# Questions tagged [linear-programming]

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

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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 ...
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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### 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 ...
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 ...
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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...
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 ...
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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### 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 ...
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### 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 ...
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### 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 ...
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 ...
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 ...
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-...
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, ...
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 ...
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 ...
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 ...
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 ...
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,...
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 ...
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 ...
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### 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 ...
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### 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 ...
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### 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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### 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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### 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 ...
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 ...
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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### 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&...