edge color $G$ with minimal number of colors in polynomial time?
...
- 25.4k
1
vote
1
answer
130
views
find a PSD matrix that that verify matrices sum of equality
$A $, $ C$ $(n,n)$ are symmetric PSD matrices,
$B$ is PD symmetric matrix, and $H_i$ $\; $ $(i=[1,m])$ represent $ m $ complex matrices. $H_i$ are all one rank matrix
Our objectif is to find ...
- 377
1
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1
answer
99
views
Estimating volume of a simple object
Volume computation is $\#P$ hard.
Take the $[0,1]^n$ polytope.
Slice it by an half space inequality with $poly(n)$ bit rational coefficients into two unequal halves.
Volume of bigger section is $\...
- 13.9k
1
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1
answer
154
views
Find probability of non-stationary inputs into Turing machine?
Consider some finite string $x=(x_1,x_2,...,x_{n-1},x_n)$ that is drawn from a non-stationary process. Would it be possible to use the algorithmic probability formula, defined by Solomonoff as,
$$
P_M(...
- 91
1
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1
answer
82
views
The complexity on calculation of the Graev metric on the free Boolean group of a metric space
For a set $X$ by $B(X)$ we denote the family of all finite subsets of $X$ endowed with the operation $\oplus$ of symmetric difference. This operation turns $B(X)$ into a Boolean group, which can be ...
- 41.8k
1
vote
1
answer
330
views
How good is the LP relaxation?
Consider the optimization problem
\begin{align}
\max_{x\in\mathbb{R}^n}~c^Tx~, \text{ s.t. } Ax=b,~x_i\in\{0,1\}~\forall i
\end{align}where $c,b\in\mathbb{R}_{+}^n$ and $A\in\mathbb{R}_{+}^{n\times n}$...
- 1,421
1
vote
1
answer
435
views
Non-invertible Karp reduction
Karp (many-one) reducibility between $NP$-complete problems $A$ and $B$ is defined as a polynomial-time computable function $f$ such that $a \in A$ if and only if $f(a) \in B$. Berman-Hartmanis ...
- 2,993
1
vote
2
answers
233
views
On complexity of a combinatorial number theoretic problem?
Given the matrix
$\begin{bmatrix}
r_{11}&\dots&r_{1n}\\
\vdots&\ddots&\vdots\\
r_{m1}&\dots&r_{mn}
\end{bmatrix}\in\Bbb Z^{m\times n}$ with $0<r_{ij}<2^n$ and $a,q\in\...
- 13.9k
1
vote
1
answer
399
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$ are irreducible and of equal degree what is the best ...
- 13.9k
1
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1
answer
82
views
Complexity of quadratic polynomials isomorphism
Two polynomials $f,g$ are isomorphic iff $f(x_1,\ldots x_n)=g(\pi(x_1, \ldots x_n))$ for a permutation $\pi$.
$f,g$ are equivalent if there exists invertible linear transormation
$A$ such that $f(X)=...
- 25.4k
1
vote
1
answer
184
views
Do doubly infeasible Linear Programming problems always have doubly infeasible bases?
Consider a Linear Programming problem in dictionary form,
$$\max\Big\{f^\pi+\!\!\!\sum_{j\in D(\pi)}\!\! d^\pi_jx_j~\Big|~\forall~i\!\in\!B(\pi)~~~ b^\pi_i+\!\!\!\sum_{j\in D(\pi)}\!\! G^\pi_{ij}x_j\...
- 21
1
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1
answer
183
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Understanding the paper: "Guarded Fixed Point Logic"
This question is specifically about the paper "Guarded Fixed Point Logic" by Gradel and Walukiewicz. Among other things they prove the decidability of the satisfiability problem for Fixpoint Loosely ...
- 111
1
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1
answer
357
views
Analysis of a partition algorithm
EDIT:
I realized that if $J$ is not a solution so is $J^c$. I updated the algorithm because of this.
