Questions tagged [linear-programming]
Mathematical and computational method for finding the best outcome in a given mathematical model where the list of requirements is represented as linear relationships.
198
questions
3
votes
1
answer
166
views
Can the ellipsoid method be used with a randomized separation oracle?
Suppose we are trying to solve the following optimization problem:
$$
\text{maximize } ~~ c\cdot y
\\
\text{subject to } ~~ y\in S
$$
where the region $S$ is described by an exponential number of ...
- 41
0
votes
0
answers
85
views
How to prove that a given class of convex programs cannot be solved by linear programming?
Given the following program, where $f, g$ are convex functions:
$$
\text{minimize}~~ f(x)
\\
\text{subject to}~~ g(x)\leq 0
$$
the problem can be solved by convex programming algorithms, but it would ...
- 1,880
3
votes
1
answer
169
views
Solving linear programs with special structure
We have an application and at some point we need to solve a linear programming problem that looks like this:
$$
\min\ w_{1,2} + w_{3,4} + w_{5,6}\\
x_i - x_j \leq c_{ij},\ \forall\ (i,j) \in C\\
x_1 - ...
- 73
1
vote
0
answers
89
views
Max-flow while restricting flow on subsets of edges to be equal
Consider a max-flow network $G(V,E,c)$. Consider disjoint subsets of the arcs $R_1,…, R_k$. I want to find the maximum $s-t$ flow such that all the edges in the $R_i$ have equal flow.
Obviously, this ...
- 812
2
votes
0
answers
67
views
Parallel complexity of fixed dimension fixed constraints integer programming
Papadimitriou in https://lara.epfl.ch/w/_media/papadimitriou81complexityintegerprogramming.pdf shows ILP is fixed parameter tractable in number of constraints and Lenstra in https://people.csail.mit....
- 12.6k
0
votes
0
answers
35
views
What is the tightest lower bound known for the integrality gap of the Gilmore–Gomory LP for Bin Packing?
I know that it has been conjectured that the Gilmore–Gomory LP for 1-D BP (also known as configuration LP) has
Modified Integer Roundup Property, i.e., Opt ≤ ⌈Opt_f⌉ + 1. However, I could not find the ...
- 73
0
votes
0
answers
54
views
How can we convert from Robust Linear Programming to SOCP?
Problem establishment
Suppose that in inequality constraint $i$ the true values $\tilde{a}_{ij}, j\in J_i$, of uncertain data can range in the interval $[a_{ij}-\epsilon|a_{ij}|,a_{ij}+\epsilon|a_{ij}|...
0
votes
0
answers
97
views
Separation oracle for breaking cycles in directed graph
I am working on a directed graph problem and am collaterally interested to know whether there is a separation oracle for the following set of linear constraints.
We are given a directed graph $G$ ...
- 89
2
votes
1
answer
125
views
Is there an approximate version of the strong duality theorem for linear programming?
Consider the following dual linear programs:
$$
\min \mathbf{c^T x} ~~ \text{s.t.} ~~ A \mathbf{x} \geq \mathbf{b}, \mathbf{x}\geq 0;
\\
\max \mathbf{b^T y} ~~ \text{s.t.} ~~ A^T \mathbf{y} \leq \...
- 1,880
6
votes
1
answer
258
views
Is there a simplex-like algorithm that can be used with a separation oracle?
Linear programs can be solved in polynomial time using the ellipsoid method, but in practice the Simplex method is much more efficient, and the smoothed analysis framework of Spielman and Teng ...
- 1,880
5
votes
1
answer
219
views
What is known about (upper bounds on) the LP gap of the (symmetric) Travelling salesman in special instances?
What is known about the LP gap of (the natural Held-Karp relaxation of) the (symmetric) Travelling salesman in special instances?
I'm only aware of one special case where the extreme points are all ...
- 227
5
votes
0
answers
102
views
Why is the ellipsoid method numerically unstable?
In the Ellipsoid method wikipedia entry under the performance section, it is mentioned that the Ellipsoid method often times is numerically unstable in practice: "On even "small"-sized ...
- 812
0
votes
1
answer
178
views
Does an upper bound on the integrality gap imply an approximation algorithm with the same ratio?
Often, we can model combinatorial optimization problems with an Integer Program. Then there is an associated Linear Relaxation which drops the integrality constraints on the variables.
Let's say we ...
- 577
1
vote
0
answers
113
views
Is this proof of $LP$ being in $coNP$ correct?
I am referring to the natural decision version of the Linear Programming problem: given $A \in \mathbb{Q}^{m \times n}, \ b \in \mathbb{Q}^m, \ c \in \mathbb{Q}^n, \ \alpha \in \mathbb{Q}$, does there ...
- 89
2
votes
1
answer
199
views
Characterization of integral polyhedra
A rational polyhedron $P \subseteq \mathbb{R}^n$ is an integral polyhedron if it is the convex hull of its integer points. That is, if $P = conv(P \cap \mathbb{Z}^n)$.
Equivalently, $P$ is integral if ...
- 577
3
votes
0
answers
76
views
Using Baire Category to analyze the efficiency of the Simplex Method
I read from the wiki page of the Simplex Algorithm that we can "use Baire category theory from general topology, and to show that (topologically) "most" matrices can be solved by the ...
