**0**

votes

**1**answer

51 views

### Algo for covering maximum surface of a polygon with rectangles

I'm looking to an algorithm to covering maximum surface of a polygon with rectangles. Rectangles have to have a specific width, a rectangle can't overlap an other one and each one has to fit in the ...

**56**

votes

**16**answers

8k views

### Examples of algorithms requiring deep mathematics to prove correctness

I am looking for examples of algorithms for which the proof of correctness requires deep mathematics ( far beyond what is covered in a normal computer science course).
I hope this is not too broad.

**4**

votes

**1**answer

134 views

### Uniform sampling from general simplex with a twist

This is part of a question I had asked elsewhere, and then some of the links redirected me to CS stack exchange.
Given $0\leq a_1\leq\dots\leq a_D\leq1$ (all strictly positive), I want to draw points ...

**6**

votes

**0**answers

101 views

### On the use of Weisfeiler-Leman refinement in Babai's GI proof

This question is for those familiar with the methods behind Babai's recent proof that graph isomorphism can be decided in quasipolynomial time. I am a newcomer to the GI problem, so I apologize if my ...

**0**

votes

**2**answers

125 views

### Relaxed path decomposition of a graph

Definition
Given a directed connected graph $G$ without multiple edges or self loops. We call a final path of $G$ a path ending with a vertex with no successor (the path can not be extended anymore) ...

**0**

votes

**0**answers

95 views

### Finding good high-dimensional sphere coverings in Euclidean space

Suppose we want to cover the unit sphere $\mathcal{S}^{d-1} := \{\mathbf{x} \in \mathbb{R}^d: \|\mathbf{x}\|_2 = 1\}$ with spherical caps $\mathcal{C}_{\mathbf{y}} := \{\mathbf{x} \in \mathcal{S}^{d-1}...

**0**

votes

**0**answers

37 views

### How to find faces of graph? [duplicate]

I have à planair graph and I want to find an algorithm that will find all of the faces of the graph. Thanks you in advance for your answers.

**5**

votes

**0**answers

104 views

### “Factorisation” in special linear groups over rings of integers

It is known that for any number field $F$ with infinitely many units (i.e. $F$ is not $\mathbb Q$ or an imaginary quadratic field) with ring of integers $O$ the special linear group $\mathrm{SL}_2(O)$ ...

**-2**

votes

**1**answer

109 views

### Algorithm for finding numbers with an even partition number

NOTE: After edit question became about set partitions, which not was I intended, so this is second try.
Is there an algorithm for producing an infinite subset of set of integer partition numbers p(n) ...

**4**

votes

**0**answers

93 views

### In what range can we find diophantine approximations using the LLL-algorithm?

Let $\alpha_1, \ldots, \alpha_n$ be $\mathbb{Q}$-linearly independent real numbers. I want to show that for all $x_1, \ldots, x_n\in\mathbb{Z}$, $|x_i|<N$ we have some lower bound for $\left|\sum ...

**3**

votes

**0**answers

45 views

### Connectedness of semi algebraic set by c.a.d

I do not know whether there is a standard or some traditional ways to decide whether a semi algebraic set is connected or not.
One way I know is c.a.d algorithm. I have read some papers of c.a.d ...

**0**

votes

**0**answers

20 views

### Counting contingency tables with unary inputs

Given $2$ sequences $A = (a_1,\dots,a_k)$ and $B = (b_1,\dots,b_l)$ of natural numbers. A contingency table is a matrix in $\mathbb{N}^{k \times l}$ with row sums $A$ and column sums $B$.
Counting ...

**6**

votes

**0**answers

165 views

### Solving polynomial equations modulo $1$

Let $P\in \mathbb{R}\lbrack x\rbrack$ be given. (In practice, the coefficients could be given as, say, decimals to sufficient precision.) Let $M\geq 1$, and let $I$ be an interval in $\mathbb{R}/\...

**0**

votes

**0**answers

47 views

### Reconstructing a graph from set of sequences of edges

I have the following problem to solve: Given a set of sequences of edges of an undirected, planar, connected graph, find a "reasonable" reconstruction of the graph. There is an unknown number of ...

**1**

vote

**2**answers

69 views

### How can I efficiently approximate the stationary distribution of an infinite CTMC with a sparse rate matrix?

I am looking for methods to approximate the stationary distribution of an infinite CTMC with a sparse rate matrix. Each row and column of the rate matrix has a finite number of non-zero elements. ...

**4**

votes

**0**answers

152 views

### Optimal instructions for the modular construction of rectlinear Lego structures

Let $X$ be a compact (or periodic) union of integer translates of unit cubes such that the interior of $X$ is connected. (If it makes any difference, suppose that the dimension $n$ of $X$ is 3.) I am ...

**9**

votes

**0**answers

97 views

### Factorisation in $\mathbb{N}[X]$?

Do we know an efficient algorithm to factorise in $\mathbb{N}[X]$ ?
One way to do factorisation in $\mathbb{N}[X]$ is to use an algorithm to factorise in $\mathbb{Z}[X]$ and to combine some factor to ...

**6**

votes

**2**answers

851 views

### What is the Time Complexity of the Matrix Exponential?

While trying to compute the Matrix Exponential of an $n \times n$ array I decided to take advantage of a Python function called scipy.linalg.expm().
According to ...

**0**

votes

**3**answers

104 views

### How to do a clockwise ordering of a planar graph in order to define its faces?

I am currently making an algorithm for planar graphs that I need to triangulate so they become maximally planar (that is triangulated and planar) given only the lists of neighbors for each node : no ...

**0**

votes

**0**answers

17 views

### Finding orthogonal basis with constraint

Is there any fast algorithm that output an orthogonal basis $e_i,i\leq n$ of $R^n$
with $e_i\in V_i$? Where $V_i,i\leq n$ are given linear subspaces of $R^n$.
And is there any condition on $V_i,i\leq ...

**7**

votes

**1**answer

287 views

### Choosing two-colorable subgraph in a triangulation

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

**0**

votes

**2**answers

96 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 $K$. ...

**1**

vote

**0**answers

80 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 ...

**4**

votes

**0**answers

55 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

301 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

296 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 ...

**3**

votes

**0**answers

379 views

### 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

54 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

59 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 ...

**3**

votes

**1**answer

89 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

98 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

39 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

212 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

34 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 $Q$ ...

**8**

votes

**3**answers

302 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

86 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

291 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
$E_{n}((...

**4**

votes

**2**answers

211 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

214 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

107 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

96 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 $m=\...

**3**

votes

**0**answers

64 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

167 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

47 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

141 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

42 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 $k$-...

**3**

votes

**1**answer

89 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

68 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

85 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 ...