**1**

vote

**1**answer

189 views

### Redundancy and Structure of computational problems

It is widely believed that some computational problems such as graph isomorphism can not be NP-complete because it does not possess enough structure or redundancy to be computationally hard (NP-hard). ...

**1**

vote

**0**answers

89 views

### Did anybody come across a computational problem which is related to the notion of cartesian product and is at least NP-hard?

Did anybody come across a computational problem which is related to the notion of cartesian product and is at least NP-hard?
Equally interesting would be to learn about such problems with a ...

**2**

votes

**1**answer

175 views

### Does Quadratic Programming get easier when it's described by a diagonal matrix?

Generally, Quadratic Programming solves the problem
$$\text{Given }Q, c, A, b,\text{ choose }x \text{ to maximize } x^TQx + c^Tx \text{ subject to } Ax \le b$$
In this form, Quadratic Programming is ...

**2**

votes

**0**answers

162 views

### A natural problem on “cartesian union” of set families (hypergraphs). Does anybody know NP-complete problems related to the notion of cartesian product?

I'm curious whether the problem below is NP-complete.
I provide two simple definitions and one example at first.
Definition 1.
Let $\langle {\cal{S}}_i\rangle\substack{i\in I}$ and $\langle ...

**2**

votes

**1**answer

109 views

### Any result or conjecture of computaional complexity of formal languange with rational generating function?

As we know that context-free language is in P,any result or conjecture of computaional complexity of formal languange with rational generating function?And more,any result or conjecture of ...

**4**

votes

**0**answers

345 views

### Computational complexity of multiplication in a nilpotent group?

What is the computational complexity of multiplication in a Carnot group ?
Background: A Carnot group is a real nilpotent Lie group $N$ whose Lie algebra $Lie(N)$ admits a direct sum decomposition
...

**20**

votes

**0**answers

510 views

### A combination of two well-known complexity problems

Suppose you are given two graphs $G$ and $H$ and are told that one of the following two situations occurs. Either they are isomorphic, or one of the graphs contains a Hamilton cycle and the other ...

**1**

vote

**0**answers

135 views

### Randomized alternative to Buchberger's algorithm

Richard Lipton's blog describes a A New Way To Solve Linear Equations by Prasad Raghavendra.
Can the ideas in this algorithm be generalized to systems of polynomial equations to provide a randomized ...

**0**

votes

**1**answer

234 views

### Algorithm for vector space

I have $n$ vectors $e_1 \in (\mathbb Z/2 \mathbb Z)^m,\dots,e_n \in (\mathbb Z/2 \mathbb Z)^m $
and a vector $ v \in (\mathbb Z/2 \mathbb Z)^m $
I need to find the better algorithm which answers ...

**3**

votes

**0**answers

111 views

### Gaussian Elimination in terms of Group Action

Gaussian elimination makes determinant of a matrix polynomial-time computable. The reduction of complexity in computing the determinant, which is otherwise sum of exponential terms, is due to presence ...

**2**

votes

**2**answers

997 views

### Generation of All Path in a Directed Acyclic Graph

I am working on a very large dataset of a single DAG whose vertices have a low branching factor. I need to generate all possible (simple) paths starting from the source and write them to a file.
My ...

**12**

votes

**2**answers

375 views

### Optimizing the condition number

Suppose I have a set $S$ of $N$ vectors in $W=\mathbb{R}^m,$ with $N \gg n.$ 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 ...

**7**

votes

**2**answers

631 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 ...

**10**

votes

**1**answer

532 views

### A generalization of the triangle counting problem for simple weighted graphs

One nice identity is $$tr(A^3)/6$$ which counts the number of triangles of a graph represented with adjacency matrix $A.$ It also implies that triangle counting can be performed in subcubic time.
...

**10**

votes

**1**answer

241 views

### What is the largest tensor rank on matrix.

A tensor rank of tree dimentional matrix $M[i,j,k], i,j,k\in [1,\ldots,n]$ is a minimal number of vectors $x_i,y_i,z_i$, such that $M=\sum_{i=1}^d x_i\otimes y_i\otimes z_i$.
From dimension argument ...

**3**

votes

**2**answers

163 views

### Determination of rationality and computing a rational parametrization

Suppose I have a hypersurface in $\mathbb{C}P^n$ given by some $f(z_1, \dots, z_{n+1}) = 0.$ Is there an algorithm which returns a rational parametrization if there is one, and "not rational" ...

**4**

votes

**2**answers

294 views

### Sets of vectors related by a rotation

We have a two sets of vectors ($\mathbb{C}^d$), $A=\{ v_1, \ldots v_n\}$ and $B=\{u_1, \ldots u_n\}$.
The question is if there is an efficient solution (polynomial in $n$) for checking whether $A$ ...

