All Questions
1,809 questions
0
votes
2
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2k
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Time complexity of finding the GCD of a set S as a function of sum(S)
The algorithm to be used is:
Sort the set into ascending order
$x_1 = s_1$
$x_i = gcd(x_{i-1},s_i)$
$GCD = x_n$
What I'm looking for is expected run time as a function of $\sum_{i\in S}i$
As a ...
- 205
0
votes
2
answers
531
views
Any idea of solving an optimization problem with cubic constraints?
I have the following optimization problem with cubic constraints, which is hard to solve. Are there any ideas, or related references, of solving such a problem?
$$ \begin{array}{ll} \underset {y, z} {\...
- 21
0
votes
1
answer
103
views
Constrained linear optimization problem on $C^1$
I am dealing with a problem of the form ($a<b$)
$$
\displaystyle \max_{v \in C^1([a, b])} \int_a^b v(x)~\mathrm{d}x, \quad \mathrm{s.t.} \int^b_a \big(-o'(x)v(x)-v'(x)o(x)\big)f(x)~\mathrm{d}x \...
0
votes
1
answer
147
views
Is there a redundant constraint in linear programming? [closed]
From wikipedia:
But... Why do we need the $x\ge 0$ part? We can instead do $-x\le 0$, and thus saving a line in the definition (which is not a big deal but nevertheless nice).
(In order to do that, ...
- 3
0
votes
1
answer
143
views
$\mathrm{ILP}$-formulation for Minimum Maximal Matching (MMM) Problem
Despite some online searching I couldn't find examples of dedicated Integer Linear Programs ($\mathrm{ILP}$s) for determining smallest matchings, that are not contained in a larger one.
It seems that ...
- 13.2k
0
votes
1
answer
139
views
Computationally intractable orbit of a monoid action on a finite set
Suppose for each integer $n\geq 1$ we have a submonoid $M_n\subset \mathrm{Self}(\{1, \dots, n\})$ of self-maps of $\{1, \dots, n\}$.
A characterization of $M_n$ is an algorithm that takes an integer $...
- 1
0
votes
1
answer
146
views
Is minimum weight vertex cover problem NP-easy? [closed]
I think that Minimum weight vertex cover problem is NP-easy. However I don't know how to prove that. Does anyone know how to prove it?
- 91
0
votes
1
answer
128
views
What is the computational complexity of the calculation of $ \Psi(x) $?
What is the computational complexity of the calculation of $ \Psi(x) $ described below:
Let $\left\{ f_i : \{0,1,\dots,m\} \to \mathbb{R} \right\}_{i=1}^n$. For each $x \in \{0,1,\dots,m\}$ we ...
0
votes
1
answer
120
views
Complexity of solving linear equations plus disequality constraints $a \ne b$
Let $K$ be ring and $S$ linear homogeneous system with $n$ variables $x_i$ over $K$.
Add to $K$ linear disequalities of the form $x_k \ne x_l$
and let the final system be $S'$. If $K=\mathbb{F}_2$,
$...
- 25.4k
0
votes
1
answer
126
views
An otherwise linear matrix equation with the presence of a signum function : reference request
Consider the equation $$\pmb{c}+\text{sign}(G\pmb{c}) = L$$
$\pmb{c}$ is a $n\times1$ matrix.
$G$ is a $n\times n$ matrix which is also positive definite.
matrices $G$ and $c$ are real.
$L$ is a $n\...
- 698
0
votes
2
answers
210
views
Limited sum for whole sum approximation
Let $d_n, n\in\{1,2,\cdots,N\}$ be $N$ realizations drawn independent and identically from uniform distribution on $(0,L)$ where $L=\gamma\sqrt{N}$ with constant $\gamma$. Suppose that we need to ...
- 287
0
votes
1
answer
405
views
Computing discrete optimal transport
I am trying to find a combinatorial approach to solve the following optimization problem.
\begin{align}
&\max_{x_{ij}} C_{ij} x_{ij}, \\
&\text{such that},\\
&\sum_{j} x_{ij} \leq r_i~\...
