**0**

votes

**1**answer

138 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 ...

**2**

votes

**0**answers

408 views

### NP-hard proof of optimization version of exact cover [closed]

Exact cover is NPC.
http://en.wikipedia.org/wiki/Exact_cover#Equivalent_problems
Given a collection $\mathcal{S}$ of subsets of a set $X$, an exact cover is a >>subcollection $\mathcal{S}^*$ ...

**8**

votes

**1**answer

242 views

### Compute an arbitrary decimal place of $\pi$

Is there a method to find the value of the $n$-th decimal place of $\pi$ which is more efficient than having to compute all decimal places before as well?

**1**

vote

**0**answers

167 views

### Both NP-hard but different [closed]

What's the fundamental difference between the Knapsack problem and the travelling salesman (TSP) problem both of which are NP-hard, while the reality is that TSP could be solved much much faster?

**2**

votes

**0**answers

257 views

### Computing the chromatic polynomial of graph modulo $x-3$

The chromatic polynomial of graph $P(G,x)$ is univariate
polynomial which counts the number of colorings of $G$
with $x$ colors for natural $x$.
Graph is not $k$ colorable iff $P(G,k)=0$.
The ...

**3**

votes

**1**answer

663 views

### How hard is a variant of graph automorphism problem?

I'm interested in a variant of graph automorphism problem (which is prime candidate for $NP$-Intermediate problem).
Restricted GA
Input: Given an undirected graph $G(E, V)$, and $\epsilon |V|/2$ ...

**1**

vote

**1**answer

106 views

### A certain instance of the Set Covering problem

Is there any useful structure associated with the following instance of the Set Covering problem?
Let $G$ be a weighted graph and let $\mathcal{P}$ denote the set of all shortest paths between all ...

**1**

vote

**1**answer

87 views

### Implications of the impossibility of efficient sampling from random non-Hamiltonian graphs

Nisan's answer to this question shows the Impossibility of efficient sampling from random non-Hamiltonian graphs (unless $NP=coNP$). I am interested in the implications of this conjecture.
Does ...

**5**

votes

**2**answers

233 views

### Generating Hard Instances

Assume NP$\neq$P and let $L$ be an NP-complete language. Is there a polynomial time computable function $f:\{0\}^*\longrightarrow\{0,1\}^*$ with $|f(0^n)|=n$ for every $n$; such that L $=\{0^n: f(0^...

**-2**

votes

**1**answer

196 views

### are all NP problems made up of P problems? [closed]

are all NP problems made up of P problems? that is, can NP problems be thought of as an accumulation of P problems? or can NP problems be divided up into a series of P problems?

**1**

vote

**0**answers

57 views

### Polynomial problems in graph classes defined by forbidden induced cyclic subgraphs

Let $C$ be a graph class defined by a finite
number of forbidden induced subgraphs, all
of which are cyclic (contain at least one cycle).
Are there graph problems that can be solved in
...

**0**

votes

**0**answers

141 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 ...

**1**

vote

**0**answers

222 views

### Is finding a single vector in the null space as difficult as discovering the whole null space?

Let $A \in \mathbb R^{k\times n}$ be a matrix of rank $k$, where $k \ll n$. One can use Gaussian eliminations to discover $\operatorname{null}(A)$ at the cost of $O(nk^2)$. My question is:
Is the ...

**37**

votes

**6**answers

3k views

### Is it easy to produce hard-to-color graphs?

This question arises from my recent visit to my daughter's second-grade class, where I led some discussion and activities on graph coloring (see Math for seven-year-olds). In one such activity, each ...

**14**

votes

**1**answer

790 views

### Who first dubbed them “expander graphs”?

Expander graphs
("sparse graphs that have strong connectivity properties")
burst onto the mathematical scene around the millennium, but I have not
been successful in tracing the origin of
(a) the ...

**10**

votes

**2**answers

1k views

### How to determine if there exists a non-zero vector in the kernel

If you are given a $0$-$1$ circulant matrix with $n$ rows and $n$ columns, is there an efficient way of determining if there exists a non-zero $\{-1,0,1\}$-vector in its kernel?
Could this problem ...

**2**

votes

**3**answers

812 views

### An established proof in Wang Tile which I doubt

When I was reading the paper:
Wang, Hao. "Notes on a class of tiling problems." Fundamenta Mathematicae 82.4 (1975): 295-305.
from http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82119.pdf
I could not ...

**1**

vote

**0**answers

74 views

### What is the generic complexity of First Order Predicate Calculus?

I suspect that it should be the same as that of the Turing machine halting problem, which wikipedia gives as GenP and attributes this result to Hamkins and Miasnikov, but I am not sure. Is the generic ...

