The integer-programming tag has no wiki summary.

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### integrality of a linear program — binary equality constaints

Consider the following linear program:
$\left\{
\begin{array}{l}
\underset{x}{max} \;\;c^Tx\\
[I, \;B]x = \mathbf{1}\\
x\geq 0
\end{array}
\right.$
where $c$ is a vector ...

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185 views

### Finding closest point to a set of circles

My requirement is to find the point closest to three circles. So lets say the three circles are C1, C2, C3. I want to find the point in the space such that the SUM of its distance from C1, C2 and C3 ...

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81 views

### Linear system with many solutions from a finite set

Basically I am looking for a linear system with
many solutions from a finite set.
Choose a finite set of rationals $S$ and fix
positive integer $k$.
Let $A$ be a linear system with $n$ variables ...

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41 views

### Are there any known bounds on the value of solutions of linear integer programming?

Given a linear objective function and a system of linear constraints; are there any known bounds on the values of (positive) integral solutions in terms of the coefficient matrix of the constraints?
...

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### Multi-objective set-cover optimisation problem

I'm looking for an algorithm to solve the following multi-objective set-cover problem.
We start with a 'universe' (set) of items $\mathcal{U}$, along with a partitioning $P = \{p_0,\ldots,p_m\}$ ...

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79 views

### Find the optimal set of subsets

Consider a set of $N$ individuals and let their distance be given by $R$, a $N\times N$ matrix. In that, $R(1,2)$ is the distance between individual 1 and 2. Now lets say that I want to separate the ...

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101 views

### Finding the maximum of a multivariate polynomial of degree one

I need to find the global maximum of the function
\begin{align}
f\left(x\right) & = p_1 \max\left(\sum a_{1i} x_{1i}, \sum b_{1i} x_{1i}\right) - \sum c_{1i} x_{1i} \\
&+\ldots \\
&+ p_n ...

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### Complexity of Nested Linear Optimization

My question is motivated by the fact, that among other ways, it is possible to restrict a variable to two discrete values, e.g. the prototypical $0$ and $1$, via an optimization constraint:
...

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128 views

### Subtour Elimination in Travelling Salesman Problem using MTZ

I am trying to formulation a problem similar to a Traveling Salesman with Time Window constraints.
To eliminate subtours, I need to use some constraint similar to a generalization of MTZ constraints ...

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332 views

### Simplified knapsack problem

There is a problem that I can not solve.
Given a set of items (each item has some integer weight) we have to fill bag with some number of copies of these items, with the only restriction that the ...

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235 views

### Finding integer points inside of a parallelogram

Suppose $P = \{p_1,\ldots,p_4\} \in \mathbb{R}^2$ defines a quadrilateral (here, specifically, a parallelogram). In the particular case I'm dealing with, I know that there exists at least one point ...

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111 views

### existence of lattice point in polytope

This question was probably asked before but here goes. I have a convex polytope given by $Ax\leq b$ for a specific integer matrix $A$ and integer vector $b$. I need a simple method/result on how to ...

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158 views

### Beating Kadane's Algorithm

I am seeking some reference on already existing work for the following problem.
Given an $n$-dimensional square matrix $A=DP$ where $D$ is a diagonal and $P$ is a permutation matrix (think of Gaussian ...

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93 views

### LP constraint enconding

I have an objective function to be maximized
$obj(x) = \sum_i \gamma_i x_i$ with $x_i \in \mathbb{R}$
With multiple constraints of the form:
$\min_{y \in 0,1} (\sum_{i \in A} \alpha_i x_i + \sum_{i ...

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**1**answer

101 views

### Separation of Anti-Hole Inequality

Given an undirected graph $G=(V,E)$ with no loops or multiple edges, a stable set is a set of vertices for which no two vertices are adjacent.
An induced subgraph $H$ of $G$ is called an odd-antihole ...

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### What is the solution to \min_k k \frac{b^k/n}{\lfloor b^k/n \rfloor}

(I've posted this question at Math.SE but got no answer, so I hope I can get a solution here.)
This problem looks familiar, but I don't remember its solution:
$$ \min_k \ \ \frac{b^k/n}{\lfloor ...