- 133
0
votes
1
answer
99
views
Finding dual of a scheduling LP formulation
Suppose I have an LP formulation as such:
$\min\ \ \sum\limits_{i,j,t}\ w_{ij}x_{ijt} (\frac{t-r_j}{p_{ij}}+0.5)$
$\sum\limits_{i,t}\frac{x_{ijt}}{p_{ij}}=1\,\forall\ j$
$\sum\limits_{j}x_{ijt}\leq ...
- 103
0
votes
1
answer
61
views
Variant of the linear programming problem
Good afternoon, my experience in mathematical programming is low. I would like to know if there is any general method to address the following problem:
$$\text{Minimize }\sum_{i=1}^n d_i(x_j)$$
$$s.a....
- 193
0
votes
1
answer
94
views
Is there any solution that currently exists for the graph automorphism problem in the general case?
I was reading the Wikipedia pages on the graph automorphism, but I could not find any solution to the problem (Not even a brute force one). So, is it indeed true that no solutions exist for the ...
- 101
0
votes
1
answer
145
views
How to solve this optimization problem efficiently? [closed]
Let, $D\in\mathbb{C}^{1\times M}$ is a row vector with $M$ elements
$V\in\mathbb{C}^{3^M\times M}$ is a given matrix
$T$ is a scalar (real and $>1$)
$\textbf{The problem at hand is as follows:}$
...
- 111
0
votes
1
answer
212
views
Is an exact violated inequality constraint met as equal constraint in optimal solution?
We have a solution which does not satisfied exactly one inequality constraint in linear program. The corresponding dual solution is also feasible. Is it correct this constraint is in equal form in the ...
- 25
0
votes
1
answer
245
views
Modification of the subset sum problem - "perfect coverage" of the set with good solutions
I have a problem.
We have a set.
$S = (a_1, a_2 ... a_k)$ and an integer $x$.
We know that there is a sum of elements to $x$ in it.
We also know that:
if there is only one sum to $x$ then it must ...
- 133
0
votes
1
answer
79
views
algorithms and tools available for a particular polytope computation
Let me define each half space i as:
$${H_i}:{c_i}{\bf{x}} \le {b_i}$$
The intersection of all such ${H_i}$ gives a polyhedron (bounded or not). Suppose I am interested in if ${H_i}$ is active (...
- 867
0
votes
1
answer
201
views
Recursive linear programming on a linear subset of a simplex
The problem I am working on is:
Given an $n$ dimensional vector $r \in \mathcal{R}^n$, and a convex set $G=\{\mu \in \mathcal{R}^n | \mu_i \ge 0, ~ \mu^T \mathbf{1}=1, ~ A\mu =0 \}$ where $\mathbf{1}...
0
votes
1
answer
204
views
Is the linear production game a convex game?
In cooperative game theory, the linear production game (LPG) is defined by letting the characteristic function have the form of a linear programming problem.
Does anyone know if the LPG is a convex ...
- 17
0
votes
1
answer
543
views
Convert general optimization problem to LP problem
I am trying to convert the following problem into a linear programming problem:
There are $M\times N$ matrix $T$ of real numbers between 0 and 1 and $N\times 1$ vector $w$ of real numbers between 0 ...
- 99
0
votes
1
answer
82
views
Introducton books for $\frak{E}_p(I)$
Are there any good books different from abstract harmonic analysis by hewitt to study $\frak{E}_p(I)$. where $\frak{E}_p(I)$ is: Let $I$ be an arbitrary index set. For each $i\in I$ let $H_i$ ...
- 209
0
votes
2
answers
118
views
Inner Product of Given Sum Positive Sequence
Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in A,\,\sup\...
