# Tagged Questions

**10**

votes

**1**answer

304 views

### Harvey Friedman's strict reverse mathematics vs. Cook-Nguyen's V$^0$

Harvey Friedman posted several manuscripts [1] proposing a program for "strict" reverse mathematics, in the sense that the base theory should be mathematically natural and coding-free.
In them he ...

**5**

votes

**1**answer

292 views

### Subsets of all Diophantine's sets

I have asked this question on math.stackexchange already:
http://math.stackexchange.com/questions/627461/subsets-of-all-diophantines-sets
Function $\mathbb{N}^k \to \mathbb{N}^m$ is computable ...

**2**

votes

**1**answer

241 views

### Computation complexity of calculating the cdf of an n-th dimensional gaussian random vector

Suppose you have a general $n$-th dimensional random Gaussian vector with probability distribution function $\mathcal{N}\left(\mathbf{x}|\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$.
What is the ...

**4**

votes

**3**answers

870 views

### Can you efficiently solve a system of quadratic multivariate polynomials?

Given a system of 2nd-degree polynomials, $P=\{p_1,\dots,p_m\}$ where $p_i: \mathbb{R}^n \rightarrow \mathbb{R}$, can you efficiently find a common zero of all of these polynomials? In other words, ...

**2**

votes

**0**answers

114 views

### Reference Request: Properties of the Integer Factorization Polytope

The complexity of Integer Factorization is to my knowledge still an open problem, whereas deciding, whether a given integer is a prime number is known to be in $P$ and a proof is available online ...

**9**

votes

**5**answers

875 views

### Examples of ubiquitous objects that are hard to find?

I've been wrestling with a certain research problem for a few years now, and I wonder if it's an instance of a more general problem with other important instances. I'll first describe a general ...

**26**

votes

**10**answers

2k views

### Can We Decide Whether Small Computer Programs Halt?

The undecidability of the halting problem states that there is no general procedure for deciding whether an arbitrary sufficiently complex computer program will halt or not.
Are there some large $n$ ...

**2**

votes

**0**answers

220 views

### cyclotomic polynomials of given degree

Is there a fast algorithm to generate all cyclotomic polynomials $\Phi_n$ for which the degree of $\Phi_n$ is a fixed constant $d?$ This is obviously related to the "inverse totient" function: compute ...

**6**

votes

**0**answers

125 views

### An explicit IP algorithm for chess?

If I have 2 large graphs to be tested for isomorphism, and can communicate with some (powerfull but untrusted) machine, I can choose graph at random, permute vertices, ask machine to guess which one ...

**1**

vote

**1**answer

176 views

### Graph classes where Hamiltonian Cycle and Hamiltonian Path problems have different complexity

While searching The information System on Graph Classes and their Inclusions, I stumbled on several graph classes for which the Hamiltonian Cycle problem is $NP$-complete while the complexity of ...

**0**

votes

**1**answer

275 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 $m$? ...

**5**

votes

**0**answers

119 views

### Complexity of finding three perfect matchings with no edge in common in a bridgeless cubic graph

According to a conjecture:
Conjecture (Fan & Raspaud, 1994) Every bridgeless cubic graph contains three perfect matchings with no edge in common.
Equivalent statement here
Main question:
...

**4**

votes

**0**answers

280 views

### About “natural proof” of Razborov and Rudich

The famous "Natural Proof" paper ,http://www.cs.umd.edu/~gasarch/BLOGPAPERS/natural.pdf , of Razborov and Rudich gives a barrier for any proof that try to separate P and NP. It mainly shows that if ...

**2**

votes

**0**answers

135 views

### Odds of projections of a point not on the hyperplane

Let $\mathcal{L}=\{\Bbb x\in\Bbb R^n:x_1+x_2+\dots+x_n=0\}$ be a specific hyperplane.
Let a projection of $c\in\Bbb R^n$ be $p(c)=[p_1,p_2,\dots,p_n]$ where $p_i\neq c_i\implies p_i=0$.
Let ...

**3**

votes

**2**answers

269 views

### Efficient representations of natural numbers via arithmetical expressions

A given natural number $n \in \mathbb{N}$ has many representations
as expressions mixing other natural numbers and the operators and punctuation symbols
$\{+,-,\times,/,\exp,(,)\}$, where '$\exp$' ...

**3**

votes

**1**answer

199 views

### Minimum distance between Hamiltonian cycles in cubic Hamiltonian graph

It is $NP$-hard to find constant factor approximation of longest cycle in cubic Hamiltonian graphs. Therefore, finding a Hamiltonian cycle in a cubic Hamiltonian graph is NP-hard.
By Smith's theorem, ...

**1**

vote

**0**answers

227 views

### How to prove the NP-hardness of this scheduling problem? [closed]

Suppose there are a set of $m$ jobs $J= \{J_1, J_2, \ldots, J_m\}$ and $n$ machines $M=\{M_1, M_2, \ldots, M_n\}$. Each job $J_i$ consists of $k_i$ unit operations, and there are totally K operations ...

