# Tagged Questions

**4**

votes

**2**answers

154 views

### Choosing two-colorable subgraph in a triangulation

Consider a planar graph which is a triangulation.
Is it possible to find a two-colorable subgraph which has common edge with every face of our graph?
It is known that such spanning tree not always ...

**0**

votes

**2**answers

49 views

### Shortest path in a weighted graph with coloured edges

I have a weighted undirected graph with $N$ veritces and $M$ edges with 10 different colours in the whole graph. Each time I pass edges of different colour I have to pay additional fee equal to $K$. ...

**0**

votes

**0**answers

64 views

### maximal sets of vertices that avoids a clique

I am looking for some known algorithm that finds, for a given graph, all the maximal sets of vertices that avoid a clique of some given size $k$. I'd prefer one written in MATLAB, but other languages ...

**3**

votes

**0**answers

48 views

### Efficient CW structures on squarefree semi-algebraic set

General Setup
Given a collection of $k$ polynomials (with real coefficients) in $n$ real variables, say $f_i(x_1,\ldots,x_n)$, let $V \subset \mathbb{R}^n$ correspond to those $x$-values for which ...

**2**

votes

**2**answers

255 views

### Irreducible algebraic sets via irreducible polynomials

There are many results about irreducible polynomials over finite fields:
we know a cardinality of all irreducible polynomials with given degree, we know explicit examples of irreducible polynomials, ...

**4**

votes

**1**answer

279 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 ...

**0**

votes

**1**answer

94 views

### Algorithm to check if a number is the sum of another number and its reverse [closed]

So im looking for an algorithm that checks (in about 10 sec) if a natural number M (1≤M≤10^100000 -yes, the range is that big) can occur by the sum of another natural number N and its reverse Nr. For ...

**3**

votes

**0**answers

332 views

+50

### Finding the number of rational points effectively

Consider $\# P$ and $\oplus P$.
There is a $\# P$-hard problem: to find number of rational solutions of a system of polynomial equations over $\mathbb{F}_2$. The corresponding $\oplus P$-hard ...

**1**

vote

**2**answers

45 views

### Practical permutation search problems resilient to backtrack techniques

I asked a similar question here: Permutation search problems with no known $o(n!)$ algorithms; to which one-way functions over a space of permutations gives a problem with no $o(n!)$ solution ...

**1**

vote

**1**answer

41 views

### Is there analogs of perlin noise algorithm?

I want to create procedure generated map, but all resources that I found talks about using of "perlin noise" algorithm. Maybe better (higher perfomance, more realistic terrain generation) analogs ...

**2**

votes

**1**answer

85 views

### A modified bipartite assignment problem

Consider the following optimization problem. I have $n$ advisors and $dn$ students. I want to assign each student an advisor so that each advisor has exactly $d$ students. Each advisor/student pair ...

**7**

votes

**1**answer

89 views

### Fast Fourier Transforms for non-trigonometric bases

The fast Fourier transform allows decomposition into a sin/cos basis in $N \log(N)$ complexity. Can one generalize the algorithm (or the ideas used) to other bases, e.g. orthogonal polynomial bases ...

**2**

votes

**0**answers

38 views

### smallest ball enclosing one point of each color

I have a set of colored points (say 10 colors with 50 points of each color) in a 100-dimensional space. I want to choose one point of each color so that the 10 points are as close to each other as ...

**12**

votes

**2**answers

199 views

### Permutation search problems with no known $o(n!)$ algorithms

I am looking for problems for whose solution no known subfactorial algorithms are known. I am particularly interested in questions of isomorphism; that is, is there a permutation that converts one ...

**0**

votes

**0**answers

31 views

### Comparing product of positive affine functions over integers

Problem
Let $f_i: \mathbb Z^n \mapsto \mathbb Z$ and $g_i: \mathbb Z^n \mapsto \mathbb Z$ affine functions and $\mathcal D \subseteq \mathbb Z^n$ a set on which they are all positive. Let $P$ and $Q$ ...

**6**

votes

**3**answers

217 views

### A simplified Art Gallery Problem in a matrix

Let's take a $m \times n$ matrix as an area with $m \times n$ blocks (likes a 2D-version of the world in Minecraft). We have to put some lamps in this matrix to illuminate the whole matrix. Here is ...

**3**

votes

**0**answers

83 views

### State of the art for univariate complex polynomials factorization with algebraic coefficients

Let $\mathbb{K}:=\overline{\mathbb{Q}}$ be the field of algebraic numbers. We choose to represent an element of $\mathbb{K}$ as its minimal monic polynomial, which is a vector in some $\mathbb{Q}^n$.
...

