Questions tagged [linear-programming]

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

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58
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3answers
37k 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 ($0 \le x_i \le 1$)...
24
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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, ...
14
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2answers
468 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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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 ...
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 ...
11
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1answer
1k views

Does linear programming admit a strongly polynomial-time algorithm?

The linear programming problem: find a strongly-polynomial time algorithm which for given matrix A ∈ Rm×n and b ∈ Rm decides whether there exists x ∈ Rn with Ax ≥ b. I know that Steve Smale's lists ...
10
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5answers
14k 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 ...
10
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3answers
2k views

Cast to boolean, for integer linear programming

I want to express the following constraint, in an integer linear program: $$y = \begin{cases} 0 &\text{if } x=0\\ 1 &\text{if } x\ne 0. \end{cases}$$ I already have the integer variables $x,...
10
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1answer
4k 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: & \...
8
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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 ...
8
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2answers
608 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 ...
7
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3answers
9k views

Linear programming with absolute values

I know that sometimes we can use absolute values into the objective functions or constraints. Is it always possible to use them, anywhere ? Example of use of absolute values: ...
7
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3answers
2k views

Finding all solutions to an integer linear programming (ILP) problem

My problem is to find all integer solutions to an ILP. As an example, I'm using an ILP with two variables, but I may have more than two variables. I describe the method I currently use to solve this ...
7
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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 ...
7
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2answers
197 views

Reducing linear programming to positive linear programming

Suppose we have an oracle that solves problems of the form \begin{align*} \text{maximize} ~~ & c^T x \\ \text{subject to} ~~ & A x = b, x\geq 0 \end{align*} when $c\geq 0$ (all coefficients ...
7
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1answer
234 views

Given 2 sets of n points: minimize sum of traveled distances

I am given two sets $S, T$ each of $n$ points in $\mathbb{R}^k$, I want to find a bijection $a : S \rightarrow T$, such that $$\sum_{s \in S} d(s, a(s))$$ gets minimized, with $d$ being the Euclidean ...
7
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0answers
156 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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0answers
252 views

Can the solution to a POMDP be found using linear programming?

It is known that Markov decision processes (MDPs) can be solved using linear programming (see page 24 of Carlos Guestrin's PhD dissertation). The linear program is: $$min_{V(x)} \sum_x \alpha(x)V(x)\\...
6
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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. ...
6
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2answers
952 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 ...
6
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1answer
8k 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 \...
6
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1answer
219 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, ...
6
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1answer
396 views

How does cycling happen in the simplex method?

I'm reading Schrijver's Theory of Linear and Integer Programming, and I have a problem understanding cycling happens in the simplex method. The simplex is described as below: Solving $\max\{cx\mid x \...
6
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1answer
8k 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(...
6
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1answer
445 views

How to find the supremum over all the “good” (interior) polytopes for a given set of 3D points?

Let $S \subset \mathbf{R}^3$ be a set of points in 3D and let $O=(x_0,y_0,z_0)$ be the origin/point of reference. We consider a convex polytope $P$ good / interior if: $P$ is wholly contained ...
6
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0answers
234 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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0answers
276 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 ...
5
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2answers
369 views

Why does this not prove $P\neq NP$?

Fiorini, Massar, Pokutta, Tiwary and De Wolf (Exponential Lower Bounds for Polytopes in Combinatorial Optimization, Journal of the ACM 62(2):article 17, 2015; PDF, ArXiv) show any linear program that ...
5
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2answers
564 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 ...
5
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2answers
282 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 $\mathbb{Z}$). ...
5
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1answer
210 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 ...
5
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3answers
796 views

Maximum set packing and minimum set cover duality

I read that the maximum set packing and the minimum set cover problems are dual of each other when formulated as linear programming problems. By the strong duality theorem, the optimal solution to the ...
5
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0answers
68 views

Uniform Sampling on Intersection of Faces of Simplices [closed]

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints $$A\vec{w}=\vec{b} \ \ and \ \ \vec{w} \geq \vec{...
5
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0answers
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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2answers
1k 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 ...
5
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0answers
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 ...
5
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1answer
463 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 ...
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 ...
4
votes
1answer
78 views

Unfeasible linear program becomes feasible if a variable is removed

Apologies, not a computer scientist by trade but I'm playing with linear programming these days. Let $\{x_i\}$ be $N$ optimization variables with bounds $$l_i \leq x_i \leq u_i$$ I'm interested in ...
4
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1answer
88 views

Finding a minimal width strip which encloses a set of points in the plane

Problem: Consider a set of $n$ points in the plane, how could we find a strip of minimal vertical distance that contains all points? Definitions: A strip is defined by two parallel lines and the ...
4
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2answers
90 views

Does real linear programming produce bipartite perfect matching using maxflow reduction?

Given a bipartite graph the standard reduction to max flow is with the construction similar to following diagram: We can formulate max flow as an linear programming problem with integer variables in ...
4
votes
1answer
124 views

Mistake in a proof of termination phase of Simplex algorithm in CLRS?

There is a pseudo-code for Simplex algorithm in CLRS: The proof consists from three-part loop invariant: Proof We use the following three-part loop invariant: At the start of each iteration ...
4
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1answer
55 views

Is there an algorithm that can find a solution that solves the most number of equations in a linear system of equations?

My apologies if this question makes no sense. I am trying to find an algorithm that can solve a linear system of equations. Unlike most problems like this, this algorithm does not need to find a ...
4
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1answer
435 views

Linear programming restricted to rational coefficients

I'm reading the appendix A of Williamson's "the design of approximation algorithms" about linear programming. In the definition of a linear programming it restricted the coefficients of cost function ...
4
votes
3answers
200 views

Help wrapping my head around a combinatorial optimization problem

Here's the problem I'm trying to solve: I have a bunch of widgets, whose weights vary over a small range. I would like to find the optimal grouping of them such that each group meets a minimum weight ...
4
votes
1answer
184 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 ...
4
votes
1answer
737 views

Maximum matching using linear programming

In a bipartite graph $G = (V,E)$, there is a neat algorithm for finding a maximum matching (or even a maximum-weight matching) using linear programming. It is explained here. The first step is to ...
4
votes
3answers
79 views

Need help optimizing the loading of passengers on small airplanes

I work for a small non-profit that provides transportation for people who need medical treatment. We connect volunteer private pilots who fly people in their own (small) aircraft, typically 3-5 seats. ...
4
votes
3answers
5k views

“Greater than” condition in integer linear program with a binary variable

How can one model the following condition in an integer linear program? $$A = \begin{cases} 1 & \text{if } B > C\\ 0 & \text{otherwise}\end{cases}$$ where $A \in \{0,1\}$ and $B, C \in \...
4
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2answers
392 views

Dynamic Shortest Path with Linear Programming

Consider a grid with $x=5$ columns, $y=5$ rows, and $T$ timesteps. There are $N=2$ agents in this grid, which can move vertically or horizontally. The positions of each agent $x$ at timestep $t$ is ...