Questions tagged [linear-algebra]

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-1
votes
1answer
61 views

A max-even subset problem

I want to know if there is any polynomial algorithm for the problem, or any NP-completeness result. Given a set $S$ and $m$ subsets $C_1, \dots, C_m$ of $S$, we want to find a non-empty set $X\...
3
votes
1answer
129 views

Conditions for Linear Diophantine Equations to always have a solution

Given a set of $n$ linear equations in $v$ integer variables, where $v > n$, we can say that this system of equations will always have an integer solution. Over $\mathbb N$, is there any such ...
3
votes
2answers
293 views

Solving/Optimizing a linear system in a finite field (Z/2Z)

I'm trying to solve the following optimization problem. A is a rectangular matrix with coefficients in the finite field Z/2Z (size less than 1000 X 1000). I have a system of the form A.X = Y (X and Y ...
2
votes
0answers
40 views

mx2m modulo-3 matrix solution

Is there an efficient algorithm for the following problem? Given: a $m$-vector $b \in \{0,1,2\}^m$, and a $m \times 2m$ matrix $A$, with the promise that for every $b' \in \{0,1,2\}^m$, there exists $...
9
votes
1answer
361 views

What algorithms exist for solving natural number linear systems?

I'm looking at the following problem: Given $n$-dimensional vectors of natural numbers $v_1, \ldots, v_m$ and some input vector $u$, is $u$ a linear combination of the $v_i$'s with natural number ...
8
votes
1answer
258 views

Generate algorithmically all grid points inside a hypercube

$\def\R{\mathbb{R}}\def\Z{\mathbb{Z}}\def\n#1{\|#1\|_\infty}$The problem comes directly from computational mathematics, and can be stated as follows: Given a regular matrix $M\in\R^{d\times d}$, find ...
6
votes
2answers
148 views

Can this system of polynomial equations be solved in polynomial time?

I have these $n$ equations, with $n$ variables. Variables are first $n$ positive integers, constants can be any rational number including zero. Given that there is always a solution, how do we find a ...
6
votes
1answer
338 views

Testing whether a determinant polynomial is identically zero

Suppose we are given matrices $A_1, \ldots, A_k$ which are $n \times n$ matrices with rational entries and are asked to determine whether the polynomial ${\rm det}(\alpha_1 A_1 + \alpha_2 A_2 + \cdots ...
6
votes
1answer
7k views

Checking Feasibility of Linear Program in Polynomial Time

Given a linear system of the form: $$\begin{array}{c} x_r = a \quad x_j = b \\ c_1x_1 + c_2x_2 + \ldots + c_nx_n = N \\ x_1+x_2 + x_3 + \ldots + x_n = k\\ 0 \le a,b,x_1,x_2,x_3...x_n \le 1\\ k \ge 0 \...
2
votes
1answer
687 views

Locality-sensitive hashing random projection

I'm trying to understand how the LSH works for Cosine Similarity metric. For instance, let's say you have $\vec{v} \in \mathbb{R}^d$ and the random vectors $\vec{r_{i}} \sim \mathcal{N}(0, 1)^d$ that ...
2
votes
1answer
166 views

How to find subset of vectors whose sum has certain characteristics

Let's say you have list of $n$ vectors with entries from $\{0,1,x\}$ and $x$ is > $n$: $$ \begin{align*} L_0 &= [1,0,x] \\ L_1 &= [1,1,1] \\ L_2 &= [1,0,0] \\ L_3 &= [x,1,0] \\ L_4 &...
1
vote
1answer
704 views

Complexity of matrix inverse via Gaussian elimination

I'm trying to determine the exact complexity of finding an $n\times n$ matrix inverse of $A$. If it is known that the complexity of Gaussian elimination is $\frac{2}{3}n^3 + \frac{1}{2}n^2+O(n)$, then ...