- 131
0
votes
0
answers
86
views
Prove that this linear relaxation has half-integral extreme points
Given a graph $G=(V,E)$, here is a Linear Relaxation of the edge cover polytope:
(1) For each $v \in V, \sum_{e \in \delta(v)} x_e \geq 1.$
(2) For each $e \in E$, $0 \leq x_e \leq 1.$
Here $\delta(S)$...
- 577
4
votes
0
answers
133
views
Complexity of real coefficients Linear Programs
I would like to know if there are known any polynomial time algorithms for deciding the feasibility of linear programs with real (not integers) coefficients.
I know that for linear programs with ...
- 141
3
votes
0
answers
150
views
(Integer) Linear Program formulation of planarity?
Q: Is there an efficient (I)LP formulation of planarity?
More specifically, I am looking for a set of constraints that are satisfied by exactly all planar graphs on $n$ vertices, in order to optimize ...
- 241
2
votes
1
answer
184
views
Is this homework problem on T-joins wrong? [closed]
In Question 9.3a, it states that if $T=V$, then the minimum cost perfect matching is the minimum cost T-join. Is this actually true? I think I have a counterexample which I have drawn below.
- 577
4
votes
1
answer
295
views
Proof of $LP$ is in $coNP$ without showing it is in $P$?
Is there a proof that linear programming is in $coNP$ without showing it is in $P$?
If so what is the strategy?
- 12.6k
5
votes
1
answer
84
views
Generate cut $(A,B)$ in edge-colored graph $(V,E_1 \cup E_2)$ such that there are more red than white crossings, i.e $|E_1(A,B)| > |E_2(A,B)|$
Let $G=(V,E)$ be graph. Recall that a cut of $G$ is (or can uniquely be identified with) a pair $(A,B)$ of nonempty subsets of $V$ which partition it. Given a cut $(A,B)$, let $E(A,B) := \{(a,b) \in ...
- 227
5
votes
0
answers
85
views
reference request: greedy algorithm for fractional interval covering
Reference Request
I've found a natural greedy algorithm for the problem below. My question is: what is already known about fast algorithms for this problem (faster than general linear programming, ...
- 9,223
1
vote
1
answer
111
views
Ensuring integral maximizer from integral linear program
An integral linear program is one that has a maximizer that is integral. Sometimes it's possible to prove that a particular LP has this property, for example by proving that it's constraint matrix is ...
- 690
1
vote
0
answers
119
views
Is the edge cover polytope integral on graphs with self-loops?
It is well known that the edge cover polytope is integral on simple graphs. I am wondering whether this also holds for graphs with self-loops.
Here is a Linear Relaxation of the edge cover polytope, ...
- 577
1
vote
0
answers
155
views
Is the matching polytope integral?
In this document https://courses.engr.illinois.edu/cs598csc/sp2010/Lectures/Lecture9.pdf
they prove the integrality of the matching polytope using the integrality of the perfect matching polytope.
The ...
- 577
1
vote
0
answers
144
views
Separation oracle for hitting all small cut on a graph?
We are given as input an undirected graph $G=(V,E)$, weights $w_e \ge 0$ for all $e\in E$ and an positive integer $k$. We aim to select a set of edges with the minimum weight, such that the cut set of ...
- 451
-2
votes
2
answers
432
views
Proof that optimal solutions of LP Relaxation of independent set are half-integral
I saw somewhere that optimal solutions of LP Relaxation of independent set are half-integral, by what I mean the possible values of a solution are ${ \{0,0.5,1 \} }$. I'm looking for proof of that.
...
1
vote
1
answer
228
views
Intuition behind the Charikar's LP formulation for densest subgraph problem
I understand why the LP gives the optimal solution for the densest subgraph problem. But don't understand the intuition behind the LP in this paper.
Just mentioning the LP for maximum density of a ...
- 23
1
vote
1
answer
182
views
Minimum non-zero variable in the optimal solution of linear programming
Suppose we have a linear programming about the vertex packing of a hypergraph G=(V,E), with size $n = \sum_{e \in E}|e|$. We introduce a variable $x_v$ for each vertex $v \in V$.
$$\max \sum_{v} x_v ...
- 49
0
votes
0
answers
152
views
linear programming with non-integer constraints
Suppose we have a linear progamming about vertex packing of a hypergraph (V,E), with size $n = \sum_{e \in E} |e|$. We introduce a variable $x_v$ for each vertex $v \in V$. The fractional version is ...
- 49
-1
votes
1
answer
79
views
Is it possible to approximate the solution of NP-Hard problems in polynomial time using linear programming? [closed]
Suppose we have a NP-Hard problem such as the k-col, which is meant to determine if a graph may be colored using at most ...
- 103
3
votes
1
answer
195
views
Minimal number of hyperplanes needed to separate sets of points from one other set
Let $\mathbb{R}^d$ be our space. We have a single good set of points $g$, and a collection of bad sets of points $B$.
We assume that for all $b \in B$ the convex hulls of $g$ and $b$ are disjoint. ...
