All Questions
154 questions
46
votes
8
answers
5k
views
Can a problem be simultaneously polynomial time and undecidable?
The Robertson-Seymour theorem on graph minors leads to some interesting conundrums.
The theorem states that any minor-closed class of graphs can be described by a finite number of excluded minors. As ...
- 12.7k
21
votes
3
answers
7k
views
What are the current breakthroughs of Geometric Complexity Theory?
I've read from Wikipedia about Geometric Complexity Theory (GCT) which (if I understood correctly) is a program for coping with the $ P=NP $ problem using algebraic methods.
That program seems ...
19
votes
2
answers
1k
views
Explicit invariant of tensors nonvanishing on the diagonal
The group $SL_n \times SL_n \times SL_n$ acts naturally on the vector space $\mathbb C^n \otimes \mathbb C^n \otimes \mathbb C^n$ and has a rather large ring of polynomial invariants. The element $$\...
- 148k
110
votes
10
answers
15k
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 ...
54
votes
2
answers
8k
views
Walsh Fourier transform of the Möbius function
This question is related to this previous question where I asked about ordinary Fourier coefficients.
Special case: is Möbius nearly orthogonal to Morse
August Ferdinand Möbius (November 17, 1790 – ...
30
votes
5
answers
14k
views
Can all convex optimization problems be solved in polynomial time using interior-point algorithms?
Just a new guy in optimization. Is it true that all convex optimization problems can be solved in polynomial time using interior-point algorithms?
- 303
17
votes
2
answers
4k
views
Optimizing the condition number
Suppose I have a set $S$ of $N$ vectors in $W=\mathbb{R}^m,$ with $N \gg m.$ I want to choose a subset $\{v_1, \dots, v_m\}$ of $S$ in such a way that the condition number of the matrix with columns $...
- 96.4k
113
votes
11
answers
18k
views
On mathematical arguments against Quantum computing
Quantum computing is a very active and rapidly expanding field of research. Many companies and research institutes are spending a lot on this futuristic and potentially game-changing technology. Some ...
15
votes
3
answers
4k
views
Complexity of computing matrix rank over integers
Does computing the rank of an integer matrix have complexity polynomial in the size of the input?
The Gaussian elimination algorithm is polynomial in the number of elementary operations (addition and ...
- 481
12
votes
1
answer
5k
views
Closest 3D rotation matrix in the Frobenius norm sense
Given a 3 by 3 matrix $M$ I would like to find the rotation matrix $R$ minimizing the Frobenius norm:
\begin{equation}
\|R-M\|_F
\end{equation}
Is there a closed form solution for $R$, or is it ...
- 561
11
votes
3
answers
6k
views
Random Sampling a linearly constrained region in n-dimensions...
Hi,
So here is my problem:
Given a nonlinear, discontinous, cost function $f(x_1,x_2,..,x_N)$ along with linear constraints $x_n \ge 0, \forall n$
$x_n \le c_n$
and $\sum_{n=1}^N x_n = 1$ find an ...
- 113
6
votes
2
answers
908
views
A Query regarding the Halting Problem (Omega): Halting Probability for Given Input Size
I was studying the Halting Problem in context of the Probability and had a few doubts regarding it. Hope someone could help me out.
I am aware of the probability of a Random program halting on a ...
- 81
6
votes
6
answers
3k
views
Circumference of Convex Shapes
Here is a puzzle I found in Mitteilungen der DMV (roughly, "Letters of the German Society of Mathematicians"), issue 19/2011. It was posed by Alfred Schreiber in "Wie man Hasen fangt" (How to catch ...
3
votes
1
answer
328
views
LP Constraints for Connected Subgraphs of Fixed Size
Question:
how can the connectedness-constraint for a subgraph, that is induced by a proper subset $W\subset V$ of the vertices of $G(V,E),\ |V|=n,\ |W|=m$, be formulated in a $LP$ or $ILP$?
...
- 13.2k
1
vote
0
answers
185
views
Maximum independent set in dense graphs
Let $0 < A < 1$ and $G$ be connected d-regular graph
with degree $d=[A n]$. The density of $G$ is about $A$.
Q1 Are there constraints on $A$ such that finding maximum
independent set of $G$ is ...