**7**

votes

**1**answer

1k views

### Can the Legendre symbol be calculated in polynomial time?

Is there an algorithm which on input "$(a,p)$" (where $0\leq a<p$ are integers) takes time polynomial in $\log p$ and outputs "NOT PRIME" if $p$ is not prime and otherwise outputs the Legendre ...

**2**

votes

**1**answer

163 views

### QBF of exponential length?

We consider a slightly extended version of a nondeterministic finite automaton, call it a "propositional nondeterministic finite automaton". It is defined as follows. Consider a fixed propositional ...

**1**

vote

**1**answer

61 views

### How to select a subset of points from a universal to minimize the distance from outside to inside?

Here is the detailed problem.
I have a set of N points in K-dimension space, called U, and I want select M points of them, called S. For each point p in U, we define the distance from p to S as
$$ d(...

**25**

votes

**1**answer

942 views

### How strong is this conjecture? $(Z/nZ)^*$ is generated by “small” elements

Conjecture: There are constants $c,k$ such that every $(Z/nZ)^*$ is generated by its elements smaller than $k (\log n)^c$.
Where $(Z/nZ)^*$ is the multiplicative group of integers mod $n$. My main ...

**2**

votes

**0**answers

123 views

### characterization of all periodic tiling of a simple set of Wang Tile

Consider a set of Wang Tile such that all the edges are either 1 or 0.... there are 16 elements in such a set.
Now, I wish to characterize all the periodic tilings of this set (better if they are ...

**1**

vote

**1**answer

101 views

### simple cycle analog in 2D (with application in tiling)

We know that any closed cycle of a graph could be decomposed into sum of simple cycles. To translate this theorem into tiling of 1D (Wang tile). We know that any 1D periodic tiling could be ...

**1**

vote

**2**answers

340 views

### Computational complexity of solution of Pell equation and more

What is computational complexity for computing integral solution of Pell equation .It seems to be in P ,and could any one give an algorithm and reference for proof of it's complexity?
And more,could ...

**1**

vote

**1**answer

276 views

### relationship between corner tile and edge tile of wang tile

It is clear that any corner type of Wang Tile could be converted to edge type of Wang Tile by defining the edge color according to the corner color.
However, could we convert edge type of Wang Tile ...

**1**

vote

**0**answers

127 views

### Graph theoretical representation of Wang Tile

We note that for one dimensional tiling problem of Wang Tile could be represented by a graph. Each cycle on the graph represents a periodic solution.
However, is there a well established counter-part ...

**1**

vote

**1**answer

131 views

### Why can't there be a problem both in P and NPC [closed]

In this illustration, P and NPC are two disjoint set.
We know that NPC is non-empty. If P $\cap$ NPC $=\varnothing$, then there are elements in NP which are not in P. Doesn't this imply that P $\neq$...

**1**

vote

**0**answers

61 views

### TSP: Approximation Ratio of the Double Tree Heuristic after Diagonals have been Removed

In their article "Double-Tree Approximations for the Metric TSP: Is the Best One Good Enough?", Vladimir Deineko and Alexander Tiskin derive a lower bound for the approximation ratio of the double-...

**7**

votes

**1**answer

188 views

### What is the Essential Reason that allows a PTAS for the EUCLIDEAN TSP?

Questions:
Is there some understanding of the reason, why the euclidean TSP allows a PTAS, whereas the metric TSP in general does not and, is the PTAS stable under sufficiently small perturbation ...

**1**

vote

**1**answer

128 views

### infinitary logic and partial fixed point logic

Is there a property definable in finite-variable infinitary logic $L^{\omega}_{\omega\infty}$ but not definable in partial fixed point logic FO(PFP) ?

**9**

votes

**1**answer

419 views

### Fast checking that overdetermined polynomial system does not have a solution

As a result of some inductive procedure for each $n$ I have a system of about $n^2$ polynomial equations with $n$ variables with integer coefficients, which can be precisely computed. As the system is ...

**3**

votes

**1**answer

158 views

### Approximate the square root of (1-X) efficiently through (nested) products

Currently, I encountered a problem of approximating the following
series:
$$
(I-X)^{-\frac{1}{2}}=I+\frac{1}{2}X+\frac{1\cdot3}{2\cdot4}X^{2}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6}X^{3}+\ldots
$$
where ...

**3**

votes

**1**answer

429 views

### Application of Combinatorics, Logic and computability theory in physical science: Tiling of Wang Tile with proportionality

The original problem of Domino Tiling and Wang Tile has great theoretical interest on computability theory... However, the great emerging problem on application of Wang Tile in material science and ...

