**4**

votes

**1**answer

155 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

89 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

321 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

438 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

47 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

397 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

151 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

87 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 ...

**1**

vote

**0**answers

118 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

106 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

175 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

123 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 ...

**8**

votes

**1**answer

233 views

### Fold-and-cut problem in three dimensions

The fold-and-cut theory states that "Any shape with straight sides can be cut from a single (idealized) sheet of paper by folding it flat and making a single straight complete cut. Such shapes include ...

**26**

votes

**1**answer

948 views

### How hard is reconstructing a permutation from its differences sequence?

My interest in combinatorially motivated computational problems led me to search for simple problems that turn out to be computationally hard. In this pursuit, I came up with a problem which I hope is ...

**2**

votes

**1**answer

297 views

### Problem to a solution

Consider an NP hard problem $\frak P$ which takes an input of length n
$\frak P$ can be solved partially by a factor $ p_i = p(n,i)\in$ [0,1)... by a polynomial time algorithm $\mathcal A(i)$ ...

**5**

votes

**0**answers

164 views

### Feasible Type Theories

I am looking for references about efficient type theories,
efficiency in the sense of computational complexity,
and type theory in the sense of Martin-Lof's type theories.
Has there been any studies ...

**1**

vote

**1**answer

123 views

### Finding reducible polynomials with restricted factors

Given $f(x),g(x) \in \mathbb{Z}[x]$, two irreducible polynomials, is there a polynomial $h(x) \in \mathbb{Z}[x]$ coprime to $f(x)$ such that $f(x) + g(x)h(x)$ is reducible over $\mathbb{Z}[x]$ with ...

**17**

votes

**1**answer

534 views

### An NP-hard $n$ fold integral

We are given rational numbers $[c_1, c_2, \ldots, c_n]$ and $v$ from the interval $[0,1]$.
Consider the $n$-fold integral
$$
J = \int_{\theta_1 \in I_1, \theta_2 \in I_2 \ldots, \theta_n \in I_n} ...

**4**

votes

**3**answers

398 views

### What new primitive recursive functions are needed to reconcile Turing time complexity with Gödel time complexity?

Let me begin with an example.
Consider the computable function $f(x) = 2x$. A Turing machine can implement this function in $O(|n|)$ steps: simply walk to the end of the input string, write a $0$, ...

**11**

votes

**1**answer

395 views

### Are there very strongly pseudorandom permutations?

A pseudorandom permutation can be defined formally as a function $\phi$ from $\{0,1\}^k\times\{0,1\}^n$ to $\{0,1\}^n$ such that for every $x\in\{0,1\}^k$ the function $\phi_x:y\mapsto\phi(x,y)$ is a ...

**2**

votes

**0**answers

132 views

### Intermediate $\mathsf{NP}$-complete problems?

Partition problem is weakly NP-complete since it has polynomial (pseudo-polynomial) time algorithm if input integers are bounded by some polynomial. However, 3-Partition problem is strongly ...

**4**

votes

**1**answer

108 views

### About infinite subset of halting probability and 1-random set

Let $\Omega$ be the halting probability (see (http://en.wikipedia.org/wiki/Chaitin's_constant) and R. Downey, and D. Hirschfeldt (2010), Algorithmic Randomness and Complexity for reference). If A is ...

**2**

votes

**0**answers

55 views

### Is the $d$-dimensional Arrangement of Trees still $NP$-hard?

The $d$-dimensional Arrangement Problem for general graphs is known to be $NP$-hard since the special case $d=1$ (OLA) already is (Garey et al, [1976]). For Trees however, the one dimensional case can ...

**23**

votes

**1**answer

710 views

### Are sums of sequences decidable?

Suppose that $f,g$ are rational functions with integer coefficients such that $\sum_{n=0}^{\infty}f(n)$ and $\sum_{n=0}^{\infty}g(n)$ both converge. Is it decidable whether
...

**7**

votes

**1**answer

410 views

### Can we invert barycentric subdivision?

With apologies to fellow algebraic topologists, I confess that I have no idea how to answer this innocent-looking question:
(1) Let's say we know that a finite simplicial complex $S$ is the ...

**7**

votes

**1**answer

438 views

### Normality of Chaitin's constant

Can anyone provide an overview of the proof that Chaitin's constant is normal, or better yet, the guiding intuition?
Even if we replace the existential quantifiers in the assertion of non-normality ...

**12**

votes

**0**answers

316 views

### Splay trees and Thompson's group $F$

( I apologize for only indicating some easy to find references, but new users are not allowed to link more than five). This is very speculative, but:
Question: Is there a reformulation of the Dynamic ...

**1**

vote

**0**answers

93 views

### Schönhage's SMM with only one instruction

It is possible to implement $\lambda$-calculus in Schönhage's storage modification machine using an infinite set of nodes and one single program consisted exclusively of (about hundred) instructions ...

**4**

votes

**2**answers

238 views

### Hardness of approximation of Dominating Set

It is stated throughout the computational complexity literature that the Dominating Set problem is NP-hard to approximate within a factor of $\Omega(\log n)$. To my knowledge, the first and only proof ...

