Questions tagged [linear-programming]
Optimization with a linear objective function, subject to linear equality and linear inequality constraints.
433 questions
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The length of the shortest $s$-$t$ path equals the maximum tension between $s$ and $t$
I am stuck at the following exercise:
Consider a directed graph $G = (V, A)$ with start vertex $s ∈ V$, target vertex $t \in V$ and weights $w_{ij} \in \mathbb{R}$ for each arc $(i, j)\in A$. For any ...
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Maximize enclosed area of given figures on 2d grid
I need to solve an optimization problem for a given set of polyominoes, for example the five Tetrominoes known from Tetris. The goal is to place each one of the figures on the 2d grid, so the area ...
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Balanced Assignment Problem with updatable cost
I have a problem that can be reduced to an assignment problem. (this is related to some cryptography problems)
Which means we have a set $A$ of $n$ agents and an equal size set $T$ of tasks as well as ...
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How to find optimizers with computer in this kind of minimax problem [closed]
I have a minimax problem of the form $$\max_{\substack{u_1,\dots,u_n \ge 0 \\ u_1+\dots+u_n = 1}} \min_{\substack{v_1,\dots,v_m \ge 0 \\ v_1+\dots+v_m = 1 \\ v_{j_1} \le v_{j_2} \hspace{1mm} \forall (...
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Cutting stock problem upper bound with gilmore and gomory
I am trying to implement this article https://arxiv.org/pdf/1905.04897.pdf (the article is there only for information, no need to read it to answer my question)
At some point they say when talking ...
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Integer linear programming formulation of boolean selection
Given a boolean variable $x$ and nonnegative integer variable $s$, I want to select $y = \begin{cases}
0 & \text{if} \ x = 0 \\
s & \text{if} \ x = 1
\end{cases}$.
Currently in the ...
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Why do we round from 1/2 when converting the LP to ILP for the weighted vertex cover problem?
I understand that to approximate a solution to the weighted vertex cover, we need to relax the integer linear program to a linear program which can be solved in polynomial time, but why do we round ...
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Dual Linear Program of the Densest-Subset Problem
In the densest-subset problem, given an undirected graph $G = (V, E)$, the goal is to
maximise the “edge-density” ratio $|E(S)|/|S|$
over all non-empty sets $S ⊆ V$ , where $E(S)$ denotes the
set of ...
- 453
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Advantages of Integral over Non-integral Linear Program?
I have a linear program over real variables for which it can be shown that all the vertices of the polytope describing its feasible region are integral.
Obviously I can just solve this using a ...
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Proof that using residual network from Ford-Fulkerson will get you min-cut
So I'm following this article and they use the following algorithm to find the min-cut.
Algorithm:
Run Ford-Fulkerson algorithm and consider the final residual graph.
Find the set of vertices that ...
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If greater than or equal to zero then binary variable equals 1: integer linear program
I have a variable $d_{i} \in \mathbb{Z}$ with an upper and lower bound. I also have a binary variable $v_{i}$ which I want to $=1$ if $d_{i} \geq 0$; else $v_{i} = 0$. How do I enforce this as a ...
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ILP - Maximize the number of pairs of variables with the same value
I have a 0-1 integer linear program for a set of $2n$ variables $S = \{x_1, ..., x_n, y_1, ..., y_n\}$. My objective is to maximize the number of pairs $(x_i, y_i)$ such that $x_i = y_i$, $i = 1, ..., ...
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Shortest path as a linear program
I just encountered this formulation of the shortest $s$-$t$ path problem as a linear program in a homework. I don't understand exactly the meaning of the variables and restrictions. Here, $G = (V, E)$ ...
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How are vertex capacities defined in a flow network?
I'm new to network flows and I'm reading this topic from Cormen's Algorithms book (3rd edition) from 26 chapter. I came across this problem from the 26.1 section
Suppose that, in addition to edge ...
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Maximum matching with social distancing
Let $G = (X\cup Y, E)$ be a bipartite graph. Suppose $X$ contains people, $Y$ contains seats in a theatre, and each edge $(x,y)$ has a weight representing how much person $x$ is willing to pay for ...
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How to get the highest score in this game?
I would like some advice in this homework question. There is a three players game, in which each player ($A, B$, and $C$) is given a $n$-length array of integer values. There are $n$ rounds in this ...
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Encoding a binary sequence with shift in MILP
I would like to know if it's actually possible to encode a (binary) sequence with rotations in MILP/MIP.
Given a binary sequence $(0,1,1,0,0,0,0,1)$ and variables $x_0,x_1,x_2,x_3,x_4,x_5,x_6,x_7$
I ...
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Is there an algorithm to solve the following point clustering problem?
According to this post
Given $n$ points $P=\{p_1,p_2,\dots,p_n\}$ in 2D space, and a matrix $D^{n\times n}$ with the distances between each pair of points, we want to partition the points into two ...
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Recommendations on where to learn and practice linear programming?
[CLOSED] Thanks!
I am studying Linear Programming in college but I am facing some difficulties to assimilate some concepts.
So do you have any recommendations of materials to learn or practice Linear ...
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Maximizing an integer Linear problem
i am taking a basic Linear programming class this semester and we've recently started solving integer linear problems. I have a question regarding how to solve one of these exercises.
$$
z = max\: 5x_{...
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Find optimal play by optimizing orders of each player alternatingly
A zero-sum game for two players allows a player to take no action during a turn. Can I reach optimal play (where both players always choose the best possible action in each turn) by the following ...
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Solving linear programming problem with mixed type of constraints
I have a query in solving the problem below:
An automobile company has two factories. One factory has 400 cars (of a certain model)
in stock and the other factory has 300 cars (of the model) in stock. ...
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1/2 Approximation to MAX-DICUT by rounding a linear program
Consider a graph $G=(V, A, w)$, where each arc $(u,v)\in A$ has a non negative weight $w_{u,v} \in \mathbb{R}^+$, partition $V$ into $U$ and $W$, $W=V-U$ such that $\sum_{(i,j)\in A} w_{i,j}z_{i,j}$ ...
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Linear program for min-length pair of edge-disjoint paths problem
Consider a problem: we have an undirected graph $G = (V, E)$, function $l: E \to \mathbb{Z}_{+}$ where $l(e)$ is edge's length $e \in E$, and two vertices $s$ and $t$. And we want to find a pair $(A, ...
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Lower bound on positive coefficients of the optimum of 0,1-linear programming problem
I have an instance linear programming such that the coefficients and the constant terms are 0 or 1.
Formally, the set of variables is denoted as $V$ and $|V| = n$. There are $m$ constraints, formed as
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