Given some positive integers $x_1,\cdots, x_n$.
The following algorithm is for solving the ...
user6671
1
vote
1
answer
831
views
What is the fastest way to sort numbers lexicographically?
I have $N$ sequences of numbers. None of them is longer than $10^6$. I want to sort those sequences lexicographically. For example, given sequences {1, 2, 4}, {1, 2, 3}, {2, 5, 7}, {2}, I want to have ...
- 129
1
vote
2
answers
256
views
Maximum subgraph edge distance greater than given number
I have a weighted graph G with approximately 75000 nodes. I would like to find subgraph G' induced on a subset of nodes, such that all edge weights in G' are greater than a given constant C and the ...
- 283
1
vote
1
answer
225
views
About expectation norms on graphs
Let $S \subseteq V$ of a $d-$regular graph $G$ such that $\mu = \frac{\vert S \vert }{\vert V \vert } $. Let $A$ be the adjacency matrix of the graph. Then define the quantity $\phi(S)= \frac{E(S,\bar{...
- 1,893
1
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1
answer
207
views
Computing probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s
This question came up in my research: What is the probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s?
So far I only figured out that I can do Monte ...
- 113
1
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1
answer
181
views
A particular argument in the review on expanders by Hoory-Linial-Wigderson
I am thinking about the third bullet point on page 455 here, http://www.ams.org/journals/bull/2006-43-04/S0273-0979-06-01126-8/
Can someone explain what is the argument there which seems to conclude ...
- 1,893
1
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1
answer
1k
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Linear programming with exponentially many constraints and variables [closed]
From class, I learnt that problems like traveller salesman have a Linear programming representation with exponentially many constraints. Using method of separation, this problem is solved rather ...
- 867
1
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1
answer
116
views
Implications of the impossibility of efficient sampling from random non-Hamiltonian graphs
Nisan's answer to this question shows the Impossibility of efficient sampling from random non-Hamiltonian graphs (unless $NP=coNP$). I am interested in the implications of this conjecture.
Does the ...
- 2,993
1
vote
1
answer
70
views
Heuristic for choosing n-vectors from n-sets
my given problem is:
choose n-vectors from n-sets (one vector from each set) so that the biggest element in the sum of the chosen vectors is minimal. Unfortunately the problem is NP-hard. So I'm ...
- 11
1
vote
1
answer
321
views
Cycles of Permutation Related to Rectangular Matrix Transposition
let the entries of a rectangular matrix $A\in\mathbb{C}^{m\times n}; m,n\in\mathbb{N}$ be stored in row-order in a linear vector $v$, i.e. $A_{i,j}=v_{i*m+j}$
Question:
How can the first element ...
- 13.2k
1
vote
2
answers
375
views
When is a power of an indeterminate in an ideal with 2 generators?
If I have an ideal ${\frak a} \colon= (f(X,S), g(X,S))$ of height $2$ in ${\Bbb C}[X, S]$, is it easy to know what power of $S$ is contained in $\frak a$? For example, what is the minimal number $m$? ...
- 87
1
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1
answer
3k
views
Minimizing sum of absolute deviations
Suppose we want to find coefficients $b$ in $\underset{b}{\operatorname{argmin}} \displaystyle\sum\limits_{i=1}^n | y_{i}-b_{1}x_{i}-b_{0}\mid$.
If we rewrite this problem in terms of linear ...
- 99
1
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1
answer
246
views
Eigenvalue problem with quadratic constraints
$\circ$ Consider the following eigenvalue problem : $$Ax=\lambda x \hspace{0.5cm} (1)$$
where matrice $A \in \mathbb{R}_{n \times n}$ is a positive semi-definite with eigenvectors $x = (x_{1},x_{2},.....
- 45
1
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1
answer
992
views
How to prove the NP-hardness of this set covering problem
In the Set Covering problem, we are given a ground set $U$ and a collection $S$ of subsets of $U$, where each subset is associated with a non-negative cost, the Set Cover problem asks to find a ...
