All Questions
1,809 questions
0
votes
1
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139
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Linear programming with exponential inequalities and rational variables
If we are given a set of real linear inequalities then using elimination theory or just linear programming we can decide. If the program also has inequalities of form $2^x\leq g$ in addition to linear ...
- 1,826
0
votes
0
answers
132
views
How can GL(n) acts on the determinant polynomial?
I'm reading Landsberg's paper, which provide an introduction to geometric complexity theory. At chapter 2 of this paper, the author defined the following objects:
Let $W = \mathbb {C}^{n^2}$, $det_n \...
- 47
0
votes
0
answers
56
views
Reference about Relation between Probabilistic Turing Machine and Turing Machine of every hierarchy
What are the relation between Probabilistic Turing Machine and Turing Machine of every hierarchy, for instance, are the Probabilistic PDA and NPDA equivalent? the Probabilistic LBA and LBA equivalent?...
- 3,094
0
votes
0
answers
123
views
Complexity of transforming a polynomial to its canonical form
Why transforming the polynomial $F(x)=\Pi_{i=1}^{d}{(x-a_i)}$ to its canonical form (@) requires $\theta(d^2)$ multiplications of coefficients?
(@) The canonical form is defined as $\Sigma_{i=0}^{d}{...
- 45
0
votes
0
answers
25
views
Complexity of combined trajectory compression algorithm
Is it possible to calculate the complexity of the combined algorithms (TD-TR-SP, TD-SP-TR) provided here:
A new perspective on trajectory compression techniques
In more details, TD-TR-SP algorithm ...
- 101
0
votes
0
answers
287
views
Complexity of pseudo-inverse of random matrix
Assume that $\mathbf{A}_{M\times N}$ is a sparse complex matrix. Then, what is the complexity of computation of its pseudo inverse, i.e.,
$$\mathbf{A}^{\mathrm{H}}(\mathbf{A}\mathbf{A}^{\mathrm{H}})^{-...
- 287
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votes
0
answers
43
views
Minimizing along independent directions, nonlinear programming
Good afternoon, I am studying the book Nonlinear Programming: Theory and Algorithms (by Mokhtar S. Bazaraa, Hanif D. Sherali, C. M.) particularly the Theorem $7.3.5$. I'm not sure I understand this ...
- 193
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votes
0
answers
101
views
How can we analytically solve this max-sum-min problem?
Let $I$ be a finite set, and $A_{ij},B_{ij},x_i,y_j\ge0$. I want to find the choice of $x_i,y_j$ maximizing $$\sum_{i\in I}\sum_{j\in J}A_{ij}\min\left(x_i,B_{ij}y_j\right)\tag1$$ subject to $$\sum_{i\...
- 167
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votes
0
answers
35
views
Converting a vector in a cone statement to inequality constraints
I would like to convert the following condition for $x$
\begin{align}
x = N \lambda, \lambda \geq 0
\end{align}
to a pure linear inequality form, i.e. find an $L$ and eliminate $\lambda$
\begin{...
- 1
0
votes
0
answers
82
views
Proving Vizing's and Brooks' theorem using the polynomial approach
It is known that the graph polynomial defined by $\prod_{i<j}(x_i-x_j)$ where the vertices $x_k\ \ , \ \ k=\{1,2\ldots,n\}$ are ordered with respect to some order; can be used to verify the proper ...
- 2,089
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votes
0
answers
232
views
What do square roots as minimums have to do with Harmonic numbers?
In an earlier question where I conjectured (and GH from MO confirmed) that the von Mangoldt function is the limit at s=1 of a certain Dirichlet series:
$$\Lambda(m)=\lim_{s\to 1+}\zeta(s)\sum_{d\mid ...
- 1,183
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votes
0
answers
84
views
Deterministically finding a subsequence of integers matching modular roots?
Take two integers $n$ and $m$ with $0<\log_2m<n<m$ and let $r_1=f_1(n)\bmod m$ and $r_2=f_2(n)\bmod m$ for functions $f_1,f_2:\mathbb Z\rightarrow\mathbb Z$.
Denote the $\ell$ roots of $(f_i(...
- 13.9k
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votes
0
answers
129
views
Do many homogeneous polynomials help in faster integer root extraction?
Given $n$ homogeneous algebraically independent total degree $2$ polynomials with no $x_1^2,\dots,x_n^2$ variable in $\mathbb Z[x_1,\dots,x_n]$ with promise that it has non-zero integer roots with ...
- 1,826
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0
answers
89
views
Why there isn't lexicographically smallest solution to a bounded linear program?
I am learning computational geometry when I run into this confusion. "A bounded 2D linear program may not have a lexicographically smallest solution", as the book says. I wonder why? I think we can ...
- 1
0
votes
1
answer
490
views
Relax a rectangular linear assignment problem
I wonder if there is any literature on the following problem
$$\begin{array}{ll} \underset{X \in \mathbb R^{m\times n}}{\text{minimize}} & \displaystyle\sum_{i,j} C_{i,j} X_{i,j}\\ \text{subject ...
- 205
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votes
0
answers
99
views
Is this Graph Iteration Already Known?
