Questions tagged [linear-programming]

Optimization with a linear objective function, subject to linear equality and linear inequality constraints.

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7
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0answers
169 views

Formulating shortest path as submodular minimization

I'm curious about the general question of whether any combinatorial optimization problem with polynomial time solution can necessarily be reformulated as minimizing a submodular function. The answer ...
7
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319 views

How are basic feasible solutions in linear programming related to vertices in its corresponding polytope?

In Section 2.3.3 "Polytopes and LP" of the book "Combinatorial Optimization: Algorithms and Complexity" by Christos H. Papadimitriou, Theorem 2.4 establishes the relation between bfs's (basic feasible ...
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104 views

Time complexity of linear programming with small number of variables

I have a linear program with $n$ variables, $m$ constraints and $O(nm)$ bit total length (the constraint matrix contains only zeros and ones). I am interested in finding a polynomial time algorithm ...
6
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248 views

What's the complexity of solving a packing LP?

Linear Programming is in polynomial time weakly (when numbers are encoded in unary). AFAIK it remains open if it is possible to solve LP in polynomial time strongly (when numbers are encoded in ...
6
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1answer
503 views

Exponential example for simplex used in SMT solvers

The original simplex algorithm requires an exponential number of pivot operations in the worst case, e.g., if run on the Klee-Minty example [3,4]. What about the simplex algorithm used in SMT solvers ...
5
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43 views

Authors of Complementary Slackness

Who were the first researchers to prove the Complementary Slackness condition for linear programming? I believe that strong optimality was proved by Gale, Kuhn, and Tucker in 1951, but I couldn't ...
5
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39 views

Does it make sense to examine the dual of a feasbility problem?

Consider a standard feasibility problem. The goal is to examine the state of feasible solutions for $Ax=b$ to find an $x$ that satisfies some property. Does the dual of this problem tell us anything ...
4
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206 views

Intuitive self-contained proof of Farkas' Lemma

I've been studying the proof of Farkas' Lemma, and given my rather fuzzy memory of Linear Algebra, am having some trouble with it. One version of Farkas' lemma states: For any convex cone generated ...
4
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121 views

Showing a linear program is infeasible or finding a feasible solution

I'm aware that for any given maximize/minimize LP problem, if its dual is unbounded then the primary is infeasible and vice versa. But what if there is no maximize/minimize objective function? For ...
4
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244 views

Algorithm to optimize polling frequency between producer and consumer

I am trying to optimize what we call AJAX request polling frequency in the domain of web design. Here's a general version of the problem in simple lingo: Problem Statement: Suppose there are 3 ...
3
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415 views

Minimum clique cover

How can the problem of finding the minimal clique cover be solved using linear/integer programming in a reasonable amount of time? Having an undirected graph, I am trying to partition all its ...
3
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158 views

2-approximation edge-cover algorithm using primal-dual method

The problem Given an undirected graph $G=\left(V, E\right)$ and positive edge weights $w_e$, design a 2-approximation algorithm based on the primal-dual principle. So far I managed to represent the ...
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66 views

Computing the line equations of two crossing tangents in a point set separated by a vertical line?

I have provided a picture as an example. We have two point sets, P and Q. P is to the left of this vertical line (named x = x0), and Q is to the right of it. The goal is to compute the line equations ...
3
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53 views

Heuristic for making set of indexes in an array/matrix with generating functions/patterns

I am trying to find a lead on how to solve or find a heuristic the following kind of problem: Given an array/matrix with entries of only 1s and 0s, using a set of looping functions/patterns of a ...
3
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30 views

What is the most efficient algorithm for finding the bounding inequalities of a cone given the extremal rays?

Say that I am given a cone, as specified by the extremal rays whose facets form its convex hull, what is the most efficient algorithm that finds the linear-rank inequalities whose intersection defines ...
3
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97 views

Start simplex method from feasible internal point

I have one algorithm that generates a feasible solution to a linear programming problem. However, it is very likely that this is not a corner point. This makes it not suitable for direct use as an ...
2
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0answers
37 views

Half-integral linear programs

What are some of the known properties of half-integral linear programs? That is, linear programs for which the solution vector always takes its values in the set $\{0, \frac{1}{2}, 1\}^n$. I'm asking ...
2
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93 views

PTAS for Multiple Knapsack with Uniform Capacities, fixed number of Knapsacks

Consider the following problem: We are given a collection of $n$ items $I = \{1,...n\}$, each item has a size $0 < s_i \le 1 $ and a profit $ p_i > 0 $. There are $m$ (a fixed number) of unit-...
2
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28 views

LP - Dual variable is zero implies primal constraint unnecessary?