- 25.4k
83
votes
7
answers
7k
views
Computational complexity of computing homotopy groups of spheres
At various times I've heard the statement that computing the group structure of $\pi_k S^n$ is algorithmic. But I've never come across a reference claiming this.
Is there a precise algorithm ...
- 44.3k
58
votes
2
answers
18k
views
How fast can we *really* multiply matrices?
Background: The Strassen Algorithm, described here, has a computational complexity of $\text{O}(n^{2.807})$ for the multiplication of two $n \times n$ matrices (the exponent is $\frac{\log7}{\log2}$). ...
- 15.5k
47
votes
7
answers
5k
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 ...
- 236k
46
votes
7
answers
13k
views
What is the time complexity of computing sin(x) to t bits of precision?
Short version of the question: Presumably, it's poly$(t)$. But what polynomial, and could you provide a reference?
Long version of the question:
I'm sort of surprised to be asking this, because ...
- 6,694
35
votes
1
answer
2k
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 ...
- 2,993
30
votes
1
answer
2k
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 ...
- 4,529
29
votes
2
answers
1k
views
Determining if a rational function has a subtraction-free expression
This question was first asked by Mehtaab Sawhney in Alex Postnikov's combinatorics class.
Given a rational function $F=P(x_1,...,x_n)/Q(x_1,...,x_n)$ with (say) integer coefficients, it is often of ...
- 2,743
29
votes
7
answers
8k
views
Solving NP problems in (usually) Polynomial time?
Just because a problem is NP-complete doesn't mean it can't be usually solved quickly.
The best example of this is probably the traveling salesman problem, for which extraordinarily large instances ...
- 2,383
27
votes
5
answers
2k
views
Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?
Suppose we have a $(2m-1) \times (2m-1)$ matrix defined as follows:
$$\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}.$$
For example, if $m=3$, the matrix is
$$\begin{pmatrix}6 & 20 & 6& 0 ...
- 1,121
23
votes
2
answers
2k
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 ...
- 151k
19
votes
2
answers
2k
views
Is the tensor product of polyhedra a polyhedron?
Conventions: A polytope in a finite-dimensional $\mathbb R$-vector space $V$ is defined to be a convex hull of finitely many points in $V$. A polyhedron in a finite-dimensional $\mathbb R$-vector ...
- 33.8k
18
votes
7
answers
3k
views
SAT and Arithmetic Geometry
This is an agglomeration of several questions, linked by a single observation: SAT is equivalent to determining the existence of roots for a system of polynomial equations over $\mathbb{F}_2$ (note ...
- 1,368
17
votes
3
answers
2k
views
The minimum of a sum of absolute values of inner products in $\mathbb{R}^d$
Consider a collection of unit vectors $v_1, \ldots, v_n$ in $\mathbb{R}^d$ (we think of $n$ being much larger than $d$). I would like to minimize the sum:
$$\sum_{i\neq j}|\langle v_i,v_j\rangle|.$$
...
- 2,288
17
votes
1
answer
2k
views
Forcing over set theory versus forcing over arithmetic
I've been trying to understand better some of the research on forcing over bounded arithmetic and its connections with lower bounds in complexity theory. For example, Takeuti and Yasumoto have some ...
- 82.6k
12
votes
1
answer
547
views
Seeking references for finding primes infinitely often
I've been pondering this weakened version of the finding primes problem for a while:
Is there an algorithm which given $k$ outputs a prime $p > 2^k$ in time $F(\log_2(p))$?
This differs from ...
- 2,302
11
votes
3
answers
496
views
Finite objects for which isomorphism is NP-hard or harder?
Are there finite objects for which deciding isomorphism
is NP-hard or harder?
Graphs and groups are not solutions.
Searching the web didn't return answer for me.
Partial result based on Chow's ...
- 25.4k
7
votes
2
answers
814
views
Building a Physical Model to Solve Sudoku
Before asking my questions, allow me to begin with a separate example to help clarify what I'm driving at. For terms that are not defined formally, please interpret them as you feel would be most ...
- 7,831
7
votes
2
answers
3k
views
Conic hulls and cones
Suppose I have a number of vectors in $\mathbb{R}^n.$ The first question is: what is the most efficient algorithm to compute their "conic hull" (the minimal convex cone which contains them)? The next ...