**8**

votes

**1**answer

280 views

### How long does the slow inefficient algorithm for computing the product in classical Laver tables take?

Let $(A_{n},*)$ denote the $n$-th classical Laver table. Let
$X_{n}$ be the set of all finite sequences of elements from $A_{n}$.
Define a function $E_{n}:X_{n}\rightarrow X_{n}$ by letting
...

**4**

votes

**2**answers

194 views

### Algorithms for finding graph isomorphisms

I was wondering if anybody knows where I can find some information about the current (practical) algorithms for finding graph isomorphisms. I've joined the bandwagon and wrote my own which I would ...

**3**

votes

**1**answer

205 views

### Periodic strings

I wish to ask a problem in periodic strings, it might be well-known but I am a beginner in this subject, so I am very glad if someones can show me. My problem is that can we add some string to the end ...

**3**

votes

**2**answers

97 views

### Algorithms for Sorting Subset Sums

In this question the number of unique sortings has been discussed.
As a follow-up, I would like to know, whether the problem of sorting the sequence of subset sums has ever been studied.
There ...

**4**

votes

**0**answers

94 views

### How to enumerate a discrete group of matrices by their Frobenius norm?

Suppose I have a discrete group $G<\mathrm{SL}_2(\mathbb{C})$,
and it is finitely generated by some known generators.
That is, $G=\langle g_1,\dots,g_n\rangle$.
The Frobenius norm of a matrix ...

**3**

votes

**0**answers

58 views

### Algorithm to calculate moments of uniform distribution on convex polyhedra

There is system of linear inequalities
$$
Ax \leq K,
$$
$$
x\geq a, x\leq b.
$$
$A$ is $(n\times m)$-matrix, where $n\approx 100$ and $m\approx 10000$, $rank(A)=n$.
Suppose that on set of solutions ...

**1**

vote

**0**answers

154 views

### “Kolmogorov complexity” of models of computation

This question was partly inspired by my learning about John Tromp's binary lambda calculus and similar minimal languages such as Jot. A more detailed discussion of some of these ideas is in Michael ...

**0**

votes

**0**answers

45 views

### Distinguishing (possibly lower dimensional) $1$-skeleton of a regular graph inscribed in a sphere

Consider you have two (possibly same) convex $1$-skeleton of a regular graph $A$ and $B$ in $m$-dimensions inscribed in a sphere with possibly exponential number of vertices in $n$-dimension with ...

**1**

vote

**0**answers

46 views

### Expected number of forward jumps to reach a given quantile of a rv [closed]

I'm a noob in randomized algorithm and ran into a problem(definitely not home work. I'm doing a self study out of my interest with help of my friends. I'm pursuing research career in a machine ...

**0**

votes

**1**answer

131 views

### Find special elliptic curves from j-invariant

Let $E$ be the elliptic curve defined over $GF(p)$ and $j$ be j-invariant of $E$, where $p$ is a big prime number. Also suppose $l$ be small prime number (for example $l<5000$) and $\#E$ denote ...

**1**

vote

**0**answers

39 views

### Calculating a Combinatorial Generalization of Planar Convex Hulls

In this question I have suggested a generalization of the notion of a set of points in the Euclidean plane being in convex configuration to the set of vertices of symmetric weighted graphs via ...

**3**

votes

**1**answer

55 views

### Generating Uniquely k-Optimal Point Sets

This question is motivated by the observation that finding an optimal tour through a set of points in the Euclidean plane is especially simple, if the points are in convex configuration and, that the ...

**0**

votes

**0**answers

66 views

### Algebraic operations with memory hardness properties

In cryptography, there are password hash functions like scrypt and argon2 for which the fastest known algorithms employ large ...

**2**

votes

**1**answer

68 views

### Area of an irregular, n-sided, non-intersecting (edges) polygon algorithm

I need to generate an irregular, n-sided polygon of non-intersecting edges (n= 200, for example) with the smallest area possible. The position of the vertex is random and I've tried designing a couple ...

**-1**

votes

**1**answer

53 views

### Computing the inverse of a Cholesky decomposition [closed]

I have chol(A) and I would like chol(A^-1). One way to do this is to construct the inverse positive definite symmetric matrix and then take its Cholesky decomposition (with Dpotri and Dpotrf for ...

**2**

votes

**1**answer

100 views

### Minimal Support Solutions of a Linear System (Dissertation)

For a given $n \times m$ matrix A with $m>>n$ and a given vector $\vec b \in \mathbb{F}^{n \times 1}$, and given that $A\vec{x}=\vec{b}$ for at least one $\vec{x} \in \mathbb{F}^{m \times ...

**1**

vote

**2**answers

56 views

### Maximal Minimum Weight DAGs

In the case of undirected, connected graphs the name for the maximal cycle-free subgraph of minimal weight is called Minimum Spanning Tree, and the efficient algorithms for their calculation are well ...

