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 ...

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71 views

Sufficient condition for solvability of linear diophantine system

I would like to know under what conditions does an integer solution exist to the under-determined linear system: Ax = b. (without constraints) Where A is m x n matrix with positive integers entries ...
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79 views

Complexity :: Integer Programming :: Non-Poly Example [closed]

When learning about computational complexity I find that when discussing the NP-Complete problems authors always give examples of such problems that can in fact be solved in poly time. I understand ...
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85 views

Global and local maxima in a weighted sum of logarithms of linear functionals?

Constrained Optimization Problem Is is possible to describe, and locate efficiently, the maxima of the function $f$, as described below in the parameters $\mathbf{x} = (x_1,...,x_m)$. The constraints ...
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0answers
38 views

Least square problems with binary variables

I want to solve the heat equation $T_t(x,t) = - L_x . T(x,t) + F(x,t)$ in an edge-weighted graph where $L_x = \sum_i x_i e_{ij}$ is weighted Laplacian matrix of the graph. Then I conclude to the ...
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1answer
128 views

basis of the lattice generated by the integer points inside a subspace of R^L

Consider $K$ linearly independent vectors $\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{a}_K \in \mathbb{Z}^L$, where $1 \leq K<L $. Hence, the span of $\lbrace\mathbf{a}_1, \mathbf{a}_2, ..., ...
5
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1answer
202 views

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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2answers
146 views

Combinatorial optimization problem involving infinite spin system

In material science research, I am developing an algorithm to solve an infinite combinatorial optimization problem which I believe is the most natural problem when the system size goes to infinity. ...
0
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1answer
74 views

Generalized assignment problem with no integrality gap

Suppose I am solving the generalized assignment problem, so that I am given matrices $U$ and $W$ and a vector $c$ (all three of which have, say, positive entries), and I want to solve ...
6
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1answer
201 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 ...
2
votes
1answer
80 views

Find base of kernel with as many 0 as possible

I have a 400x132 rectangular matrix with only 0 and 1. I am looking for the linear combinations of the columns of the matrix that sum to 0. For example C1 + C2 - C3 = 0. I want to find the linear ...
2
votes
0answers
51 views

Reference request: Edmond's Algorithm for integer hull

I'm looking for a good reference for the algorithm (supposedly by Edmonds) to compute the integer hull of a polytope, not by cutting plane methods but by starting with a set of integer points and then ...
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0answers
161 views

Both NP-hard but different [closed]

What's the fundamental difference between the Knapsack problem and the travelling salesman (TSP) problem both of which are NP-hard, while the reality is that TSP could be solved much much faster?
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2answers
296 views

Mixed integer programming formulation for Ising model

I want to implement a minimisation on a 2D spin Ising model with 30x30 grid. The spin variables is 0,1 and the objective is to minimize the sum of products of spins. For simplicity, I only include NN ...
4
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0answers
45 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 ...
0
votes
2answers
147 views

Approximate solution to large mixed integer programming problem

What are the available approaches to find an approximate solution to a large mixed integer programming problem? I ran my problem in the Gurobi MIP solver. It can find a feasible solution in ...
3
votes
0answers
77 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. ...
2
votes
1answer
303 views

Name search for special Linear Integer Program

I am looking for a name for the following question in literature! (and if you know it, then it would be great) I couldn't find it and due to wide audience here, presumably you know more. Thank you ...
1
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0answers
64 views

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 ...
3
votes
2answers
359 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 ...
0
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0answers
89 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 ...
1
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0answers
62 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? ...
2
votes
2answers
106 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 ...
2
votes
2answers
207 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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0answers
133 views

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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0answers
281 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 ...
4
votes
2answers
456 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 ...
2
votes
1answer
435 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 ...
2
votes
0answers
133 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 ...
3
votes
0answers
210 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 ...
2
votes
2answers
98 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 ...
1
vote
1answer
118 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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0answers
59 views

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 ...
5
votes
2answers
487 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 ...
1
vote
1answer
369 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 ...
1
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2answers
262 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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1answer
3k views

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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0answers
197 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 ...
0
votes
1answer
228 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 ...
2
votes
2answers
1k 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 ...
2
votes
1answer
1k 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 ...
0
votes
0answers
231 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 ...
4
votes
0answers
299 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 ...
2
votes
2answers
468 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 ...
1
vote
1answer
1k 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 ...
3
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0answers
856 views

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 ...
4
votes
3answers
804 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 ...
0
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0answers
121 views

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 ...
3
votes
1answer
274 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 ...
5
votes
2answers
465 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 ...
5
votes
1answer
780 views

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 ...