**16**

votes

**1**answer

2k views

### How fast can we *really* multiply matrices?

Background: The Strassen Algorithm, described here, has a computational complexity of $\text{O}(n^{2.8})$ for the multiplication of two $n \times n$ matrices. However, the constant is so large that ...

**0**

votes

**1**answer

184 views

### graph to tree and graph isomorphism problem

Sorry if the following are stupid questions (i do not know much about the graph theory).
1. Motivation
we do not know the graph isomorphism problem in class P or NP complete and it is P in the ...

**3**

votes

**1**answer

330 views

### Complexity of convex polytope volume calculation ? (Volume of Voronoi cell) (Error probability)

Assume I have polytope in R^k given by N (k<< N) linear inequalities (A_i x < b_i).
I guess complexity of its volume calculate is higher than linear in "N", am I right ?
(Is the complexity ...

**13**

votes

**1**answer

561 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 ...

**2**

votes

**1**answer

263 views

### Oracle Results: P^A = NP^A

Context
In the work of Baker, Gill, Solovay, we know that there exists some oracle A s.t.
$$P^A = NP^A$$.
Now, in CCAMA, this oracle $A$ is given as an EXP complete language.
Question:
Can we do ...

**1**

vote

**0**answers

76 views

### Deciding / Approximating Parity of Small Depth Decision Trees

Let C be a circuit such that:
C: $\{0,1\}^n$ to $\{0,1\}$
the top most gate is a parity gate
all the inputs to the parity gate are small depth decision trees
there is a total of $2^{ log^k n}$ ...

**3**

votes

**2**answers

1k 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: ...

**1**

vote

**3**answers

1k views

### How to get the largest subset of a set of sets of intervals with no overlapping intervals

Given: Set of Set of Intervals. Example {{(1,2), (3,4)}, {(1, 3)}, {(13,14)}}
Wanted Result: Largest Subset, in which no Interval overlaps with another from every Set pairwise.
Example:
Input ...

**2**

votes

**3**answers

270 views

### Generating a set of integer passwords that can be securely authenticated

First, apologies for the title. This is an odd question, and I couldn't come up with a simple title for it.
My question is as follows.
Given a positive integer $k$, determine a set of properties ...

**4**

votes

**2**answers

231 views

### Computational complexity of Knot polynomials

What's known about computational complexity of different types of knot invariant polynomials?
For example, Evaluating Jones Polynomial is known to be #P hard.
Is there any reference that surveys such ...

**5**

votes

**3**answers

551 views

### computational complexity of primitive recursive functions

If we have a rewrite system for primitive recursive functions, which simplifies each term according to how the function was defined, then what is the computational complexity of this calculation? That ...

**4**

votes

**2**answers

251 views

### How small can a language in NP\P be?

How small can a language in $NP$ but not in $P$ be? Of course, I don't expect a proof that there exists a language in $NP\setminus P$, so instead I'll ask: Can we rule out any of these conjectures?
...

**17**

votes

**1**answer

2k views

### Möbius Randomness of the Rudin-Shapiro Sequence

The Rudin-Shapiro sequence (also known as the Golay-Rudin-Shapiro sequence) is defined as follows.
Let $a_n = \sum \epsilon_i\epsilon_{i+1}$ where $\epsilon_1,\epsilon_2,\dots$ are the digits in the ...

**13**

votes

**3**answers

475 views

### Complexity of equitable partitions

We are talking about undirected simple graphs and partitions of their vertex sets into disjoint non-empty cells. Such a partition is equitable if for any two vertices $v,w$ in the same cell, and any ...

**3**

votes

**1**answer

232 views

### Diagonalization and classes of computable functions

Fix a standard effective listing $(\phi_e)_{e\in\omega}$ of the partial computable functions from $\omega$ to $\omega$. Let $\mathcal{C}$ be a class of computable total functions $\omega\rightarrow ...

**2**

votes

**2**answers

503 views

### sparsity of QR decomposition

Hi, everyone!
I have a sparse $n \times n$ matrix $A$ with $nnz(A)$ denoting the number of non-zero entries in $A$. Now I use QR factorization to decompose $A$ into an orthogonal matrix $Q$ and ...

**3**

votes

**1**answer

221 views

### Complexity of Labeled Graph Homomorphism

The following recreational math problem has been floating around work:
We're given an $m \times n$ grid ($m,n$ positive integers). We wish to label the elements of the grid with letters so that we ...

**2**

votes

**1**answer

121 views

### Maximizing positive definite quadratic using the eigendecompoisition

Consider the problem:
$\textrm{max}\;\; x^T Q x$
subject to $||x||_\infty \leq 1$, where $Q$ is a positive definite matrix.
I believe this problem is NP-hard (although I have only found hardness ...