Say I have a primal program P with n variables and c constraints. Let's say that I have an optimal solution for the dual program D, in which the y1, the variable related to the first constraint in P, ...
2
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67 views

Goemans' Extended Formulation of the Permutahedron And Comparator Networks that are not Sorting Networks

I am interested in using Michel Goemans' extended formulation first developed for the permutahedron to study comparator networks that are not sorting networks. In his paper "Smallest Compact ...
2
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62 views

Relaxations for MILP with logical constraints

I have an LP with a (non-fixed) number of logical constraints in the form of $X_1 \rightarrow X_2$ (where $X_1$ and $X_2$ are linear functions inequalities of the $n$ input variables). To express ...
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76 views

Implementing a linear programming feasibility test in 3D

I have a little problem which requires determining if a system of linear inequalities in 3D is infeasible. The constraints (or oriented planes) are added one by one, so there is an opportunity to stop ...
2
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164 views

How to setup a model for a guillotine cutting stock problem?

Backgroud. I'm reading papers about cutting stock problem (CSP). Said Ben Messaoud, Chengbin Chu, Marie-Laure Espinouse (2008) Characterization and modelling of guillotine constraints. European ...
2
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59 views

Adding linear constraint to continuos LP to improve performance

Consider a standard LP minimization problem of the form $$\begin{array}{ll} \text{minimize} & c^\top x\\ \text{subject to} & A x = b\\ & x \geq 0\end{array}$$ Should I expect, on average,...
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117 views

Formalizing an intuitive linear programming proof

My professor has asked me to prove the following: Prove that we can use an algorithm for linear programming to solve linear inequality feasibility problems. The number of variables and ...
2
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168 views

Why is my Forrest-Tomlin update worse than recomputing LU?

I wrote a simple C++ implementation of the revised simplex method that recomputes the LU decomposition of the basis from scratch on each iteration. I have to solve problems with many variables but few ...
2
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46 views

Implications of the class of problems with parallel solutions being not P-complete for optimization of matrices

I am not a specialist on computational complexity theory. I do work on optimization and I am currently researching about the implications of the class of problems with parallel solutions (NC) being ...
2
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126 views

How to construct a network flow problem?

I have the optimization problem given below max $\sum_{i=1}^{N}\sum_{j=1}^{M} x_{ij}R_{ij}$ s.t $\quad 1)\quad \sum_{j=1}^{M} x_{ij}=1 \quad \forall i$ $\quad 2)\quad x_{ij} \in {0,1}$ $\quad ...
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357 views

General Steiner Tree Variants

In the general Steiner tree problem (Steiner tree in graphs), we are given an edge-weighted graph G = (V, E, w) and a subset S ⊆ V of required vertices. A Steiner tree is a tree in G that spans all ...
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99 views

Facility location on a Sphere with great circle distance

I am looking for an algorithm to find the point that minimizes the sum of the great circle distances to a set of fixed points on a sphere. In more detail: Given $x_1,\ldots, x_k$ fixed points in the ...
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146 views

Practical implications of strongly polynomial time algorithm for linear programming

Why do people care about whether a strongly polynomial time algorithm for linear programming exists or not? Does this have any practical improvement?
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82 views

Issues with an optimization problem

I have an expression $$Ax+By+Cz.$$ where $A$, $B$ and $C$ are positive constants $\ge1$. The variables $x$, $y$ and $z$ are non-negative integers. I am also given a number $T$. I want to find the ...
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1k views

Job assignment problem

I want to solve job assignment problem using Hungarian algorithm of Kuhn and Munkres in case when matrix is not square. Namely we have more jobs than workers. In this case adding additional row is ...
2
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0answers
163 views

Maximum feasible subsystem problem (MaxFS) in 2 variables

Topic: The maximum feasible subsystem problem, which is generally NP-hard [1]. Question: Are there special algorithms in case of only 2 variables (2D linear constraints)? The problem seems to be a ...
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0answers
25 views