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460 views

### Area of a lattice polygon in terms of its width

Let $M$ be a lattice polygon on a plane (i.e. its vertices are integer points $(i,j)\in\mathbb Z^2$).
Let us define lattice width in a direction $v=(m,n)\in\mathbb Z^2$ as $w_v(M)=\max\limits_{x,y\in ...

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**1**answer

325 views

### Reference Request for Integer factorization with LP/ILP

Have there been attempts to factor integers with Linear Programming?
Searching the internet suggests that for Integer Factorization only Number Theoretic algorithms, like sieves, are taken into ...

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199 views

### Brute force lattice problems

What are the easiest brute force algorithms for solving closest and shortest vector problems?
I want to find an arbitrary, but small ($\lesssim 20$) number of lattice vectors closest to a given ...

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### If then condition on mixed linear integer programming [closed]

Hi all. Let $a$ and $b$ be two real variables such that $0 \le a \le a_{max}$ and $0 \le b \le b_{max}$. I must write the following if-then-else condition with linear inequalities:
if $a < ...

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162 views

### LP relaxation for ILP\IP (integer linear programming)

I am familiar with LP relaxation for ILP (or IP). Assume we concern with integer minimization problem, which we formalize using ILP; we then relax the ILP into LP and we say that the LP provides a ...

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142 views

### $\ell_o$ Minimization (Minimizing the support of a vector)

I have been looking into the problem
$\min: \|x \|_0$ subject to$: Ax=b$. $\|x \|_0$ is not a linear function and can't be solved as a linear (or integer) program in its current form. Most of my time ...

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629 views

### is there a solution to system of linear Diophantine equations?

I have a matrix A \in Z^{n \by m}, where m > n and a vector b \in Z^n. Then, under what conditions does an integer solution exist to the equation
Ax = b.
Is there a way to bound the norm of the ...

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446 views

### sum of maxima vs the maximum of the sum

Consider the following integer program
$$
\begin{align}
\max &\sum\nolimits_{i}\sum\nolimits_{j} U_i(j)\cdot x_{i,j}\\
\text{subject to}& \sum_{i}x_{i,j}\cdot f\left(i,j\right)\leqslant ...

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172 views

### Maximum subset of set of Integers with minimum distance

Hi, i have a set of integers for example: {0,1,3,100,102} and i am looking for a maximum subset in which all elements have a minimum distance to all elements (or the "next" doesnt matter i guess) for ...

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278 views

### Matching a binary matrix

Given a MxN 0-1 matrix D, with the property that
both M and N are odd numbers
its row sums and column sums in the $\mathbb{Z}_2$ field are all equal to the same number (0 or 1).
How do we find M ...

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302 views

### Sherali-Adams relaxation

I am trying to find a book or a paper, which explains, how and why the Sherali-Adams relaxation method works. The original paper (1990) is difficult for me to understand. I need a more basic ...

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611 views

### Multiple disjoint subset sum problem

Given two sets of nonnegative integer numbers:
$X = {x_1, x_2, ... x_n}$
$Y = {y_1, y_2, ... y_m}$
Need to find partition of $X$ on $m$ disjoint subsets, such as sum of elements in $i$-th subset ...

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### 0,1 solution to system of linear integer equations.

I have the following problem:
$A x = b$
where $A, b$ - $m \times n$-maxtrix and $m$-vector of nonnegative intgers (respectivelly).
$x \in \{0,1\}^n $ - vector of binary variables, which need to be ...

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622 views

### Lattice points close to a line

Take a sheet of grid paper and draw a straight line in any direction from the origin. What is the closest non-zero grid point $\boldsymbol{p}\in\mathbb{Z}^2$ within a distance $\epsilon>0$ of the ...

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### Question on non-linear parametric mixed integer program

I am trying to solve a mixed integer minimization problem, where there are a number of parameters, and there are products of parameters with variables appearing in the objective function. I assume ...

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247 views

### Partially optimal solutions in integer linear programming

Linear programs with a totally unimodular system matrix are known to have an optimal integer point. They are therefore solvable via relaxing the integer constraints to intervals.
An other interesting ...