- 23
1
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1
answer
226
views
Construction of an integral point set given the set of distances, its minimal description to get a measure of its complexity and its unique identifier
Given a set of distances between every pair of points of an integral point set $P$ of $n$ points; say $D = \{{d_i}\}$.
Q1. What is the least time complexity
possible/known for recreating the
...
- 851
1
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2
answers
306
views
Name of operations on two vectors
Suppose we have two vectors $x\in \mathbb{R}^n$ and $y\in \mathbb{R}^m$.
I could define the mapping
$$
T: \mathbb{R}^n\times \mathbb{R}^m \rightarrow \mathbb{R}^{n\times m}
$$
as follows
$$
T(x,y) = ( ...
1
vote
1
answer
4k
views
What does "Vertex Solution" mean?
Hello!
I come across the word "vertex solution" in the context
" We can also assume that x and y are vertex solutions,so that the sequence {x,y} remains in a finite set."
Could anybody know any ...
- 13
1
vote
1
answer
1k
views
Longest run of heads
Consider $n$ independent tosses of a fair coin; the sample space has $2^n$ elements. Let $R_n(x)$ be the length of the longest run of heads in outcome $x$. We know that $$E[R_n]=\Theta (\log n)$$
csun....
- 51
1
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1
answer
292
views
Can all programs reducible to ones with only arithmetic operations on inputs be simulated with polynomial overhead by arithmetic machine?
I failed to get an answer at https://math.stackexchange.com/questions/364061/can-all-programs-reducible-to-ones-with-only-arithmetic-operations-on-inputs-be, so I am asking here.
In https://math....
- 11
1
vote
2
answers
1k
views
Nonstandard Hessian approximations in Gauss-Newton
The Gauss-Newton algorithm optimizes functions
$$
E(x) = \sum f(x)^2
$$
by approximating f as (locally) linear, in which case the Hessian of $E$ is approximated as
$$
H = 2 \sum {J_f}^T J_f
$$
Now ...
- 561
1
vote
1
answer
769
views
Results for minimizing the norm w.r.t a unitary matrix
Suppose $x \in \mathbb{R}^n$, $B,U \in \mathbb{R}^n\times\mathbb{R}^n$ and $U$ a unitary matrix. Define $g_{U}(x) = || BUx||$ where $||.||$ is some norm or norm-ish function on $\mathbb{R}^n$ (not ...
- 322
1
vote
2
answers
134
views
LP/QP with not-so-constant linear constaints
I have an otherwise standard LP or PSD QP problem as below:
$\min\limits_x {c}' x$ subject to $Ax\leq b$
or
$\min\limits_x \frac{1}{2}{x}' Qx + {c}' x$ subject to $Ax\leq b$
the only exception ...
- 11
1
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1
answer
1k
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k-uniform k-partite hypergraph matching in polynomial time
I have what seems like an elementary question, but google didn't throw up any answers for it. I would appreciate any pointers that MO users may provide.
It is well known that for $k\geq 3$ finding ...
- 61
1
vote
1
answer
2k
views
Can one efficiently optimize over the inverse of matrix?
Hello,
I have the following problem:
Find a non-negative matrix $L$ (i.e. $L_{i,j} \geq 0$ for all $i,j$), $L \neq I$ so that $A(I-L)^{-1}y \geq 0$ (the inequality must hold for each component), ...
- 53
1
vote
1
answer
640
views
Approximate Set Cover Problem by Rounding
Here is the simple algorithm for approximating set cover problem using rounding:
Algorithm 14.1 (Set cover via LP-rounding)
Find an optimal solution to the LP-relaxation.
Pick all sets $S$ for ...
1
vote
1
answer
257
views
Pseudorandom Generator vs Constant Depth Circuits / Branching Programs
Hi.
I am looking for a survey on the state of the art in pseudorandom generators vs
(1) constant depth circuits
and/or
(2) Branching Programs
For (1), is "Anindya De, Omid Etesami, Luca Trevisan ...