- 2,239
0
votes
1
answer
150
views
Function that dominates everything in little o
I have a function $f(n)$ that satisfies the following property: for any function $g(n) = o(n^{-2})$, we have $f(n) = \Omega(g(n))$ (the implied proportionality constant in the $\Omega$ expression ...
0
votes
1
answer
546
views
Solution of infinite dimension linear system
Suppose that ${a_n}$ and $b_n$ is decreasing sequence such that $a_0=A$, $lim_{n->\infty}a_n=0$ and $b_0=B$, $lim_{n->\infty}b_n=0$.
For fix n,
we can construct n dimension linear equation ...
- 111
0
votes
2
answers
708
views
Approximate solution to large mixed integer programming problem
What are the available approaches to find an approximate solution to a large mixed integer programming problem?
I ran my problem in the Gurobi MIP solver.
It can find a feasible solution in ...
user24451
0
votes
1
answer
452
views
Large scale least squares of non symmetric and non square problems
Given a system like $b=Ax$ with an non symmetric and non square $A$ I would like to solve it having many elements in $x$ (lets say $10^7$).
There is a large amount of algorithms for symmetric ...
- 125
0
votes
1
answer
100
views
generalization from linear programming solution [closed]
I have a series of similar linear programs that depend on an input vector $a\in A$ and whose solution is an output vector $b\in B$. I can solve them individually, but this is wasteful. I suspect that ...
- 109
0
votes
1
answer
85
views
About the suboptimality of linear estimators
Let $X$ be a random variable and $N$ a Gaussian noise independent from $X$. We observe $Y=X+N$ and want to estimate $X$ based on $Y$ to minimize the mean square error $mmse(X|Y):=E(\hat X(Y)-X)^2$.
...
- 29
0
votes
1
answer
2k
views
eigen-decomposition solution? is it unique?
Assume an N*N covariance matrix (Q) which is a positive definite matrix. The decoder X is assumed to be N*s, where s<=N. X is calculated to be s eigenvectors corresponding to s minimum eigenvalues. ...
- 1
0
votes
1
answer
205
views
SDP Algorithms/ maximally complementary solutions
Hello,
I was wondering if there are algorithms for (linear) Semidefinite Programs (SDP) out there, that converge towards a maximally complementary solution, even if strict complementary does not hold.
...
- 1
0
votes
1
answer
504
views
$\ell_o$ Minimization (Minimizing the support of a vector)
I have been looking into the problem
$\min: \|x \|_0$ subject to$: Ax=b$. $\|x \|_0$ is not a linear function and can't be solved as a linear (or integer) program in its current form. Most of my time ...
- 11
0
votes
1
answer
226
views
Continuity of Lexicographic Minimum Solution of a parametrized LP problem
Given a parametrized LP problem
find x, that minimizes F*x
such that Ax <=Bt+D
where t is a parameter.
And suppose C(t) is a set of all optimal solutions of LP with parameter t.
Let x_L(t) be ...
- 29
0
votes
2
answers
148
views
Bounding 2nd Eigenvalue of a Pseudo-Rotation-ish matrix
Let $p,q$ be arbitrary primes.
Let $N = p * q$.
Let $I$ be the $N * N$ identity matrix.
Let $R$ be the $N * N$ matrix defined as follows:
$R[x_0 * p + y_0, x_1 * p + y_1]=1$ if and only if $x_0+1 ...
- 45
0
votes
1
answer
695
views
Minimum distance between two data sets
Suppose we have two sets of data, $X$ and $Y$, each of which contains
$10$ positive numbers. Now let us order the data sets $X=\left\{ x_{1},\cdots,x_{10}\right\}$,
$x_{1}\ge\cdots\ge x_{10}>0$ and ...
0
votes
1
answer
409
views
Need help to find an efficient algorithm for the following problem!
Consider $x$ an n-dimensional vector with $x_i$ is integer in the range $[0 \dots k], k\in N$.
Given $A_{n\times n}$ is the covariance matrix of $x$.
$u$ is a given n-dimensional vector of real ...