**6**

votes

**0**answers

178 views

### Complexity of approximating the size of the range of a matrix

Given an $m$ by $n$ matrix $M$ with $m \leq n$ and elements from $\{-1,1\}$, let us define:
$$S_M = |\{Mx : x \in \{-1,1\}^n\}.$$
It is NP-hard to compute $S_M$ exactly I believe by applying the ...

**3**

votes

**1**answer

194 views

### Polynomial factoring over finite fields

What is known in general about the complexity of factoring polynomials over finite fields?
For instance given $\Bbb F_q$ where $q=p^n$ and total degree $d$ polynomial in $m$ variables what can we say ...

**2**

votes

**0**answers

93 views

### Detecting Negative Cycles in Undirected Graphs

I recently faced the problem of quickly detecting negative cycles in undirected, weighted graphs. Resorting to the Bellman-Ford Algorithm, as commonly suggested, turned out to be very inefficient and ...

**2**

votes

**0**answers

51 views

### Is there a name for this variant of the MST and the TSP?

Suppose I am given a weighted graph $G$ that contains a "start vertex" $v_0$, and my goal is to construct a set of paths that all originate at $v_0$ and touch all of the vertices of $G$, with as ...

**9**

votes

**1**answer

185 views

### Exact determinant of a circulant matrix

The wikipedia gives us a formula for the determinant of a circulant matrix. That is:
$$\mathrm{det}(C)
= \prod_{j=0}^{n-1} (c_0 + c_{n-1} \omega_j + c_{n-2} \omega_j^2 + \dots + c_1\omega_j^{n-1})= ...

**3**

votes

**0**answers

79 views

### Is there a Havel-Hakimi for geometric graphs?

Suppose that we are given $n$ points in the plane, with a degree prescribed for each, and the question is whether we can place a geometric graph on them. Is there an efficient algorithm for this?
...

**2**

votes

**1**answer

104 views

### Does anyone have the correct link to treewidth.com?

Several posts (and this) on StackExchange sites like MO have some link-rot. For example, I've been looking into tree decomposition and keep coming across references to treewidth.com, but the link ...

**3**

votes

**0**answers

21 views

### Complexity of OBDD isomorphism (representing same function after permutation of variables)?

According to wikipedia Ordered Binary Decision Diagarams (OBDD) are a data structure that is used to represent a Boolean function.
OBDD is a DAG with two sinks $0,1$.
The size of the BDD is number ...

**11**

votes

**1**answer

673 views

### Reasons for difficulty of Graph Isomorphism and why Johnson graphs are important?

In http://jeremykun.com/2015/11/12/a-quasipolynomial-time-algorithm-for-graph-isomorphism-the-details/ it is mentioned:
'In discussing Johnson graphs, Babai said they were a source of “unspeakable ...

**2**

votes

**0**answers

57 views

### Blossoms and Colorings

There is a striking analogy between finding maximum matchings in graphs and determining the chromatic number of graphs: both problems are fairly easy for bipartite graphs, but harder, resp. too hard ...

**3**

votes

**0**answers

83 views

### Algorithm for finding the fewest cartesian products to partition a set of points

The question Algorithm for finding the fewest rectangles to cover a set of rectangles was already answered here and a similar question was answered here. Those questions were about regular geometric ...

**1**

vote

**1**answer

39 views

### Test Instances for Perfect Matchings in Graphs

Are there any graphs with a known set of perfect matchings and other predefined properties, such as vertex connectivity, which can be used for testing the implementation of matching algorithms?
...

**3**

votes

**0**answers

68 views

### What is the influence of unreliable comparisons on the results of sorting

Considering sorting algorithms based solely on binary comparisons of the elements to be sorted(algorithms such as insertion sort, selection sort, quicksort, and so on), what problems do we face when ...

**3**

votes

**2**answers

175 views

### Bellman-Ford for Matching Problems?

I am looking for a simple way of calculating minimum-weight perfect matchings in complete graphs with an even number of vertices.
I know that there are implementations that are based on Edmond's ...

**1**

vote

**1**answer

73 views

### What is the relation between size of maximum clique and branchwidth?

Let $bw(G)$ be the branchwidth of graph $G$ and $\omega(G)$ be the size of maximum clique in $G$. I think the following inequality holds:
$$
\omega(G)\leq bw(G)
$$
Intuition: Assume (in reverse of ...

**8**

votes

**1**answer

650 views

### Some Questions on the Collatz conjecture

The set of all positive whole numbers is denoted by $\mathbb{N}_+$.
Let $f\colon\ \mathbb{N}_+\to\mathbb{N}_+:n\mapsto
\begin{cases}\frac{n}{2}&\text{$n$ even}\\3n+1&\text{$n$ ...