Seeking guidance on what to read for Feasibility Binary IP with ''almost total unimodular'' (-1, 0, 1)-Coefficient Matrix and No Obj Function

I am working on an algorithm in graph theory which I wish to prove it's polynomiality/NP-hardness. I am investigating a binary variable (0, 1) integer program which has the coefficient matrix ...
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0answers
15 views

Integrality gap in Online Problems and adaptation to competitive ratio

As we all know, in offline problems it is common practice to calculate the integrality gap to get some bound on the approximation ratio of the integral solution. Now this gap ($IG:=\frac{OPT_{frac}}{...
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0answers
29 views

Gaussian Elimination, find all rows equal to zero

I have a system of integer linear equalities modulo P a large prime. Using GE, I place the matrix in row echelon form to check if there are solutions. After that, I want to check if there is any ...
1
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1answer
34 views

Gaussian Elimination: Check an assignment

This question is similar to Check if a row is in the span of a matrix Suppose I have a matrix $M$ over $GF(2)$ with rows that represent a system of linear equations: A xor B xor C = 1 A xor B xor ...
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0answers
18 views

Why maximum-matching algorithm falls into the category of fill-reducing algorithms?

My understanding is that "maximum matching" (or "maximum transversal") are algorithms to pre-order matrix to increase the numerical stability. In Timothy Davis' book Direct Methods for Sparse Linear ...
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64 views

Can this graph-ordering problem be solved with LP?

I have a modelling problem that I am trying to solve with LP. (More specifically, in Python using PuLP or Pyomo). I am not terribly knowledgeable in this area and have been struggling to find the ...
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0answers
37 views

About Steiner tree problem in graphs

In the paper (p. 3) and the slides presents the formulation of the Steiner problem on graphs via so called Steiner cuts. But according to the definition, the number of Steiner cuts and so the ...
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0answers
20 views

Minimal subset of rows that generate smaller polyhedron

Given a matrix $[A|B]$ I want to find a minimal matrix $[A'|B'] \subseteq [A|B]$ (i.e. the rows in $[A'|B']$ are also in $[A|B]$) such that $A'x < B' \Rightarrow Ax < B$. Geometrically, I want ...
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36 views

Solving a linear program with simplex algorithm, matrix not full rank

I need to solve the following LP $$\begin{array}{ll} \text{minimize} & c^T x\\ \text{subject to} & A x = b\end{array}$$ where $$A = \begin{bmatrix} 1 & 3 &1&0&0 \\ -2&...
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0answers
44 views

An LP with two covering constraints - how to round

I came across an LP with two covering problems, and I wonder how to find a good approximation. For the relevant part of the LP: We have a set $E$ , for each $e\in E$ we have a corresponding set $Y_{e}\...
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0answers
39 views

Time complexity: Using linear programming to solve a system of linear equations

As far as I know, most direct methods for solving linear systems of equations have time copmlexity $O(n^3)$ (where $n$ is the number of variables), with the few methods being faster having huge ...
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0answers
113 views

Confusion about the geometric interpretation of the simplex method for linear programming

In Section 7.6.2 of the textbook "Algorithms" by Sanjoy Dasgupta, Christos Papadimitriou, and Umesh Vazirani, the authors provide a geometric interpretation of the two main tasks of each iteration of ...
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0answers
24 views

Given a set of solutions, find an IP formulation with the same solution set

Input: A list of integer variables $x_1, ..., x_n$. A finite set of feasible solutions $S \subset \mathbb{Z}^n$. Task: Find an integer linear program (IP) on the integer variables $x_1,...,x_n$ ...
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41 views

Optimizing library dimensions

Say I have a library that looks like that: ...
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0answers
77 views

Big M method for continuous variables

Is there any way to model the big M method for continuous variables? Something similar to this but $B, C \in \mathbb{R}_{\geq 0}$ and $A\in\{0,1\}$. Due to the precision problem, when the $B$ and $C$ ...
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0answers
295 views

maximizing absolute value in linear programming

I know that similar questions have been answered several times, and based on the answers, I attempted a solution to my problem. But I simply do not get the right results. The problem is as follows. I ...