When attempting to set up an ILP formulation for a weight-minimal cubic spanning tree (i.e. one with vertex degrees either 1 or 3) I needed connectivity constraint, but misremembered the contents of ...
- 13.2k
0
votes
0
answers
46
views
linear inequalities and reference request
I have proved and am using the following simple lemma in my current research problem:
Let $\{a_1,...,a_m\}$ and $\{b_1,b_2,...,b_n\}$ be set of positive numbers such that $\sum_{i=1}^m a_i < \sum_{...
- 1,269
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votes
0
answers
149
views
Is 3-coloring bounded degree graphs subexponential: $O(\exp{(\sqrt{n}\log^2{n})})$?
We got an argument that 3-coloring bounded degree graphs is subexponential
with complexity $O(\exp{(\sqrt{n}\log^2{n})})$.
The treewidth of a planar graphs on $n$ vertices is $O(\sqrt{n})$
and 3-...
- 25.4k
0
votes
0
answers
369
views
Finding a point in the relative interior of the convex hull of a set of integer-valued vectors
Let $X \subset \mathbb{Z}^n$ be the set of integer-valued vectors satisfying a system of linear constraints. We can suppose that $X$ is the set of integral points in a given polyhydral set $Y \subset \...
- 136
0
votes
1
answer
345
views
Is there fast algorithm for 3SUM?
Is there algorithm for 3SUM which have complexity O(n) or O($n^{3/2}$) for randomly chosen input with bit length of maximum number approximately equal to count of input numbers?
- 195
0
votes
0
answers
257
views
Hercules and the Hydra with time constraints
The game of Hercules vs. the Hydra can be put in terms of a single number in hereditarily-factorized form. For example, if the Hydra is $2^{19^3} \cdot 5^{11^7}$, Hercules must choose between two ...
- 2,302
0
votes
0
answers
74
views
Transformation or correspondence between language and real number
As we know, formal language can be regarded as a set of strings of alphabet, and real number can be regarded as sequence generated by set of integers, for example, denominators of the simple continued ...
- 3,094
0
votes
1
answer
81
views
Can convex combinations of indicator functions for pairwise non-disjoint sets unordered by inclusion dominate one another?
Let $N$ be a finite subset of the naturals. Let $P$ be a set of subsets of $N$ such that:
1) $P\neq \varnothing$,
2) $\forall x\in P, |x| >1$,
3) $\forall x,y\in P,$ if $x\neq y$, then $x\not\...
- 3
0
votes
0
answers
68
views
A seemingly easy integer programming question
Let $k, m \in \mathbb{Z}_{ > 1}$. Let $a \in \mathbb{Z}_{> 0}^m$ and $t \in \mathbb{Z}^k$. Let $\varepsilon = (\varepsilon_{i,j})_{1 \leq i \leq m \\1 \leq j \leq k}$ be a matrix with entries in ...
- 501
0
votes
1
answer
128
views
Continuous backpack with multiple choice items. NP prove
Continuous backpack with multiple choice items is the problem where you need to collect items by one from each of distinct sets and associate them with rational numbers so that their sum of weights ...
- 1
0
votes
0
answers
890
views
Maximum shortest path problem
I have the following problem. You have a graph and every edge has a certain set of possible weights. The question is to find the assignment of those weight which will maximize the shortest path.
In ...
- 342
0
votes
0
answers
2k
views
Probability two random intervals overlap
I'm working on an algorithm for orthogonal line intersection detection and am trying to analyze some things about it. For simplicity, we can consider the problem as follows:
Given N randomly ...
- 101
0
votes
0
answers
104
views
Surd Partition Problem
Could the following "Surd Partition" problem be NP complete? Note that if the square roots are omitted in the following then the problem is well known to have a polynomial solution.
Surd Partition
...
0
votes
0
answers
633
views
Prove that the subset sum problem with fixed size and number reusability is NP complete
I'm trying to solve the following problem:
There are B lists of unspecified size containing integers. Pick a number from each list so that the sum of all the picks is exactly A. Prove that this ...
0
votes
1
answer
270
views
Generalized assignment problem with no integrality gap
Suppose I am solving the generalized assignment problem, so that I
am given matrices $U$ and $W$ and a vector $c$ (all three of which
have, say, positive entries), and I want to solve
$$\text{...
0
votes
0
answers
165
views
Resources about integral maximization problem
I am looking at the following problem. Given an interval I, and a function f over that interval, find sub-intervals for which:
The sum of the length of the sub-intervals is < k;
The sub-intervals ...
- 109
0
votes
0
answers
917
views
Inverse problem with a rank-1 update
I hope you can help me out with this. I have to find the solution x to an inverse system
$$
x=A^{-1}b
$$
This inverse problem is basically a least square problem with a rank-1 update.
$$
x=[uv^{T}...
0
votes
0
answers
87
views
maximum weight k-edge problem
Given positive integer $k$ and an undirected graph $(V,E)$, with nonnegative (non-uniform) weights on the nodes. Find $k$ edges whose spanning nodes have the maximum weight.
Is this in P or NP? I ...