**0**

votes

**1**answer

209 views

### Does this algorithm terminate in all scenarios?

Let $x \in \mathbb{R}^p$ denote a $p$-dimensional data point (a vector). I have two sets $A = \{x_1, \dots, x_n\}$ and $B = \{x_{n+1}, \dots, x_{n+m}\}$, so $|A| = n$, and $|B| = m$. Given $k \in ...

**3**

votes

**0**answers

170 views

### Nesting big-O with big-Omega $O(g(\Omega(h(n))))$: is it $O$ for all $\Omega$ or for one $\Omega$?

I want to express the following statement about a function $f(n)$: there exists $f_\Omega\in\Omega(h(n))$ such that $f\in O(g(f_\Omega(n))$. What's the correct notation for this? Is it $f\in ...

**17**

votes

**2**answers

853 views

### Deep theorems and long proofs

I ran across this discussion by Daniel Shanks,
"Is the quadratic reciprocity law a deep theorem?."
Solved and Unsolved Problems in Number Theory. Vol. 297. AMS, 2001. p.64ff.
which made me ...

**2**

votes

**1**answer

207 views

### Are there any efficient (polynomial time) algorithms for finding if a multivariate quadratic polynomial has a root?

I know that in general, polynomial satisfiability is NP; however, I'm curious to know what work has been done on special classes of polynomials, and in particular quadratic polynomials of multiple ...

**-2**

votes

**1**answer

104 views

### “logical distance” link algorithmic complexity to statistical information [closed]

Someone mentioned what I think was referred to as 'logical distance'.
My hard drive crashed and I dont have the link anymore.
I do recall that I ran across it on this site, in response to linking ...

**2**

votes

**1**answer

107 views

### Dimension independent computational complexity of singular value decomposition

Suppose $X$ is a $m \times n$ real matrix, which has only $k$ number of nonzero elements ($k \ll mn$).
Given a vector $y$, the sparsity of $X$ allows $X y$ to be computed in $O(k)$ time
which is ...

**8**

votes

**0**answers

214 views

### Computing van Kampen diagrams

If G is a finitely presented group (with generating set X) and w is a word over X such that
w=1 in G, then the latter can be witnessed by a so called van Kampen diagram for w, which is
a planar ...

**6**

votes

**4**answers

518 views

### How long does it take to compute a class number?

I was wondering if there are any known (upper and lower) bounds for the complexity of computing the class-number of a finite extension of the rationals. (A general bound should be in function of the ...

**11**

votes

**2**answers

480 views

### How hard (P, NP, NP-hard) is it to compute Schur norms of matrices (as multipliers)?

Given a matrix $A\in M_n(\mathbb{C})$, I will denote by $||A||_\infty$ the operator norm of $A$, as seen acting on the Hilbert space $\mathbb{C}^n$. This makes $M_n(\mathbb{C})$ into a Banach space ...

**3**

votes

**2**answers

218 views

### Complexity of a problem remotely related to the discrete logarithm $A=x g^x$

Let $x,g \in \mathbb{F}_p^\ast$.
Given $g$ and either
$$ A = x g^ x$$
or
$$ A = x g^{x^2-1}$$
find $x$.
What is the complexity of solving this?
Is there a reduction to the discrete ...

**4**

votes

**2**answers

438 views

### Simplified knapsack problem

There is a problem that I can not solve.
Given a set of items (each item has some integer weight) we have to fill bag with some number of copies of these items, with the only restriction that the ...

**3**

votes

**1**answer

169 views

### Hamiltonian circuit

Let us consider a disk with one labelled point on the boundary and $n$ labelled points in the interior.
Let T be a triangulation of the whole disk with vertices on the labelled points such that T ...

**0**

votes

**1**answer

424 views

### Proof of the lower bounds of time of algorithm working [closed]

I have asked this question on math.stackexchange already: http://math.stackexchange.com/questions/515920/lower-bounds-on-the-running-time
There are some problems, when there is non-trivial lower ...

**1**

vote

**1**answer

148 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 ...

**3**

votes

**2**answers

274 views

### Is There An Algorithmic Complexity Of A Random Distribution

Has anyone studied an equivalent to algorithmic complexity for probability distributions?
This would be a measure which was similar to Kolmogorov complexity but look at the complexity of a (discreet ...

**1**

vote

**1**answer

552 views

### The smallest altitude amongst the triangles formed by points in the unit circle

Let $S$ be a finite set of points inside the unit circle. Consider all possible triangles formed by three distinct points in $S$, and among all such triangles find the smallest altitude. Denote this ...

**0**

votes

**0**answers

71 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 ...