- 96.4k
7
votes
2
answers
7k
views
Computational complexity of calculating the nth root of a real number
Several sources state that the computational or time complexity of square rooting is the same as that of multiplication (or division). See for example:
Jean-Michel Muller, "Elementary Functions: ...
- 385
6
votes
1
answer
2k
views
Given a set of 2D vertices, how to create a minimum-area polygon which contains all the given vertices?
Not sure whether this question belongs here or math.stackexchange.
You can assume that all the vertices are unique. The given vertices can be the vertices of the polygon, thus they do NOT have to be ...
- 163
6
votes
1
answer
368
views
Time functions of non-deterministic Turing machines
Let $M$ be a non-deterministic Turing machine which recognizes a language $L$, that is, for every input word $u$ there is an accepting computation with input $u$ if and only if $u\in L$. The smallest ...
user6976
6
votes
1
answer
4k
views
Complexity for solving linear equations?
What is the best known complexity for finding a vector $x \in \mathbb{R}^n$ to minimize $||Ax - b||^2$ and/or to solve (when possible) the system of linear equations $Ax=b$?
I am interested in ...
- 544
6
votes
1
answer
1k
views
Exact arithmetic for real algebraic numbers
There was a reply to a question (that I can't find) which mentioned SARAG (Some Algorithms
in Real Algebraic Geometry) see http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted2.html. This ...
- 9,132
5
votes
1
answer
453
views
an algebraic variety for a boolean circuit
There is a polynomial reduction from a $3-CNF$ $SAT$ problem to some system of polynomial equations over $\mathbb{F}_2$.
I mean there is polynomial reduction $F$ such that for every boolean ...
- 2,576
5
votes
3
answers
1k
views
Algorithm for the intersection of a vector subspace with a cone of non-negative vectors
Hi,
I would like to know whether there is some more effective way of how to compute an intersection of a vector subspace of $\mathbb{R}^{n}$ with a cone of vectors with non-negative entries than the ...
4
votes
1
answer
8k
views
Detection of Redundant Constraints
Suppose I pose the following query to a constraint logic programming
system:
?- Y <= 6 - X, Y <= (- 4) + 4 * X, Y <= 4 + X / 3.
Are there systems that would recognize the last inequality as
...
Countably Infinite
4
votes
1
answer
3k
views
optimization of inverse matrix with constraint on matrix elements
everyone! I have this optimization problem with constraint.
$D$ and $T$ are symmetric matrices, where T is known and D is the unknown parameter.
$x$ and $v$ are two known p-dimensional vectors.
The ...
- 49
3
votes
1
answer
293
views
Relationship between P-noncomputable and P-random sets
$P$ means polynomial complexity.
$S_p$ is the class of all $P$_random set, and $S_{pc}$ is the class of all $P$ noncomputable sets, is $S_p \bigcap S_{pc}$ empty? If not empty, any example?
what is ...
- 3,094
2
votes
0
answers
105
views
Optimization over a convex cone generated by a set is equal to optimization over the set
Within my research I found an important doubt and that prevents me from advancing, the context of my doubt is as follows:
We considerer the following optimization problem
$$
\left\{\begin{array}{cl} \...
- 159
1
vote
1
answer
631
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 ...
- 867
1
vote
1
answer
7k
views
Finding the square root modulo n, when the factors of n are known
Last month, I asked whether there is an efficient algorithm for finding the square root modulo a prime power here: Is there an efficient algorithm for finding a square root modulo a prime power?
Now, ...
- 2,649
89
votes
28
answers
20k
views
Which popular games have been studied mathematically?
I'm planning out some research projects I could do with undergraduates, and it struck me that problems analyzing games might be appropriate. As an abstract homotopy theorist, I have no experience with ...
59
votes
10
answers
26k
views
Problems known to be in both NP and coNP, but not known to be in P
One such problem I know is integer factorization.
What are other interesting cases?
43
votes
10
answers
7k
views
What is the shortest program for which halting is unknown?
In short, my question is:
What is the shortest computer program for which it is not known whether or not the program halts?
Of course, this depends on the description language; I also have the ...
- 23k
33
votes
1
answer
1k
views
Is this conjecture strictly weaker than P=NP?
My three computability questions are related to the following group theory question (first asked by Bridson in 1996):
For which real $\alpha\ge 2$ the function $n^\alpha$ is equivalent to the Dehn ...
user6976