My question is about a certain combinatorial game. The game works as follows. We have $n$ urns, each of which contains $m$ balls, where $m$ and $n$ are positive and satisfy $m < n$. A move consists ...

Translating to English from a non-English physics book about measurements:Anif has $8$ big marbles and $15$ small marbles. The weight of the big and small marbles are $37.5$ and $12.5$ respectively. ...

Hi I am a math major and I am trying to solve a challenging math problem. The tools needed to solve this are in the field of computer science and coding. I am okay at programming, however I need some ...

Create a closed tour visiting the numbers. The tour must start and end at the same cell. During the tour, you can only visit the neighboring cell (up,down,left,right). The tour cannot contain a number ...

I am working on a 2D covering problem and would appreciate some insights or references to relevant mathematical concepts.The ProblemThe goal is to completely cover a $98 \times 98$ grid using the ...

Let $d\in \mathbb{N}$ be a fixed constant. Let $V\subseteq \{-1,0,1\}^d$ be some set of vectors and let $w\in \mathbb{Z}^d$. Consider the linear programfind $ x\in \mathbb{R}^V $ such that: $\sum_{v\...

I am searching for the definition of a minimal system of equations for a polytope. Specifically, I am studying a variant of the graph colouring problem, and in Braga et al.$\color{magenta}{^\star}$ ...

Given a matrix $A \in \mathbb{R}^{m \times n}$, I want to find a vector $\vec{x} \in \mathbb{N}_{+}^{m}$ so $\vec{y} = {A}^{T} \vec{x}$ is nearly a constant vector, i.e., the values of $\vec{y}$ are ...

Consider a matrix $\mathbf{A} \in \{0, 1\}^{K \times N}$ and a vector $\mathbf{b} \in \mathbb{Z}_{>0}^{K \times 1}$ (i.e., elements of $\mathbf{A}$ are either $0$ or $1$, and elements of $\mathbf{b}...

I'm working on an ILP formulation that's supposed to find the best scoring set of routes on a given turn in a 18xx-style board game (in this particular case, Shikoku 1889)Problem descriptionTo ...

I'm struggling to model this constraint for a problem:$$x_C^4 = 1 \implies (x_A^4 + x_B^4 \geq 1 \land x_A^1 + x_B^1 = 0) \;\lor\; x_A^2x_B^3 = 1 \;\lor\;x_A^3x_B^2=1.$$where all variables are ...

Consider the following LP program$$ \min x_{n+1} $$subject to:$$ \sum_{i=0}^n 2 x_i + x_{n+1} = n $$$$ x_i \in \{ 0, 1 \} $$And $n$ odd.The claim is that using the standard B&B algorithm ...

Is there a standard optimal way to enforce the finite lattice (order) definition in an Integer Linear Program, for a lattice with a given number of elements $n$?I have tried a web search but with no ...

I confused myself into a hole here, so just need to see what I am missing. So, in the Gomory fractional cut cutting plane algorithm you use the basis to generate a new cut. How come it does not ...

I have a lot of integer vectors $v_1=(a_1,b_1),v_2=(a_2,b_2),\dots,v_n=(a_n,b_n)$ and vector $(a,b)$. I need to find integer linear combination, s.t. $c_1v_1+\dots+c_nv_n=(a,b)$. Is there a smart way ...