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

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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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1answer
25 views

ILP and number of variables in constraints

This question is about the time impact of the length (i.e. number of variables) of the constraints in an Integer Linear Programming formulation. Most people try to reach the minimum number of ...
3
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1answer
71 views

Linear programming formulation of cheapest k-edge path between two nodes

Given a directed graph $G = (V,E)$ with positive edge weights, find the minimum cost path between $s$ and $t$ that traverses exactly $k$ edges. Here is my attempt using a flow network: \begin{align} ...
3
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1answer
82 views

Given an optimal solution to the LP, show how it can be used to construct a directed cycle with minimal directed cycle mean cost

Let $\mathcal G = (\mathcal V, \mathcal A)$ be directed graph with associated edge costs $c_{i,j}$ that has at least one directed cycle. Define the directed cycle mean cost to be $\frac {\{\text {sum ...
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1answer
25 views

Introduction to Linear Optimization: Driving the artificial variables out of the basis (case: no entries in the $j$-row are nonzero)

Reading the book Introduction to Linear Optimization by Bertsimas and Tsiklisis, I've come across the following subject: Driving the artificial variables out of the basis. The description is as ...
2
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1answer
32 views

How can the max-flow and min-cut problems, if dual to one another, both have unbounded optimal value?

The max-flow min-cut theorem states that the value of the maximum flow is equal to the minimum cut capacity. It is possible that the max-flow and min-cut is equal to $\infty$. However, reading ...
3
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1answer
20 views

Dual problem of a maximization primal problem $P$?

Suppose we have a primal problem $P$ which is stated as a maximization problem $\max c^{T} x$. The dual problem is defined (Introduction to Linear Optimization by Dimitris Bertsimas) only for a ...
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1answer
45 views

Assuming finite optimal cost of a specific LP, find an optimal solution directly

Minimize $\sum^n_{i=1} c_i x_i$ subject to $\sum^n_{i=1} a_i x_i = b$ (a single constraint), $x_i \ge 0$. Derive a simple test for feasibility of this problem Assuming the optimal cost is ...
4
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1answer
38 views

Finding a perfect matching using an LP

I have a basic question about the power of Linear Programming that has been bothering me for some time. I believe there is something simple I am missing. Linear Programming is $\mathsf{P}$-complete, ...
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1answer
63 views

Feasible solution existence

I wonder what is the fastest way to check whether the intersection of a set of half-spaces is empty. Right now I'm using a linear programming formulation (with Gurobi as solver) to check if there is ...
4
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1answer
182 views

Proof of Strong Duality Via Farkas Lemma

I am trying to prove what is often titled the strong duality theorem. There is a hint in the book that I'm following, and I want to follow the method they have outlined for me. I will outline the ...
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1answer
62 views

Check constraint under some condition in linear programming

I would like to minimize linear pseudo-boolean function $$\mathrm{obj} = \sum_i c_i \mathrm{sel}_i$$ subject to $$\sum_i c_i sel_i \geq \mathrm{Value} \qquad\qquad(1)$$ where $c_1,\dots c_5, ...
2
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1answer
178 views

Advantage of MTZ problem formulation of TSP

In class, we saw the Miller-Tucker-Zemlin formulation of the Travelling Salesmen Problem (TSP). MTZ is a way of formulating the TSP as an integer linear programming instance. I understand how MTZ ...
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0answers
32 views

What is linear relaxation in the context of bayesian networks? [closed]

To add onto the question, how are elliptical differential equations applicable in this context? I was listening to a talk about Bayesian networks and someone asked if they were using differential ...
2
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0answers
67 views

Boat riddle as a Combinatorial optimization problem?

Hello I study computer science and I just digged into combinatorial optimization and ILP. I remember a riddle abotu a bunch of people on a river bank, and a boat with limited capacity (lets say the ...
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2answers
110 views

Applications for boolean logic operations in zero-one integer linear programming (ILP) [closed]

It is nice to know that every boolean formula can be expressed by zero-one integer programming by this answered question. But are there any applications? To be more precise: Are there papers which ...
2
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1answer
53 views

What's the dual problem of stable matching?