1
vote
1
answer
290
views
Proof that any NP problem can be reduced (in P time) to any problem in NPC?
Given the seemingly broad definition of NP, it is very interesting that one can prove that any member of NP can be reduced in polynomial time to any member of NPC. (I guess this is true by definition ...
1
vote
1
answer
531
views
Split sum into equal terms
Given a sum of $l$ integers $r_1+...+r_k+...+r_l$ and an integer $t$.
Find indices
$1 < p_1 <...< p_h <...< p_{t-1} < l$
such that in sum
$(r_1+...+r_{p_1})+...+(r_{p_{h-1}+1}+......
- 11
1
vote
1
answer
7k
views
Finding the square root modulo n, when the factors of n are known
Last month, I asked whether there is an efficient algorithm for finding the square root modulo a prime power here: Is there an efficient algorithm for finding a square root modulo a prime power?
Now, ...
- 2,649
1
vote
4
answers
978
views
Maximum average value within a rectangular bounding box
The goal is to expedite detection using the sliding window approach. In other words, an object classifier is known and I need to find where the possible locations of this object are in an image. This ...
- 111
1
vote
2
answers
1k
views
Inequality-constrained linear-regression, what is the covariance of the estimator?
If you do a linear regression: $||Ax - e ||^2$, where e is iid Gaussian, mean 0 and variance 1, then your answer is $x_{hat} = (A' A)^{-1} (A' * e)$ and the covariance of $x_{hat}$ is $(A' A)^{-1}$
...
1
vote
0
answers
61
views
Reference request: Proof theory in $W_1^1$
Buss defined $V_2^1$ as a second-order bounded arithmetic corresponding to $\mathsf{PSPACE}$.
Later, Skelley introduced $W_1^1$, a third-order bounded arithmetic of $\mathsf{PSPACE}$.
Since the ...
- 11
1
vote
0
answers
29
views
Integral hull of a polyhedron Q is polyhedron
Let $Q \subseteq R^n$ be a rational polyhedron and let $Q_I=Convexhull(Q \cap Z^n)$. By finite basis theorem, we have $Q=P+C$ for some rational polytope $P$ and finitely generated cone $C$ where $C=R_+...
- 21
1
vote
0
answers
37
views
Computing all roots of a function with square-root terms
Given $3n$ positive numbers $a_1, \ldots, a_n$, $b_1, \ldots, b_n$, and $x_1, \ldots, x_n$, we are given a function
$$f(x) = \sum_{i = 1}^n \frac{a_i}{\sqrt{(x - x_i)^2 + b_i}}.$$
Can we find all the ...
- 11
1
vote
0
answers
55
views
How computationally efficient are kernel tricks? [closed]
"If we compare to non-kernel polynomial regression it is O(Tnp) where is p is dimension of polynomial while kernel polynomial is O(n^2d) + O(T*n^2) where d is original number of attributes, ...
1
vote
0
answers
114
views
Simultaneous elimination of variables in multiple polynomials
I have a system of $n=O(1)$ non-homogeneous polynomials of total degree $d=O(1)$ $p_1,\dots,p_r\in \mathbb Z[x_1,\dots,x_n]$. I would like to eliminate $n-1$ variables simultaneously from the $n$ ...
- 13.9k
1
vote
0
answers
92
views
Proof for non-existence of short integer program for squares
We do not know if $P=NP$ or not or if there is a superfast integer mutiplication algorithm. But I do not think either assumption is necessary to answer this question.
Is there a way to show within an ...
- 13.9k
1
vote
0
answers
69
views
Is it in theory possible to perform general Miller’s algorithm inversion as used with the optimal ate pairing with large trace in subexponential time?
Let’s I have the following :
2 curves $G_1$ defined on $F_p$ and $G_2$ being the $G_1$ curve’s twist defined on $F_p^2$ both having the same prime order ; a large trace ; and $F_p^{12}$ as their ...
- 87