- 690
1
vote
0
answers
77
views
Time complexity of alternation free quantified linear program with no free variables and only existential quantifications
We know $\exists x\in\mathbb R^n:Ax\leq b$ is standard linear program.
I am mainly looking at following case of quantified linear program with no free variables with only existential quantifications ...
- 529
1
vote
0
answers
66
views
Sherali-Adams lowerbound instance of Unique Games constructed via CLT
The question comes from the following paper I have been reading:
[1] Integrality Gaps for Sherali–Adams Relaxations. SODA'09. Moses Charikar, Konstantin Makarychev, Yury Makarychev.
Theorem 6.1 of [...
- 191
2
votes
1
answer
171
views
Finding whether $n$ polytopes have nontrivial intersection from pairwise comparisons
I have a set of $n$ convex polytopes of the form
$$\mathcal{L_i} = \{ \beta \mid C_i \beta \leq 0 \}$$
where $C$ is a matrix and $\beta$ is a vector. I know that for each pair of polytopes $$(\...
0
votes
1
answer
229
views
finding maximum weight subgraph
My graph is as follows:
I need to find a maximum weight subgraph.
The problem is as follows:
There are n Vectex clusters, and in every Vextex cluster, there are some vertexes. For two vertexes in ...
- 57
5
votes
1
answer
182
views
Complexity of Finding Largest Set of Intersecting Convex Polytopes
I have a set of $n$ convex polytopes, and I wish to find the largest subset of those polytopes that shares at least one point in common. I think that this problem should be NP-hard, but I am ...
1
vote
0
answers
276
views
Does simplex algorithm run in polynomial on Bipartite Perfect matching polytope?
It is well known that simplex algorithm runs in exponential time in worst case.
However are there situations (necessary and sufficient conditions) where simplex algorithm runs in polynomial time?
In ...
- 12.6k
4
votes
1
answer
66
views
Minimizing a convex piece-wise linear function of short $(\max, +)$ circuit length
If $a_{ij}$ is an $m \times n$ matrix of real numbers, and $b_j$ are $n$ more real numbers, then
$$\max_i \sum_j (a_{ij} x_j + b_j) \qquad (\ast)$$
is a convex piecewise linear function of $(x_1, \...
- 361
15
votes
1
answer
755
views
Can one efficiently uniformly sample a neighbor of a vertex in the graph of a polytope?
I have a polytope $P$ defined by $\{ x : Ax \leq b, x \geq 0\}$ .
Question: Given a vertex $v$ of $P$, is there a polynomial time algorithm to uniformly sample from the neighbors of $v$ in the graph ...
- 1,439
4
votes
1
answer
183
views
How small can extension complexity be?
In this article on extension complexity of regular polygons https://arxiv.org/pdf/1505.08031.pdf it is mentioned that extension complexity of $n$ regular polygons should be $\theta(\log n)$. This is ...
- 529
3
votes
1
answer
217
views
How is SDP an extension of spectral algorithms?
In one of his lectures, Uri Feige described semidefinite programming (SDP) as
... an algorithmic technique that extends both linear programming and spectral algorithms.
I know the basic ...
- 600
3
votes
0
answers
84
views
Reference request: strong polynomial-time for LP
A follow-up of sorts on this question:
Complexity of finding a consistent hyperplane
What is a good survey of partial results on the strong poly-time status of the general LP problem?
- 10.1k
4
votes
1
answer
155
views
Generalizations of linear programming
Linear problems can be solved in polynomial time. So can semidefinite programs and, presumably, many other useful classes of optimization programs.
Is there a survey/lecture notes describing ...
- 600
2
votes
0
answers
108
views
Best algorithms for real linear programming
Linear Programming asks for $x\in\mathbb R^n$ such that $Ax\leq L$ holds where $A\in\mathbb R^{m\times n}$ and $L\in\mathbb R^m$ are given. Karmarkar has shown that $\ell$ is the number of bits of ...
- 12.6k
4
votes
1
answer
79
views
Solving an LP with at most m-1 nonzeros
Consider the linear program:
$$
A x = b, ~~~~~~ x\geq 0
$$
where $A$ is an $m$-by-$n$ matrix, $x$ is an $n$-by-1 vector, $b$ is an $m$-by-1 vector, and $m<n$.
It is known that, if this ...
- 1,880
3
votes
2
answers
173
views
Minimum relevant variables in linear system - additive approximation
In the problem Minimum Relevant Variables in Linear System (Min-RVLS), the input is a linear system, e.g.:
$$ A x = b $$
and the goal is to find a solution $x$ with as few nonzero variables as ...
- 1,880
5
votes
1
answer
443
views
Is there a counterexample to this work?
Is there a counterexample to this claim https://arxiv.org/abs/1610.00353? They claim a $O(n^6)$ LP model with simulations to support. I think asking validity is not a reasonable problem. However ...
- 12.6k
10
votes
1
answer
489
views
How "hard" is it to maximize a polynomial function subject to linear constraints?
General Problem
Suppose we have a multivariate polynomial function $f(\mathbf{x})$, and several linear functions $\ell_i(\mathbf{x})$. What is known about the complexity of solving the following ...
- 464