So the dual problem of max-flow is min-cut. What's the dual problem of stable matching?
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1answer
261 views

Is 0-1 integer linear programming NP-hard when $c^T$ is the all-ones vector?

Karp's 21 NP-complete problems show that 0-1 integer linear programming is NP-hard. That is, an integer linear program with binary variables. If we set the $c^T$ vector of the objective $\text ...
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2answers
240 views

Find a binary matrix so that no vector from {-1,0,1}^n is in its kernel

Given integers $n,m$, I want to find a $m \times n$ binary matrix $X$ such that there does not exist any non-zero vector $y \in \{-1,0,1\}^n$ with $Xy=0$ (all operations performed over $\mathbb{Z}$). ...
7
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1answer
562 views

Short and slick proof of the strong duality theorem for linear programming

Consider the linear programs \begin{array}{|ccc|} \hline Primal: & A\vec{x} \leq \vec{b} \hspace{.5cm} & \max \vec{c}^T\vec{x} \\ \hline \end{array} \begin{array}{|ccc|} \hline Dual: & ...
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26 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 ...
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1answer
420 views

How to minimize the sum of difference of element in sub-sequence of array of length k from given sequence of length n

How to minimize the sum of difference of element in sub-sequence of array of length k from given sequence of length n ? for example : for n=10 1 2 3 4 10 20 30 40 100 200 the sub-sequence of length ...
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1answer
264 views

Linear Programming: algorithm to check if ratios can be combined with n bottles to equal a given ratio

Say you have n bottles. each with a ratio of $(a_i: b_i: c_i)$. ($a_i/(a_i+b_i+c_i),\cdots$ swap out the numerator for $b_i$ and $c_i$ respectively). Now you are given a ratio of $(a:b:c)$. Use ...
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2answers
299 views

Filling Rows of a Matrix Subject to Conditions

I'm seeking to write an algorithm which, given a value of N, will fill a matrix consisting of (N+1)(N+2)(N+3)/6 rows and 4 columns with the integers from 0, ... , N, subject to the conditions that: ...
3
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1answer
831 views

Checking Feasibility of Linear Program in Polynomial Time

Given a linear system of the form: $$\begin{array}{c} x_r = a \quad x_j = b \\ c_1x_1 + c_2x_2 + \ldots + c_nx_n = N \\ x_1+x_2 + x_3 + \ldots + x_n = k\\ 0 \le a,b,x_1,x_2,x_3...x_n \le 1\\ k \ge 0 ...
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1answer
155 views

Nash Equilibrium of 2-players game

I have a rather interesting exercise in Game Theory. Assume there is a 2-players game, and player $i$ has $n_i$ pure strategies. The game is given by listing the payoffs for each player for each $n_1 ...
4
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1answer
91 views

Use complementary slackness to prove the LP formulation of max-flow only need polynomial number of path constraints

This is a homework problem for a class that ended 2 years ago, I'm learning it by myself. Consider a directed graph $D=(V,A)$, $s,t\in V$. $A=\{a_1,\ldots,a_n\}$. Let $P=\{p_1,\ldots,p_m\}$ be the ...
2
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0answers
62 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 ...
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175 views

Algorithm to solve job assignment problem

Can someone suggest an algorithm to solve job assignment problem with condition? With condition means that some jobs cannot be done by some workers. For example table as shown below: In this table ...
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1answer
2k views

Express boolean logic operations in zero-one integer linear programming (ILP)

I have an integer linear program (ILP) with some variables $x_i$ that are intended to represent boolean values. The $x_i$'s are constrained to be integers and to hold either 0 or 1 ($0 \le x_i \le ...
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70 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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2answers
84 views

Finding the required value of an algebric expression

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 ...
6
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1answer
677 views

NP complete problems that are solvable in polynomial time if the input (e.g. number of variables) is fixed?

