# Tagged Questions

Informally, an algorithm is a set of explicit instructions used to solve a problem (e.g. Euclid's algorithm for computing the greatest common divisor of two integers). For more specific questions on algorithms, this tag may be used in conjunction with the approximation-algorithms, algorithmic-...

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### 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 ...
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### 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 ...
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### 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 $\vec{x} \in \mathbb{F}^{m \times 1}$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Monotonicity per dimension of multivariate scattered data

For my thesis, I am working on interpolation using the RBF method (Radial Basis Functions). Before interpolating, I want some a priori insight into the data, for example check in which dimensions it ...
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### Extraction of Coefficients in the Exponential Function of a Series

Question: Let $f(x) \in x\mathbb{C}[[x]]$. What is the (asymptotically) fastest algorithm for calculating the coefficient of $x^n$ in $e^{f(x)}$? Naive Solution 1: Using fast polynomial ...
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### G-Correlation of Vectors

Let $\vec{a},\vec{b} \in \mathbb{R}^{n}$. Consider the function $f: S_n \to \mathbb{R}$ given by $f(\sigma):= \sum_{i=1}^{n} a_i b_{\sigma(i)}$. Let $G$ be a subgroup of $S_n$, given by $O(\log n)$ ...
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### Algorithm to count the number of perfect matchings in non planar graph

I need to count the number of perfect matchings of a certain family of graphs. This family of graph is non planar and a type of snark. For the initial cases, it seems that this number is growing ...
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### Comparison of Constrained Optimization Methods

I am trying to solve a constrained optimization problem using filter methods and came across two papers on the topic that I am having some problems with. The original filter method paper is the ...
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### Is there an algorithm to find a linear dependence between points on elliptic curves?

Let $E$ be an elliptic curve over a finite field $\mathbb{F}_p$ of characteristic $p$. Let $P,Q\in E(\mathbb{F}_q)$, such that $Q=mP+n\tau(P)$, where $\tau$ is the p-th power of frobenious map and $m$ ...
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### Seeking for abelian subalgebra of fixed dimension in finite Lie algebra

The problem is: I want to know if there is abelian subalgebra of dimension $k$ in Lie algebra of dimension $n$. My Lie algebra is given by its structure constant table. There are some algorithms ...
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### The cost function in the Weighted Bipartite Matching Problem (a.k.a the Assignment Problem)

In the definition of this problem, the weight/cost function generally takes value in $\mathbb{Z}$ (or sometimes $\mathbb{Q}$). This is what I observed from some books (e.g. "Combinatorial ...
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### Travelling salesman: can the furthest-neighbour algorithm beat the nearest-neighbour?

This is a problem that has bugged me for quite some time, and I have not been able to find any documentation about it online. It is well known that the NN algorithm can yield the worst possible route -...
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### Performance guarantee of RLF [closed]

I cannot manage to find the performance guarantee of the Recursive Largest First (RLF) algorithm for approximating the chromatic number of a graph. I know DSATUR has a $\mathcal{O}(n)$ guarantee, ...
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### $k$-th subset in order of increasing sum

I have an array of integers T[N] indexed from 1. For example T[1] = 1, T[2] = 4, T[3] = 5, T[4] = 9. Let's enumerate all subsets of this set in a certain order: increasing sum of the elements. In case ...
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### Computing digits of irrational exponentiation

Let us have positive irrational numbers $a$ and $b$ represented by functions $f_a,f_b\colon\mathbb{N}\to\mathbb{N}$ respectively such that $f_a(0)=\left \lfloor{a}\right \rfloor$ and $f_a(i)$, $i>0$...
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### Algorithm to generate hyperbolic metric on a compact surface

Let $F$ be a compact surface of genus $g$ with generators $a_1,b_1,\ldots ,a_g,b_g$ with relation $[a_1,b_1]\cdots [a_g,b_g]=1$. We can also consider surfaces with boundary in which case the ...
Since $n = \frac{n(n+1)}{2}-\frac{n(n-1)}{2}$, every natural number can be represented as the difference of two triangular numbers: $n = \frac{a(a+1)}{2}-\frac{b(b-1)}{2}$. Finding such a ...