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Easy instance of set cover

I am trying to prove that a natural greedy algorithm solves the following instance of the set cover problem: for a set of elements $e\in U$ with a set of weights $w_e$, we define the cost of a subset ...
Tom Solberg's user avatar
  • 4,049
0 votes
0 answers
39 views

Max-flow modeling with unified vehicle and commodity variables

I am working on a network flow problem that involves routing through a time-space network. The network consists of: A single source node and a single demand node. A fleet of vehicles with specified ...
graphtheory123's user avatar
0 votes
0 answers
36 views

ILPs with square constraint matrix

Given the Integer Linear Programming ($\text{ILP}$) problem \begin{array}{ll} \text{minimize} & c^T x \\ \text{subject to}& \mathbf{A}^T x \ge b \\ \text{where}&c,x,b\in\mathbb{N}_0^n,\\ &...
Manfred Weis's user avatar
  • 13.2k
0 votes
0 answers
64 views

Alternatives to McCormick Envelope

I have an optimization problem for which I have the optimal solution obtained by the ILP. However, when I introduced the McCormick Envelope to replace the product of a bi-linear term in its LP ...
LyLa's user avatar
  • 3
0 votes
1 answer
169 views

How to integrate an indicator function/constraint into the cost function of a linear program?

I have a mathematical model $P$ for which I optimize two cost functions say $F_1$ and $F_2$ subject to a set of constraints $C1$–$C10$. In $F_2$, I want it to be included only when its expression ...
LyLa's user avatar
  • 3
0 votes
1 answer
114 views

Mixed integer program and continuous Diophantine approximation

Let $n\in\mathbb{N}$ such that $n\geq 2$ and let $0<r<1$ be a real number. We wish to solve the following problem. $$\min_{(t,(z_j)_{j=2}^n) \in \mathbb{R}\times \mathbb{Z}^{n-1}} t$$ subject to ...
Pathikrit Basu's user avatar
1 vote
0 answers
96 views

On optimizing a multivariate quadratic function subject to certain conditions

The problem is to maximize $f(x_1,x_2,\cdots,x_n)=\sum\limits_{i=1}^{n}\Big(x_i-k_i\Big)^2$ for $n\ge 3$ subject to the conditions (1) $\sum\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}k_i\le n(n-1)$ ...
shahulhameed's user avatar
0 votes
1 answer
36 views

Benefit of adding a trivial constraint to ILPs

let ILP be an integer linear program with constraints-matrix $\boldsymbol{\mathrm{M}}\in\mathbb{Z}^{m\times n}$ and cost vector $\boldsymbol{\mathrm{c}}\in\mathbb{Z}^n$, ${\boldsymbol{\mathrm{x}}^*}\...
Manfred Weis's user avatar
  • 13.2k
0 votes
1 answer
93 views

How quickly can this IQP or its MILP relaxation be solved

Let $A\in\{0,1\}^{(n,n)}$ be a $n$ by $n$ boolean matrix (in particular think of an adjacency matrix of a graph), and consider the following optimization problem: $$\begin{align*}&&\max_{P\in\{...
alosc's user avatar
  • 71
0 votes
1 answer
320 views

Correct way to conduct equilibrium scaling of linear/integer/MIP program

I would like to scale my linear/integer program and also mixed-integer program using the equilibrium scaling method. I have worked on two research papers and one research book. However, they did the ...
asdf's user avatar
  • 21
0 votes
0 answers
40 views

Subtour-gluing constraints for ILP formulation of TSPs

If one doesn't want to introduce additional variables to the ILP of a TSP instance, one has to add exponentially many so-called subtour-elimination constraints; in practical calculations subtour-...
Manfred Weis's user avatar
  • 13.2k
1 vote
0 answers
37 views

Sum of all integer binary solutions of a TUM linear system

I have the following problem: $A x = b$ where $A$ is a $m \times n$ total unimodular matrix (TUM) with entries in $\{0,1\}$ and $b$ is a $m$-vector of strictly positive integers. Let $\mathcal X$ be ...
Luca Savant's user avatar
2 votes
0 answers
76 views

Polyhedron coordinate bound

Given a polyhedron $$Ax\leq b$$ where we assume $A\in\mathbb Q^{m\times n}$ and $b\in\mathbb Q^{m}$ and it takes $L$ bits to represent the inequalities what is a good bound on the quantity $\|y\|_\...
Turbo's user avatar
  • 13.9k
1 vote
1 answer
628 views

Allowing an "OR" option between equations in a linear program

I am looking for a way to express an "or" option in a system of linear inequalities for a linear program I am working on. I will explain what I mean precisely: Lets say I have a set of ...
Eric_'s user avatar
  • 141
0 votes
0 answers
108 views