I have seen some problems that are NP-hard but polynomially solvable in fixed dimension. Examples, I think, are Knapsack that is polynomial time solvable if the number of items is fixed and Integer ...
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153 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 ...
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3answers
2k views

Are all Integer Linear Programming problems NP-Hard?

As I understand, the assignment problem is in P as the Hungarian algorithm can solve it in polynomial time - O(n3). I also understand that the assignment problem is an integer linear programming ...
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4answers
625 views

Solving system of linear inequalities

I am trying to solve a system of inequalities in the following form: $\ x_i - x_j \leq w $ I know these inequalities can be solved using Bellman-Ford algorithm. ...
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2answers
132 views

Finding a set of maximally different solutions using linear programming or other optimization technique

Traditionally, linear programming is used to find the one optimal solution to a set of constraints, variables and a goal (all described as linear relationships). Sometimes, when the objective is ...
9
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1answer
231 views

Maximizing a convex function with a linear constraint

The problem is $$\max f(\mathbf{x}) \text{ subject to } \mathbf{Ax} = \mathbf{b}$$ where $f(\mathbf{x}) = \sum_{i=1}^N\sqrt{1+\frac{x_i^4}{(\sum_{i=1}^{N}x_i^2)^2}}$, $\mathbf{x} = ...
3
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1answer
98 views

Why does absence of strongly polynomial time algorithm for LP imply restriction to rational instances?

From Bernhard Korte, Jens Vygen, Combinatorial Optimization, Instances of LINEAR PROGRAMMING are vectors and matrices. Since no strongly polynomial-time algorithm for LINEAR PROGRAMMING is known ...
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1answer
305 views

Find maximum distance between elements given constraints on some

I have a list of numbered elements 1 to N that fit into positions on a number line starting with 1. I also have constraints for these elements: The element 1 is in position 1, and element N must be ...
5
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2answers
196 views

Randomized Rounding of Solutions to Linear Programs

Integer linear programming (ILP) is an incredibly powerful tool in combinatorial optimization. If we can formulate some problem as an instance of an ILP then solvers are guaranteed to find the global ...
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0answers
59 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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1answer
2k views

Formalization of the shortest path algorithm to a linear program

I'm trying to understand a formalization of the shortest path algorithm to a linear programming problem: For a graph $G=(E,V)$, we defined $F(v)=\{e \in E \mid t(e)=v \}$ and $B(v)=\{ e \in E \mid ...
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1answer
797 views

Integer LP formulation and the existence of a solution

A film producer is seeking actors and investors for his new movie. There are $n$ available actors; actor $i$ charges $s_i$ dollars. For funding, there are $m$ available investors. Investor $j$ will ...
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2answers
339 views

A variant of the Assignment Problem

In my variant of the assignment problem I have a set $A$ of agents and a set (of possibly different cardinality) $T$ of tasks. Each agent needs to be assigned exactly $n$ or $n+1$ tasks, and each task ...
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35 views

A variant of the assignment problem (?) [duplicate]

Possible Duplicate: A variant of the Assignment Problem (Not a comp.scientist, but have the basic research. Please excuse me if I've overlooked anything obvious.) In my variant of the ...
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1answer
148 views

Formulating a linear program s.t. only extreme point solutions are found

If there are many solutions to a linear program s.t. the objective function is minimized/maximized (= optimal solutions are on an edge of the polytope), how can I force an LP solver to find only an ...
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1answer
729 views

Sorting as a linear program

A surprising number of problems have fairly natural reductions to linear programming problems. See Chapter 7 of [1] for examples of network flows, bipartite matching, zero-sum games, shortest paths, a ...
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2answers
577 views

Known facets of the Travelling Salesman Problem polytope

For the branch-and-cut method, it is essential to know many facets of the polytopes generated by the problem. However, it is currently one of the hardest problems to actually calculate all facets of ...
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402 views

Minimize the maximum component of a sum of vectors

I'd like to learn something about this optimization problem: For given non-negative whole numbers $a_{i,j,k}$, find a function $f$ minimizing the expression $$\max_k \sum_i a_{i,f(i),k}$$ An example ...