- 101
0
votes
0
answers
104
views
Big eigenvalues of a special stochastic matrix
Given a matrix $M$ of size $n\times n,$ we write its different eigenvalues by $x_1,x_2,\ldots,x_m$ with $m\leq n$ such that $|x_1|>|x_2|>|x_3|>\cdots|x_m|,$ and call $x_2\doteq |\lambda_2|(M)....
- 105
0
votes
0
answers
79
views
Approximation for accumulative set cover
Let $S_1,\ldots,S_m\subseteq U$ be subsets of a set $U$ of size $\lvert U\rvert=n$. Over all permutations $\pi$ of the set $\{1,\ldots,m\}$, I want to maximize the quantity
\begin{equation}
\sum_{k=1}...
- 2,966
0
votes
1
answer
2k
views
Finding linearly independent columns of a large sparse rectangular matrix
I have a problem that necessitates solving a large non-negative least-squares
problem. My matrix A is large, sparse, highly rectangular (num rows >> num cols)
and nearly binary. However, A is not ...
- 103
0
votes
0
answers
103
views
Gauss-Newton for quotient functions
I'm optimizing a function of the form
$$
\sum \frac{ \|\mathbf{f_i}(x)\|^2 }{ g_i(x)^2 + h_i(x)^2 }
$$
where $x$ is a real vector, $\mathbf{f}(x)$ is a real vector, and $g(x)$ is a scalar. My first ...
- 561
0
votes
0
answers
555
views
VC dimension and boolean hypercube subgraphs
Are there any well studied graph theoretic properties that are common to all subgraphs of the boolean hypercubes that have a given VC dimension d.
- 11
0
votes
0
answers
194
views
A linear program related question
Dear all, recently, I encountered the following problem. It is closely related to the order of growth for UMD constants of all $n$-dimensional Banach lattice.
Let $\alpha^k \in (\alpha_1^k, \alpha_2^...
- 769
0
votes
0
answers
79
views
Computing maximum point for minimal function of a family of linear functions
Let $x \in S^n $ where $S^n = ${$ [x_1,x_2,...,x_{n+1}]\in \mathbb{R}^{n+1} \mid x \ge 0 , \sum x_i = 1 $} and let $f_i : I^n \to \mathbb{R}$ be a finite $m$-sized family of LINEAR functions such that:...
- 11
0
votes
0
answers
783
views
LP relaxation for ILP\IP (integer linear programming)
I am familiar with LP relaxation for ILP (or IP). Assume we concern with integer minimization problem, which we formalize using ILP; we then relax the ILP into LP and we say that the LP provides a ...
0
votes
0
answers
118
views
sparsest cut always has solution
Hi!
How to prove that sparsest cut always has an optimal solution which is the cut for some vertex-subset.
Looks like it's should be a kind of fundamental theorem for sparsest cut. But I didn't ...
- 1
0
votes
0
answers
138
views
Bounds on reducing from NP to SAT
Let M be a non deterministic turing machine.
Suppose M is a TM that runs in T(n) time.
Given an instance of x in {0,1}^n, and the question M(x) accepts?
We can
1) convert M into an oblivious TM ...
- 19
0
votes
2
answers
102
views
Does Max Flow produce uniform results? [closed]
I am interested in using Max Flow algorithm. I want to simulate transfer of quantity. Anyway, I am unsure of some thing.
Does Max Flow algorithm produce uniformly distributed max flow?
I have ...
- 11
-1
votes
1
answer
448
views
how do you convert binary to decimal [closed]
Suppose I have computed $\pi$ to 4 trillion binary digits
(1 trillion hex digits). I want to convert to decimal. You could remove the leading $11_2=3_{10}$ and then multiply the remaining number by $...
- 365
-1
votes
2
answers
114
views
On OR condition in Linear Programming with exponentially many constraints [closed]
Suppose we have two linear programs $Ax\leq b$ and $Bx\leq c$ is there a way to combine them into one program of possibly a larger dimension $Cy\leq d$ such that projection of vectors $y$ into a ...
- 13.9k
-1
votes
1
answer
137
views
Does a half plane contain intersection of some other half planes? [closed]
I'm doing research in Optimization and I have found this obstacle in the way.
If we have set of half planes like $c_ix\leq b_i$ where $i\in \{1,\ldots ,k\}$ there is an algorithm(it would be better ...
- 19
-1
votes
1
answer
103
views
How to solve MILP problem on several linear subspaces
I have a set of close mixed-integer programming problems. More exactly, all the problems share the same set of (binary and continuous) variables, the same set of linear inequality constraints, and the ...
- 39
-1
votes
1
answer
88
views
sparse data fitting problem [closed]
I am a new learner of optimization, and I am confused by the question below, (how to change a 0-norm constrain into binary and linear constrain ?)
Given a sparse data fitting problem:
$ minimize \...
- 1
-1
votes
0
answers
41
views
Is it possible to backtrack an optimization solver? [closed]
I have an optimization problem and was using a linear programming optimizer to find solutions. However, I find that past a certain size, the problem becomes "infeasible" and has no solutions....
