Questions tagged [linear-programming]

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

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2answers
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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 ...
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1answer
1k 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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0answers
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 ...
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5answers
13k 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
3k 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. ...
8
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2answers
588 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 ...
2
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1answer
150 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 ...
4
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1answer
1k 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 ...
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2answers
549 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
147 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 ...
6
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1answer
7k 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 h(...
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1answer
1k 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
918 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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0answers
44 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 problem I ...
4
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1answer
183 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
3k views

Sorting as a linear program

A surprising number of problems have fairly natural reductions to linear programming (LP). See Chapter 7 of [1] for examples such as network flows, bipartite matching, zero-sum games, shortest paths, ...
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2answers
1k 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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2answers
1k 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 ...
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2answers
3k views

Partition of a set of integer into 3 subsets of approximately equal sum

I'm having a very hard time trying to figure out how to solve this problem efficiently. Let me describe how it goes: "A hard working mom bought several fruits with different nutritional values for ...
3
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1answer
463 views

Efficient bandwidth algorithm

Recently I sort of stumbled on a problem of finding an efficient topology given a weighted directed graph. Consider the following scenario: Node 1 is connected to 2,3,4 at 50 Mbps. Node 1 has 100 ...
14
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2answers
455 views

Does every NP problem have a poly-sized ILP formulation?

Since Integer Linear Programming is NP-complete, there is a Karp reduction from any problem in NP to it. I thought this implied that there is always a polynomial-sized ILP formulation for any problem ...
11
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4answers
1k views

Finding exact corner solutions to linear programming using interior point methods

The simplex algorithm walks greedily on the corners of a polytope to find the optimal solution to the linear programming problem. As a result, the answer is always a corner of the polytope. Interior ...