Solutions to matrix equations in the non-negative integers

For an integer matrix $S$, and an integer vector $y$, I'm looking for solutions to $xS = y$ where the entries in $x$ are in the non-negative integers. I've been doing this with Sage's mixed integer ...
JonHales's user avatar
  • 101
-1 votes
1 answer
103 views

How to solve MILP problem on several linear subspaces

I have a set of close mixed-integer programming problems. More exactly, all the problems share the same set of (binary and continuous) variables, the same set of linear inequality constraints, and the ...
Nikolay's user avatar
  • 39
1 vote
2 answers
960 views

Integer linear programming (ILP) formulation of connectivity of induced subgraph

Can anyone assist me to find out what should be the ILP formulation of a case when I try to label vertices by say $0$, $1$ and $2$ and want the subgraph of graph $(V,E)$ made by same vertex set but ...
Himanshu Khandelwal's user avatar
0 votes
1 answer
113 views

How do I solve this integer programming problem with non convex constraints?

I am not sure if this is the right place to post this question, please point me to the correct forum if I posted in a wrong place. I have an optimization problem like this ...
Aaron_Geng's user avatar
6 votes
1 answer
861 views

Is Binary Integer Linear Programming solvable in polynomial time?

The paper Solving the Binary Linear Programming Model in Polynomial Time claims that Binary Integer Linear Programming is in P. However, it seems that no subsequent literature in the mainstream has ...
aroyc's user avatar
  • 221
1 vote
0 answers
126 views

Mixed integer formulation of union of polytopes?

Given $t$ different unbounded polyhedra $P_1:A^{(1)}x^{(1)}\leq b^{(1)},\dots,P_t:A^{(t)}x^{(t)}\leq b^{(t)}$ we are looking for the representation of $\bigcup_{i=1}^tP_i$ (not their convex hull) with ...
VS.'s user avatar
  • 1,826
1 vote
0 answers
37 views

Fast certficate of negativity for objective value of mixed-integer linear program

Let $c \in \mathbb R^n$, $A \in \mathbb R^{m \times n}$, $b \in \mathbb R^m$, and $I \subseteq \{1,2,\ldots,n\}$. Consider the Mixed integer linear program (MILP) $$ \begin{split} f^* = &\max \; ...
dohmatob's user avatar
  • 6,853
1 vote
1 answer
330 views

How good is the LP relaxation?

Consider the optimization problem \begin{align} \max_{x\in\mathbb{R}^n}~c^Tx~, \text{ s.t. } Ax=b,~x_i\in\{0,1\}~\forall i \end{align}where $c,b\in\mathbb{R}_{+}^n$ and $A\in\mathbb{R}_{+}^{n\times n}$...
dineshdileep's user avatar
  • 1,421
1 vote
0 answers
74 views

How to minimize n-polytope's bounding box with linear transformation?

I am working on an exact algorithm for integer linear programming for my master's thesis: $Ax\leq b, x \in \mathbb{Z}^n$ $cx\rightarrow min$ For my idea to work out, I need a guarantee that n-...
Иван Шумилов's user avatar
0 votes
1 answer
490 views

Relax a rectangular linear assignment problem

I wonder if there is any literature on the following problem $$\begin{array}{ll} \underset{X \in \mathbb R^{m\times n}}{\text{minimize}} & \displaystyle\sum_{i,j} C_{i,j} X_{i,j}\\ \text{subject ...
Jiaji Huang's user avatar
2 votes
1 answer
502 views

Feasibility Mixed integer Linear programming with quadratic constraints?

Consider the mixed integer program $$Ax\leq b$$ $$By\leq c$$ $$\begin{bmatrix}x&y\end{bmatrix}C\begin{bmatrix}x\\y\end{bmatrix}+D\begin{bmatrix}x\\y\end{bmatrix}\leq d$$ where $x$ are integer ...
Turbo's user avatar
  • 13.9k
-1 votes
2 answers
114 views

On OR condition in Linear Programming with exponentially many constraints [closed]

Suppose we have two linear programs $Ax\leq b$ and $Bx\leq c$ is there a way to combine them into one program of possibly a larger dimension $Cy\leq d$ such that projection of vectors $y$ into a ...
Turbo's user avatar
  • 13.9k
0 votes
0 answers
369 views

Finding a point in the relative interior of the convex hull of a set of integer-valued vectors

Let $X \subset \mathbb{Z}^n$ be the set of integer-valued vectors satisfying a system of linear constraints. We can suppose that $X$ is the set of integral points in a given polyhydral set $Y \subset \...
rasul's user avatar
  • 136
4 votes
1 answer
2k views

Under what conditions does an Integer Programming problem run in polynomial time?