**10**

votes

**4**answers

858 views

### Quantum algorithms for dummies

I want to try my hand at designing quantum algorithms to solve certain problems. I feel like I understand (for example) how Grover's algorithm and Shor's algorithm work, and I'm excited to apply the ...

**1**

vote

**1**answer

188 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 $$E[R_n]=\Theta (\log n)$$
...

**1**

vote

**0**answers

77 views

### Are set covering problems with nonlinear cost functions NP-Hard?

Are set covering problems (set cover problem wikipedia) with a nonlinear cost function also NP-hard? Is there a general result about this?
To be more specific the cost function I am interested in ...

**1**

vote

**1**answer

161 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 http://math.stackexchange.com/questions/364061/can-all-programs-reducible-to-ones-with-only-arithmetic-operations-on-inputs-be, so I am asking here.
In ...

**12**

votes

**3**answers

1k views

### Will quantum computing kill cryptography ? [closed]

I apologize as this question is not really mathematical, and therefore perhaps not
well-suited for this site. Please feel free to close it if you think it is not. My reason
for asking it here is that ...

**1**

vote

**1**answer

108 views

### Separation of Anti-Hole Inequality

Given an undirected graph $G=(V,E)$ with no loops or multiple edges, a stable set is a set of vertices for which no two vertices are adjacent.
An induced subgraph $H$ of $G$ is called an odd-antihole ...

**5**

votes

**0**answers

127 views

### Computational complexity of resolution of singularities of varieties over fields with characteristic 0 [closed]

What is the computational complexity of resolution of singularities of varieties over fields with characteristics 0?

**6**

votes

**2**answers

748 views

### Approximate number of primes below a given integer?

The problem of the complexity of the exact counting problem for primes is interesting. The best result we have about primes is that it is hard for TC0. But counting the number of witnesses to a TC0 ...

**6**

votes

**2**answers

572 views

### Strassen's algorithm

I am reading Landsberg's "Tensors: Geometry and Applications". Here he mentions tensor formulation of Strassen's algorithm and shows that the rank of Strassen's matrix multiplication tensor is $7$ and ...

**0**

votes

**0**answers

58 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}
...

**5**

votes

**1**answer

265 views

### Maximal chain of 1s in binary strings

Let $S$ be the set of $2^n$ binary $n$-bit strings. For every $x\in S$, let $f(x)$ is the maximal chain of bits 1 in $x$. So Can we find a good upper bound of $$F(n)=\frac{\sum_{x\in S}f(x)}{2^n}$$
Of ...

**17**

votes

**0**answers

478 views

### Computational complexity of topological K-theory

I am a novice with K-theory trying to understand what is and what is not possible.
Given a finite simplicial complex $X$, there of course elementary ways to quickly compute the cohomology of $X$ with ...

**1**

vote

**1**answer

221 views

### definition of NP [closed]

"Popular" discussions of PvsNP often begin by characterizing NP problems as those for which a purported solution can be verified in polynomial "time". Later the definition changes to; the problem is ...

**3**

votes

**1**answer

207 views

### The existential theory of the reals

Some definitions of the existential theory of the reals (ETR) allow a real closed field and some definitions allow only rational numbers as coefficients of polynomials. Which one is correct? Will the ...

**2**

votes

**2**answers

131 views

### Size-limited oracles

I am interested in complexity of algorithms which have access to the following peculiar sort of oracle:
Suppose that an invocation of an algorithm f with an input of size n has access to an oracle ...

**6**

votes

**0**answers

246 views

### Is integer GCD in NC?

Wikipedia, in the page http://en.wikipedia.org/wiki/Greatest_common_divisor#Complexity mentions integer GCD is in NC by citing http://www.cs.cornell.edu/courses/CS6820/2012sp/Handouts/Sedjelmaci07.pdf
...

**1**

vote

**2**answers

113 views

### complexity of finding optimal matchings of given fixed size

It is known, that maximal matchings (i.e. matchings with the maximal number of edges) and optimal matchings (i.e. matchings for which the sum of edge weights is optimal) can be calculated in ...

**6**

votes

**1**answer

244 views

### Do sparse DAGs can have large min-cuts?

For a graph $G$, let $e(G)$ denote the number of its edges, and $c_k(G)$ the smallest number
of edges that must be removed in order to destroy all paths of length $\geq k+1$.
Note that $c_1(G)\geq ...

**16**

votes

**1**answer

587 views

### Polynomial-time algorithm to compare numbers in Conway chained arrow notation

I am looking for a polynomial-time algorithm which, given a character string containing two numbers in Conway's chained arrow notation for large numbers, indicates whether the first number is less ...

**1**

vote

**0**answers

129 views

### NP problem implications [closed]

Hi, i would like to have some clarification on NP-completeness.
In particular I'm reading an article where they show:
1) Partitioning the edges of a graph into connected component of 3 edges (3-path ...