**3**

votes

**2**answers

208 views

### Conjecture of a subset of Wang tile which might be decidable

From the two papers proving the undecidability of Wang tile in 1966 by Berger and in 1971 by RM Robinson, the tiles used in proving undecidability has a general common feature:
The left color and ...

**3**

votes

**0**answers

202 views

### Complexity of counting words of given length in regular or context-free language

Let $L$ be a regular or context-free language over
alphabet $\{0,1\}$.
What is the complexity of counting words of length $n$ in $L$?
Is it possible to efficiently find if for given $n$
all words ...

**7**

votes

**4**answers

893 views

### NP-hard problems in linear algebra and real analysis [closed]

I am curious about NP-hard problems in linear algebra and real analysis. An example in linear algebra would be the calculation of the permanent.
I would thus like to collect in this thread a list of ...

**2**

votes

**1**answer

184 views

### Is undirected short-simple-path-through-3-vertices decidable in polynomial time?

Consider the following language:
$L=\{\langle G=(V,E),s,v,t,l\rangle\;|\;s,v,t\in V, l\in \mathbb{N} \wedge $ There exists a simple path from $s$ to $t$, going through $v$ of length $\leq l\}$.
($G$ ...

**5**

votes

**6**answers

906 views

### practical algorithms for np complete problems

Inspired by:
Conjecture on NP-completeness of tesselation of Wang Tile up to finite size
And the practicality of this topic (solving tessellation on a lattice):
coloring in lattice
Computational ...

**5**

votes

**2**answers

2k views

### What is the time complexity of truncated SVD

Full SVD, on an m*n matrix $A$, $[U,S,V] = svd(A)$, would cost $O(m^2n + mn^2 + n^3)$ time.
But what is the time complexity if we only need the $k$ largest singular values, say, $[U_k,S_k,V_k] = svds(...

**79**

votes

**9**answers

9k views

### Analogues of P vs. NP in the history of mathematics

Recently I wrote a blog post entitled "The Scientific Case for P≠NP". The argument I tried to articulate there is that there seems to be an "invisible electric fence" separating the problems in P ...

**1**

vote

**0**answers

133 views

### Examples of languages that are in P and are not in CFL [closed]

Any examples of languages that are in P(polynomial time to recognize it) and are not in CFL(context-free language)?The more the better.

**4**

votes

**3**answers

339 views

### powers in strings

I have a feeling that the following question might have been studied: Suppose I have a finite alphabet $A,$ with $|A| = n,$ and a string $S$ of length $N.$ A string can be said to contain a $k$-th ...

**17**

votes

**2**answers

618 views

### Minimum number of variables on which a multivariate polynomial depends?

Let $p:F_2^n\rightarrow F_2$ be a multivariate polynomial, let's say of degree 3. (Both the degree and the order of the field could probably be replaced by other constants without affecting this ...

**4**

votes

**1**answer

301 views

### Is the Kolmogorov complexity of at least one string of a given length equal to its length? [closed]

Is it true that for all strings of a given length (for any alphabet with more than one symbol), at least one has a Kolmogorov complexity equal to its length?
If the answer is Yes, is there a proof of ...

**-5**

votes

**1**answer

455 views

### Given an arbitrary composite odd integer $N$, find two integers $P$ and $Q$ such that $P-Q \neq 1$ and $N=P^2-Q^2$ [closed]

Given an arbitrary composite odd integer $N$, find two integers $P$ and $Q$ such that:
$P-Q \neq 1$ and $N=P^2-Q^2$
I am assuming that the best known solution to this problem runs at $O(2^{|N|})$.
...

**8**

votes

**2**answers

1k views

### How many Complexity Classes do you know?

We can read about the main complexity classes in textbooks and online in Wikipedia:
http://en.wikipedia.org/wiki/Computational_complexity_theory
However, in papers, there are a lot of important new ...

**3**

votes

**1**answer

154 views

### Is the domination number NP for non-bipartite graphs?

Calculating the domination number is an NP-Hard problem. Does it remain NP-Hard if we restrict it to non-bipartite graphs?

**3**

votes

**1**answer

50 views

### Complete classification of complexity classes / infinite approaching sequences

http://en.wikipedia.org/wiki/Time_complexity#Table_of_common_time_complexities
For complexity as seen in the above link, complexity classes can be log, polynomial, exp, or composition of any of these ...

**7**

votes

**1**answer

427 views

### Seeming contradiction about P vs NP between graphclasses.org and at least two papers about clique in even-hole-free graphs

I believe correctness about clique in even-hole-free graphs
of graphclasses.org
and the paper Vertex elimination orderings for hereditary graph classes, Pierre Aboulker, Pierre Charbit, Nicolas ...