Given $AX\leq B$ where $A\in\Bbb Z^{m\times n}$,$B\in\Bbb Z^m$ finding $X\in\Bbb Z^n$ where $m\geq n$ is the integer programming problem. If $A$ is totally unimodular then the problem is solvable in ...
Turbo's user avatar
  • 13.9k
2 votes
0 answers
80 views

Making a polyhedron integral by selecting value for a specific co-ordinate of constraint vector

I am currently trying to solve a binary integer programming(maximization) problem, where the first row of the constraint matrix corresponds to the constraints on the total number of 1's in the vector ...
A.2's user avatar
  • 123
3 votes
2 answers
1k views

Equality constraints in mixed-integer optimization

Suppose I have a linear mixed-integer optimization problem of the form $$MIP: min_{(x,y) \in \mathbb{R}^n \times \mathbb{Z}^m} c^\top x + d^\top y \hspace{0.2cm} \text{s.t.} \hspace{0.1cm} Ax+By \leq ...
Christoph Neumann's user avatar
3 votes
1 answer
368 views

Lot sizing problem: how to add these cuts efficiently

Consider the set of constraints of the uncapacitated lot sizing problem: $$ \{(x,s,y)\in \mathbb{R}^n_+ \times \mathbb{R}^n_+ \times \mathbb{B}^n \;|\;s_{t-1}+x_t = d_t+s_t,\; x_t \le My_t,\; t=1,\...
Kuifje's user avatar
  • 225
0 votes
0 answers
68 views

A seemingly easy integer programming question

Let $k, m \in \mathbb{Z}_{ > 1}$. Let $a \in \mathbb{Z}_{> 0}^m$ and $t \in \mathbb{Z}^k$. Let $\varepsilon = (\varepsilon_{i,j})_{1 \leq i \leq m \\1 \leq j \leq k}$ be a matrix with entries in ...
Alex's user avatar
  • 501
0 votes
0 answers
890 views

Maximum shortest path problem

I have the following problem. You have a graph and every edge has a certain set of possible weights. The question is to find the assignment of those weight which will maximize the shortest path. In ...
Eugene's user avatar
  • 342
1 vote
0 answers
187 views

Strong Duality of Mixed Integer Linear Program

The problem at hand is to optimize a mixed-integer linear program closely related to the maximum flow problem. I would like to reformulate the problem with its dual and I'm concerned with the ...
Amitai G's user avatar
3 votes
2 answers
1k views

SDP relaxation vs LP relaxation

I have a question I hope you might be able to answer. Let's say we have an integer program for the stable set problem (or clique, not principal). \begin{equation} \begin{aligned} & \text{...
Eugene's user avatar
  • 342
2 votes
1 answer
529 views

Integer programming and Groebner basis

I enjoyed reading different papers about using Groebner basis to solve integer programming. Is there any literature about the complexity and/or comparison with other (more classical) methods like ...
teller's user avatar
  • 337
6 votes
1 answer
263 views

Algorithm that solves every Mixed Integer Linear Program (to optimality)?

Given a Mixed Integer Linear Program with rational coefficients (both for the objective functions and all constraints), is it always possible to solve it algorithmically? I know that you usually ...
J Fabian Meier's user avatar
2 votes
0 answers
112 views

Pcross-like, nonogram-like in near-linear time [closed]

I have a problem with a puzzle game like pcross in which I have a nxn square: At any index of rows and columns I have an integer that say the maximum numbers of points that I can place in that row/col....
Steve Xibalba Rivieccio's user avatar
2 votes
0 answers
154 views

Listing all Lattice Points in a Box

Let $B := [-1,1]^n$ be an $n$-dimensional box. Moreover, let $v_1,\ldots,v_n \in \mathbb{R}^n$ form a basis of $\mathbb{R}^n$, where the entries of the $v_i$ are explicitly irrational. We can assume ...
User133713's user avatar
1 vote
0 answers
120 views

The column generation technique on a Train Unit Assignment Problem [Linear Programming]

I am doing an assignment where I need to implement a mathematical model that I can't wrap my head around. For the technique of column generation, one would need to my understanding, a master problem ...
k88's user avatar
  • 111
5 votes
1 answer
2k 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 ...
Xoph's user avatar
  • 153
0 votes
1 answer
270 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 $$\text{...
Kevin Jenkins's user avatar
6 votes
1 answer
1k 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 ...
Robert's user avatar
  • 83
2 votes
1 answer
143 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 ...
Robert's user avatar
  • 83
0 votes
2 answers
708 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 ...
user avatar
3 votes
1 answer
340 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 $...
JensLans's user avatar
2 votes
0 answers
120 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 ...
Ali's user avatar
  • 127
1 vote
0 answers
75 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? ...
GLG's user avatar
  • 111
2 votes
2 answers
841 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 ...
user avatar
1 vote
0 answers
493 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: $$\max(\...
Manfred Weis's user avatar
  • 13.2k