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451 views

### Some weird “system” of inequalities in nonnegative integers.

Suppose I have a bunch of nonnegative integers $(a_{ijkl})_{1 \leq i \leq j \leq k \leq l \leq 17}$ such that for all 17-tuples nonnegative integers $w_t$ (for $1 \leq t \leq 17$) we have that ...

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### Minimum distance between adjacent concentric circles that cross integer lattice points

This problem looks simple, but I searched around and couldn't find any similar problems or related resources. Hope someone could provide a clue or at least a hint of what class of prolbems it belongs ...

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207 views

### Representing integer points inside a polytope using a unit hypercube

Let $D\subset\mathbb{Z}\times\mathbb{Z}$ such that $D$ is formed by the points bounded by a convex polytope. Let $f:D\to\mathbb{R}$ defined by $f(x,y)=ax+by+c$, where $a,b,c\in\mathbb{R}$ and ...

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206 views

### Spectral analysis of sparse symmetric integer matrices

Hi all,
A project I'm currently working on requires me to compute the eigenvectors / eigenvalues of sparse symmetric integer matrices. This is needed in the context of Principal Component Analysis. I ...

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139 views

### Knapsack Constraint

I'm trying to implement a recursive algorithm that I came up with that first solves the knapsack for a given objective and then cuts off the solution and then finds the next best solution.
However, I ...

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### Domination in Nice Lattices

Let an integer vector be nice when it has only two nonzero components, which sum to zero. So (0, 0, 3, 0, -3) and (-1, 0, 1, 0, 0) are examples of nice vectors in $n=5$ dimensions.
Call a lattice ...

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252 views

### The number of hyperplanes determining the integer points of a polyhedron

This question is inspired by this one.
Let $P \subset \mathbb R_{+}^n$ be a convex polyhedron whose complement in $\mathbb R_{+}^n$ has finite volume. Let $Int(P) = P \cap \mathbb N^n$. (For ...

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344 views

### How to solve this integer programming problem?

I have a sequence of matrices $\lbrace A_i \rbrace_{i=1}^N$ and I want to select a column from each of these matrices so that the following sum is minimized:
$\sum_{i=1}^N || A_{i} \vec{x_{i}}- ...

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362 views

### On special type polynomial inequalities over integers

A special monomial is a monomial of the form $C\cdot x_{i_1} \cdot \ldots \cdot x_{i_n}$, where C is an integer and no variable is repeated more than once in the monomial. For instance, $x\cdot y\cdot ...

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### Symmetry of the integer gap

Are there results that bound the asymmetry of the duality gap of an integer program? That is to say, if the difference between the LP solution and the IP (primal) solution is $a$, is there a function ...

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### How to solve Linear Programming problem with tighter Integer Programming constraints

I want to learn a bit about Linear Programming.
After some research, I decided to solve the Cutting Stock problem as an example to learn. After doing some more research, I feel like I finally ...

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### Proving that a binary matrix is totally unimodular

I'm working on a set of problems for which I can formulate binary integer programs. When I solve the linear relaxations of these problems, I always get integer solutions. I would like to prove that ...

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426 views

### When is a triangular matrix totally unimodular?

I have an {0,1}, invertible, triangular matrix, that I would like to show to be totally unimodular. Are there any known results on the total unimodularity of classes of triangular matrices?

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384 views

### non-linear mixed integer programming question

I tried this question over in the algorithms section of stackoverflow and never really got a handle on the problem. I know it concerns non-linear mixed integer programming.
[In the following, 1...n ...

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### Condition for existence of certain lattice points on polytopes

Let $a_1,\cdots, a_n$ be integers such that $a_i\geq 2$ for all $i$ and $k>0$ another integer.
I am interested in whether there exist integers $x_1,\cdots, x_n$ with $0<x_i<a_i$ satisfying:
...

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157 views

### linear program with zeros

Hi, I want to be able to solve a linear program that has constraints that are either zero or a range. An example below in LP_Solve-like syntax shows what I want to do. This doesnt work. In general all ...

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### Hermite normal form in families

How does Hermite normal form (over $Z$) vary in families? I.e. if I have an $n\times m$ matrix $M$ whose entries are integral polynomials in some integral variable $x$, how does the Hermite normal ...