- 131
0
votes
1
answer
185
views
Derandomize on nondeterministic assumptions
Is there anyone prove the results such like the follows?
If $NP\not\subseteq BP(2^{\Omega(n)}),$ then $BPP\subseteq P^{NP}$
In summary, my question it that, can we get some derandomized results ...
- 57
0
votes
1
answer
1k
views
Existence of a pseudo-polynomial time algorithm for a counting problem.
Let T={1,...,n} be a set of tasks. Each task i has associated a non negative processing time p_i and a deadline d_i. A feasible schedule of the tasks consists of a permutation of n elements pi, such ...
- 27
0
votes
0
answers
57
views
Complexity of evaluation of analytic functions
Given an analytic function $f(x)$ (say as combination of elementary functions and operators), is it possible to compute $n$ first bits of the value of the function on the whole interval $[a, b]$ ...
- 67
0
votes
0
answers
21
views
Easy instance of set cover
I am trying to prove that a natural greedy algorithm solves the following instance of the set cover problem: for a set of elements $e\in U$ with a set of weights $w_e$, we define the cost of a subset ...
- 4,049
0
votes
0
answers
74
views
Klondike Solitaire as an NP-complete game
I am not a mathematician. I am trying to understand if the paper "The complexity of solitaire" that shows this game is NP-complete also has a implicit assumption that a given hand can only ...
- 1
0
votes
0
answers
37
views
Does the same hardness with access to different oracles imply equivalent oracles?
If $\textrm{coNP}^{\textrm{NP}}=\textrm{coNP}^{L}$ for $L\in\textrm{NP}$, then does that make $L$ NP-complete?
0
votes
0
answers
60
views
Understanding "A Note on a Theorem by Ladner"
I'm reading A Note on a Theorem by Ladner by Balcazar and Diaz, trying to understand the proof of Theorem 2.1. I don't know if it's just old notation, but I'm having quite a bit of trouble.
In step (...
0
votes
0
answers
122
views
Is it in theory possible to create a subexponential algorithm for solving discrete logarithms in multiplicative subgroups or within an Integer range?
As far I understand, when it comes to finite fields, Pollard rho and Pollard’s lambda are still the best algorithm for solving discrete logarithms in a multiplicative subgroup/suborder…
Index calculus ...
- 87
0
votes
0
answers
48
views
A question on a quantitative form of Farkas' lemma
Suppose A is an $m \times n$ matrix whose entries are non-negative integers and $\mathbf{b}$ is a vector with rational entries. A version of Farkas lemma implies that if the equation $$A\mathbf{x}=\...
- 6,214
0
votes
0
answers
115
views
Software for computing polytopes
As can be inferred from the title, I want to do some computation on the facets representation of the polytopes given the vertices. My advisor recommended me Polymake, which is indeed useful even with ...
- 1
0
votes
0
answers
92
views
Algorithm that can solve or approximate the solution to a combination problem
I have a computational problem on my hands and I would like your help.
Here is my problem (simplified)
Let $X = \{x_1, x_2, \ldots, x_n\}$ represent a set of $n$ values.
Each value $x_i$ has a ...
- 1
0
votes
0
answers
39
views
Prove the NP-hardness of the following problem: Whether there exists a partition for a set of data points
Can anybody help me prove the NP-hardness of the following question:
Given $x_0, x_1, ..., x_m \in \mathbb{R}^n$, determine whether there exists a partition $S\cup [m]\backslash S$, such that $x_0 \in ...
0
votes
0
answers
91
views
Some new questions on Rademacher complexity
For $A\subset R^n$,$A=(a_1,a_2,\dots, a_n)$, $\sigma_i$ are Rademacher random variable.
Is $|\mathbb{E}_\sigma \inf_{a\in A}\sum_{i=1}^n\sigma_ia_i| \le |\mathbb{E}_\sigma \sup_{a\in A}\sum_{i=1}^n\...
- 185