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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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### 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 ...
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 $\odot$ ...
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 ...
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 $X$. ...
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 $v$. ...
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 ...
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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