**4**

votes

**2**answers

282 views

### Cubic graphs decompositions

There are many interesting computational problems related to connected cubic graph decomposition. For instance, decomposition of cubic graph into a perfect matching and a connected 2-factor ...

**3**

votes

**0**answers

151 views

### Partitioning a cubic graph into two induced cycles of equal order

I am aware that deciding the existence of a partition of the vertices of a connected graph $G(V, E)$ into two induced cycles is $NP$-complete(Theorem 2). Induced cycle is a cycle without any chord ...

**11**

votes

**1**answer

246 views

### Comparing two numbers given their factorization

I'm not an expert, but given the integer factorization of two numbers $a,b$:
$$a = p_{i_1}^{a_1}...p_{i_n}^{a_n}, \quad b = p_{j_1}^{b_1}...p_{j_m}^{b_m}$$
What is the time and space compexity of ...

**1**

vote

**0**answers

133 views

### Complexity of Nested Linear Optimization

My question is motivated by the fact, that among other ways, it is possible to restrict a variable to two discrete values, e.g. the prototypical $0$ and $1$, via an optimization constraint:
...

**1**

vote

**2**answers

120 views

### Constructing Useful SAT Instances

Given a set of binary strings, all of length $s$, is it possible to construct a SAT instance with s literals that is satisfied only by those binary strings as assignments?
For example, consider the ...

**4**

votes

**2**answers

764 views

### Efficient algorithms to determine the roots of: $p(x) = r^x $ in the finite field $GF(q)$, where $r$ a primitive root of the field

I need to make sure that no efficient (i.e., polynomial time) algorithm exists for the following problem:
Exponentiating Polynomial Root Problem (EPRP)
Let $p(x)$ be a polynomial with $\deg(p) \geq ...

**6**

votes

**1**answer

380 views

### Best ranking in tournament: polynomial time algorithm?

This question was posed by my colleague Torbjörn Lundh in his paper Which Ball is the Roundest? A Suggested Tournament Stability Index, Journal of Quantitative Analysis in Sports 2(3), 2006. We have ...

**2**

votes

**0**answers

93 views

### What are natural examples of non-relativizable proofs? [duplicate]

As I understand it, a proof that P=NP or P≠NP would need to be non-relativizable (as in recursion theory oracles).
Virtually all proofs seem to be relativizable, though.
What are good examples of ...

**12**

votes

**2**answers

528 views

### Faster multiplication with a restricted set of multiplicands?

Let $A$ be a set of $k>1$ distinct elements from a semigroup. We wish to compute the product
$$ p=b_1 b_2 \cdots b_n$$
where each $b_i\in A$.
Clearly $n-1$ multiplications suffice to compute $p$; ...

**10**

votes

**1**answer

315 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

294 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

248 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

1k 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

904 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

222 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

127 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

185 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

335 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

120 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

296 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

136 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

275 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

219 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

268 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

216 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

179 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 ...

**18**

votes

**2**answers

873 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

109 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

112 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

229 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

530 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

491 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

222 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

456 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

170 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

440 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

151 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

278 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

554 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

72 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

422 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

177 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

491 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

164 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

350 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

466 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 ...