**3**

votes

**1**answer

119 views

### Counting connected fundamental domains of actions on Cayley graphs

The following question arises, for me, from mathematical music theory:
Write $({\Bbb Z}^n,E_n)$ for the Cayley graph of ${\Bbb Z}^n$
relative to standard free generators.
Given a subgroup $L$ of ...

**1**

vote

**1**answer

460 views

### k-uniform k-partite hypergraph matching in polynomial time

I have what seems like an elementary question, but google didn't throw up any answers for it. I would appreciate any pointers that MO users may provide.
It is well known that for $k\geq 3$ finding ...

**3**

votes

**0**answers

166 views

### Kolmogorov complexity with bounded ressources

Thanks to symetry of information (i.e $\forall x,y, K(xy) = K(x) + K(y|x) - O(log(|x| + |y|)$), one can easily show that :
$ \exists N \forall x, (|x| = n^{log(n)} and |x| \geq N), \exists y, (|y| ...

**3**

votes

**1**answer

123 views

### Quick tests for Self complementary vertex transitive graphs

Are there any quick tests to determine if a graph is Self complementary vertex transitive? That is if the graph is self complementary vertex transitive the answer should be yes.

**6**

votes

**1**answer

345 views

### Vertex transitive graphs

Does having vertex transitivity make the problem of calculating independence and chromatic numbers easier?

**3**

votes

**2**answers

567 views

### symmetric difference of languages - both are in NP and coNP

I have this problem,
Let $L_1,L_2$ be languages in $NP \cap co-NP$. I want to show that their symmetric difference is also in $NP \cap co-NP$. Like:
$L_1 \oplus L_2$ = {x | x is in exactly one of ...

**3**

votes

**0**answers

152 views

### Best lower bound for proof complexity of graph asymmetry

Graph automorphism problem ( GA) of determining whether a graph has a nontrivial automorphism is a good candidate for a problem in $NP$-intermediate. I'm looking for references that study the ...

**0**

votes

**2**answers

104 views

### Bounding 2nd Eigenvalue of a Pseudo-Rotation-ish matrix

Let $p,q$ be arbitrary primes.
Let $N = p * q$.
Let $I$ be the $N * N$ identity matrix.
Let $R$ be the $N * N$ matrix defined as follows:
$R[x_0 * p + y_0, x_1 * p + y_1]=1$ if and only if $x_0+1 ...

**3**

votes

**2**answers

189 views

### Form a $\mathbb{Z}^d$ lattice cycle from given lengths

Suppose you are given a list of integer lengths,
e.g.,
$(5,3,2,2,1,1,2,1,1)$.
The task is to decide if they can form a closed cycle
in $\mathbb{Z}^d$ by connecting segments of those
lengths in order, ...

**3**

votes

**2**answers

566 views

### Computational complexity of unconstrained convex optimisation

What is known about the relationship between unconstrained convex optimisation and computational complexity? For example, for which optimisation problems and which gradient descent algorithms is one ...

**14**

votes

**4**answers

3k views

### What would be some major consequences of the inconsistency of ZFC?

I was happily surfing the arXiv, when I was jolted by the following paper:
Inconsistency of the Zermelo-Fraenkel set theory with the axiom of choice and its effects on the computational complexity by ...

**6**

votes

**1**answer

542 views

### NP-hardness of a graph partition problem?

I'm interested in this problem: Given an undirected graph $G(E, V)$, Is there a partition of $G$ into graphs $G_1(E_1, V_1)$ and $G_2(E_2, V_2)$ such that $G_1$ and $G_2$ are isomorphic? Here $E$ is ...

**4**

votes

**1**answer

438 views

### Infinite monkeys computing … triangle area?

I wonder if it is possible to specialize the question:
(a) What is the probability that a random Turing Machine program
will halt?, to: (b) What is the probability that a random Turing Machine
...

**10**

votes

**2**answers

457 views

### What is the computational-complexity-theoretic analogue of computable inseparability? For example, if P is not NP, are there disjoint NP sets with no separation in P?

Disjoint sets $A$ and $B$ are computably inseparable, if there
is no computable separating set, a computable set $C$ containing $A$ and disjoint from $B$. The
existence of c.e. computably inseparable ...

**12**

votes

**1**answer

536 views

### Does every feasible partial order relation on the natural numbers extend to a feasible linear order relation?

It is well known that every partial order on a set can be extended
to a linear order on that set. That is, for every partial order
$\lhd$ on a set $X$, there is a linear order $\prec$ on $X$ such
that ...