**5**

votes

**2**answers

421 views

### A simple language and systematic computations

The following somewhat popular simple computer language was enjoyed on sci.math, sci.math.research, pl.sci.matematyka, and perhaps before and after at several places (I wish I knew it's exact ...

**2**

votes

**1**answer

172 views

### For interior point methods of linear programming, what is the “L” in the computational complexity $\mathcal{O}(n^3 L)$?

My question is about interior point methods of linear programming. Suppose the constraint matrix $A$ has $m$ rows and $n$ columns, and $m<n$. The state-of-the-art methods, like primal dual interior ...

**5**

votes

**5**answers

447 views

### Efficient Hamiltonian cycle algorithms for graph classes

Generally speaking finding a Hamiltonian cycle is NP-Hard and so tough. But if $G=L(H)$ is the line graph of $G$ then we can reduce the finding of a Hamiltonian cycle in $G$ to a Eurler your of $H$ ...

**4**

votes

**1**answer

159 views

### NP-hardness of sparsest cut

Consider bipartitioning the vertices of a graph $(V,E)$ into $V = P \cup Q$ to minimize $$\frac{|E(P,Q)|}{|P| |Q|},$$ where $E(P,Q)$ denotes the set of edges in the cut. The usual citation for ...

**1**

vote

**0**answers

91 views

### Indecomposability of image transformations (pure algebra). Open questions

W-transformations -- definitions
We will consider a class called finite window transformations $\ T:C^\mathbb Z\rightarrow C^\mathbb Z\ $ defined a paragraph below; $\ \mathbb Z\ $ is the ring of ...

**4**

votes

**1**answer

340 views

### The relationship between P vs NP problem and “Kolmogorov complexity with time”

Let $P$ - polynomial($P(x) \ge x$), $n \in \mathbb{N}$, $l < log(n)$.
Problem1: "Is there program with length $\le l$ that print $n$ by using $\le P(log(n))$ time?"
Is it Problem1 $\in ...

**13**

votes

**1**answer

458 views

### How fast can we numerically calculate Kloosterman sums?

Define the usual Kloosterman sum by $$S(m,n;c) = \sum_{\substack{x \pmod{c} \\ (x,c) = 1}} e\Big(\frac{mx + n\overline{x}}{c}\Big),$$
where $x \overline{x} \equiv 1 \pmod{c}$, and $e(x) = e^{2 \pi i ...

**2**

votes

**1**answer

128 views

### Complexity of numerically solving systems over the reals

Basically I am interested in
What is the complexity of numerically solving systems over $\mathbb{R}$?
By solving I mean finding at least one numeric solution with given
precision.
Probably the ...

**1**

vote

**0**answers

49 views

### A criterion or algorithm for polynomial which admits Markov partition on its Julia set

For a given polynomials $P(z)$, whether there exists general algorithm to check it admits a Markov partition on its Julia sets. (in finite computation time.)
May be it is more difficult for the ...

**10**

votes

**2**answers

422 views

### What Turing-Complete models of computation carry a notion of time complexity that “agrees” with that of Turing Machines?

Certain models of computation are technically Turing-Complete, but cannot feasibly simulate a Turing Machine within the usual time constraints we hope for. One example of this is Godel's recursive ...

**1**

vote

**3**answers

169 views

### unbounded complexity

If a language L is decidable, does that imply that the is a computable function f such that L is in O(f(n)) ?
For example what would be the complexity class of the language of "provably halting ...

**5**

votes

**0**answers

88 views

### Can the isomorphism relation for countable models become harder when adding finitely many constants?

I am particularly interesting in the case where $T$ is o-minimal, but I would be interested in any theory $T$ (or even an $L_{\omega_1,\omega}$-sentence) which has this property.
Context: view the ...

**2**

votes

**0**answers

132 views

### Row subset selection of matrix to optimize condition number

Given a matrix $\mathbf{A} \in \mathbb{C}^{N\times M}$ with $N \gg M$. This matrix results from a linear equation system and has a certain structure (however, I do not think that details provide any ...

**2**

votes

**1**answer

108 views

### Computation of the mean of a random variable to estimate algorithm complexity

I made an incremental algorithm which I would like to evaluate the complexity. The algorithm works with a sliding window of size n.
To study the complexity, the window is considered full and the data ...

**9**

votes

**1**answer

198 views

### How hard is Heyting satisfiability, i.e. the constructive version of SAT? In particular, is 2-HSAT NL-complete or is it harder?

First of all, is it clear what I mean by $k$-HSAT?
I'm assuming that for $k>2$, $k$-HSAT is NP-complete, but the details of the reductions between $k$-HSAT and $k$-SAT aren't obvious to me.
I'm ...

**2**

votes

**1**answer

131 views

### $\mu$-recursive definitions for the complexity classes P, NP, etc

The standard complexity classes such as P, NP are usually defined using Turing Machines. In finite model theory those classes can be defined via the classical first-order/second-order logics.
I am ...