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      Questions tagged [algorithms]

      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-randomness and algorithmic-topology tags.

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      Algorithm to list all Kostant partitions

      Let $\Phi_+$ be the set of positive roots in some root system, and let $Q_+$ be the positive part of the root lattice, i.e., the set of elements of the form $\sum_{\beta\in \Phi_+}m_\beta\beta$ with $...
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      1answer
      206 views

      What's the runtime of this method and is this method correct?

      Let $n=pq$ be the prime decomposition. I am searching for $l$ such that there exists a $k$ with: $$n^l = a \cdot 2^k + b$$ and $$ 1 < \gcd(b,n^l) < n^l$$ Edit by comment of @GerryMyerson: If $...
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      0answers
      50 views

      How to inference the dual form of perceptron?

      The model of perceptron is a linear binary classifier, which is $f(x)=\mathbb{sign}(w^Tx+b)$. $x$ is the datapoint as $w$ as well as $b$ are the parameters. The cost function of Primal Perceptron is $...
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      42 views

      Jordan Decomposition of Sparse matrix

      Suppose we are given $n \times n$ rational matrix, $A$ with at most $k$ nonzero elements in each row and each column with $k \ll n$. What is the best algorithm to compute its Jordan decomposition? ...
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      45 views

      Calculating Minimum Spanning Trees in Very Big Graphs

      I need to determine Minimum Spanning Trees (MST) of very big complete graphs, whose edgeweights can be calculated from data that is associated with the vertices. In the planar euclidean case, for ...
      2
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      1answer
      51 views

      Intersection of sphere with triangle containing moving vertices

      First off, apologies if I cannot properly articulate my question in the most formal way. However, I believe my question should be simple enough to grasp anyhow. In $\mathbb{R}^3$, I would like to ...
      2
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      0answers
      17 views

      Difference Between Total Least Squares Plane and Plane Orthogonal to Principal Axis of Inertia Tensor

      Given a finite set $P$ of points in $\mathbb{E}^3$ , one can calculate an approximating plane either as the solution of a Total Least Squares problem or by interpreting the problem physically, ...
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      0answers
      17 views

      Complexity of weighted fractional edge coloring

      Given an edge-weighted multigraph $G=(V,E)$ with a positive, rational weight function $(w(e): e \in E)$, the weighted fractional edge coloring problem (WFECP) is to compute ($\min 1^T x$ subject to $...
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      42 views

      Planar graphs with perfect matching count in linear time?

      We can find Pfaffian orientation and take determinant to compute permanent in $O(n^\omega)$ time where $\omega$ is exponent of matrix multiplication. We know that permanent of $O(n)$ vertex planar ...
      2
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      1answer
      110 views

      Matrix completion problem with determinant condition?

      Given two $\{0,1\}^{n\times n}$ matrices $L$ and $M$ and an integer $m$ is there a polynomial in $n$ algorithm to find a $\{-1,0,+1\}$ matrix $T$ such that $$\mathsf{det}(L+T\odot M)=m$$ where $\odot$ ...
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      153 views

      Heuristic for lower bounding the time for integer factorization?

      I am posting this question here in hope that someone finds this heuristic useful, and maybe someone with more experience will make use of this: As @GerryMyerson suggested here is a statement of what ...
      5
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      1answer
      88 views

      Algorithm to sample from unknown probability distribution, given its projections?

      The Cramér–Wold theorem states that, if $X$ is a random variable living in $\mathbb{R^d}$, then the distributions of all one-dimensional projections of $X$ uniquely determine the distribution of $X$. ...
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      19 views

      Sufficient Condition for the Existence of Vertex Disjoint Shortest Paths

      Let $G(V,E)$ be a symmetric connected graph with $n$ vertices and let $D\in\mathbb{N}_0^{n\times n}$ be the matrix containing as entries $d_{uv}$ the least number of edges on a path from $u$ to $v$. ...
      1
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      0answers
      52 views

      Projection of a polytope along 4 orthogonal axes

      Consider the following problem: Given an $\mathcal{H}$-polytope $P$ in $\mathbb{R}^d$ and $4$ orthogonal vectors $v_1, ..., v_4 \in \mathbb{R}^d$, compute the projection of $P$ to the subspace ...
      3
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      1answer
      134 views

      $\sum_{i=1}^x\sum_{j=1}^xf(i\cdot j)$ Double Summing a (Not Completely) Multiplicative Function

      Let $f(n)$ be a multiplicative function that is not completely multiplicative, i.e $f(m)\cdot f(n)= f(m\cdot n)$ only if $gcd(m,n)=1$. Let $S(x)$ be the double sum over $f$, that is: $$S(x)=\sum_{i=1}...

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