# Questions tagged [integer-programming]

Integer programming regards optimization problems, where one seeks to find integer values for a set of unknowns, that optimizes the objective function. A common subset of this type of problems are integer linear programming problems, where all inequalities, equalities and the objective function are linear in the unknowns.

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### Why are optimization problems often called “programs”?

Why are optimization problems often called programs? linear programming geometric programming convex programming Integer programming ...
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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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### 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 ...
254 views

### On linear integer inequalities with infinitely many solutions

Suppose that a linear system of inequalities $Ax \le b$, where $A\in Z^{m\times n}$ and $b\in Z^m$ have integral coefficients, has an infinite number of integral solutions $x$. Can one conclude that ...
177 views

### What is the probability of interpolating the Tutte polynomial of a planar graph from the values at the two hyperbolas?

The Tutte polynomial is a bivariate polynomial with positive integer coefficient which is a graph invariant and can be defined recursively. Evaluating it is $\#P$-complete even when restricted to (...
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### Is a Parametric Integer Linear Programming Problem eventually quasi-polynomial?

I will consider a family of Integer Linear Programs parametrized by a positive integer $t$. Let $\mathbf{x} = (x_1, \ldots, x_n)$ be the indeterminates. Let $A$ an $m$ by $n$ matrix whose elements ...
813 views

### Speed up Linear programming

I have a linear programming problem like this: minimize $c^t X$ under the constraint that $AX \ge b$. I will need to solve this linear programming problem online many times. I need it to be as fast ...
220 views

### Efficient solutions to general Bézout’s identity $a_1 b_1 + \dots + a_n b_n = 1$

Suppose I have integers $a_1, \dots, a_n$ which are coprime, meaning that $$a_1 b_1 + \dots + a_n b_n = 1$$ has a solution in integers $b_1, \dots, b_n$. I would like to get a bound saying ...
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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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### Algorithm to minimally connect line segments in Euclidean plane

Suppose you have finitely many line segments in the Euclidean plane. How do you "connect them to form one chain of line segments of minimal length?" More formally and generally, what I'm looking for ...
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### Lattice paths in polytopes

Let $P$ be a polytope in $\mathbb{R}^n$. Let $A_ix = b_i$ be the defining equations of its codimension $1$ faces. Is there an algorithm or some kind of criterion to decide if the lattice points inside ...
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### 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 ...
342 views

### What are the definable sets in Skolem arithmetic?

Definable subsets of $\mathbb N$ in the language of Presburger arithmetic are exactly the eventually periodic sets and quantifier free part corresponds to Integer Programming with linear inequalities ...
129 views

### On necessary condition for no integer points in polytope

For a convex polytope $\mathcal K$ in $\Bbb R^n$ presented by $O(n^c)$ linear inequalities is it true that for $|\mathcal K\cap \Bbb Z^n|=0$ it is necessary that at least one axis of John's ellipsoid ...
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### 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 ...
263 views

### 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 ...
119 views

### Choice of MIP (mixed integer programming) solver

I would start using MIP solver for the research on the tiling. I know (heard of) the open source solver jump: https://github.com/JuliaOpt/JuMP.jl and also the gold standard solver from IBM cplex. ...
362 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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### 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 ...
225 views