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.

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1answer
113 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.
4
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1answer
207 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?
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1answer
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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 ...
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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, ...
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1answer
99 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 ...
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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, ...
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108 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 ...
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65 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 ...
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2answers
152 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. ...
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1answer
101 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 ...
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1answer
118 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 ...
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134 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 ...
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1answer
61 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 ...
3
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1answer
141 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. ...
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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 ...
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49 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 [...
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1answer
156 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 $$(\...
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1answer
108 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 ...
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1answer
159 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 ...
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143 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 ...
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1answer
62 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, \...
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1answer
663 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 ...
4
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1answer
166 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 ...
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1answer
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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 ...
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79 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?
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1answer
149 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 ...
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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 ...
4
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1answer
68 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 ...
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2answers
137 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 ...
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1answer
425 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 ...
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1answer
317 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 ...
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37 views

Monotone complexity of PLP

Blum and Nisan show Positive Linear Programming could be done in $NC$ if we only ask for approximate solutions. This paper https://pdfs.semanticscholar.org/8dc7/5aa9d72864022d848c3e599c5f24d9d527e7....
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1answer
129 views

On complexity of linear programming with quadratic equality/inequality constraints?

Feasibility test in Linear programming is in $P$ and in convex quadratic programming is in $P$. What is the maximum $k$ such that $n$-variable $m=poly(n)$ linear constraint feasibility test with $k$ ...
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1answer
295 views

Reaching the double exponential upper bound in Fourier-Motzkin elimination

One round of Fourier-Motzkin elimination may yield $n^2/4$ inequalities where $n$ is the original number of linear inequalities, whence an upper bound of $n^{2^d}/2^{2^{d+1}-2}$ for $d$ rounds of ...
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2answers
398 views

Complexity of finding a consistent hyperplane

Given $m$ binary labeled points in $\mathbb{R}^d$, it is well-known that in general it's NP-hard to find a hyperplane that minimizes sample error. A brute-force search considers all $O(m^d)$ sample ...
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70 views

Explicit Formula of Delsarte's Linear Programming Upper Bound for $A_q(n,3)$

The problem of giving an explicit formula for $A_q(n,d)$ is sometimes referred to as "the main problem in coding theory." The value of $A_q(n,d)$ is given by the maximum number of codewords in a q-ary ...
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Max and Min Equations with Linear Programing

I am given constants $c_{i,j}$ as inputs. I want to find values for the variables $x_i$ that satisfy a system of equations, where each equation either takes the form $$x_i = \max\{x_j + c_{i,j} : j \...
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1answer
176 views

Solving linear program with 1 quadratic constraint complexity

Consider the following linear program, $$\min y \\ xc_1 \leq c_2 + yz,\\ x = x_1 + \dots + x_n,\\ z \leq x_1 + x_2, \\ z \leq x_2 + x_3, \\ \vdots\\ z \leq x_{n-1} + x_n, \\ x,x_1, \dots, x_n,y,z \...
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1answer
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What is wrong with this procedure to convert quadratic programming to convex quadratic programming?

Consider the feasibility quadratic program with constraint $$\sum_{i=1}^nc_{i1}x_{i}\leq \ell_1$$ $$\vdots$$ $$\sum_{i=1}^nc_{it}x_{i}\leq \ell_t$$ $$\sum_{i,j=1}^na_{ij}x_{i}x_{j}+\sum_{i=1}^nb_{i}x_{...
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0answers
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What exactly did Lenstra prove on mixed integer linear program?

I studied Lenstra's paper https://www.jstor.org/stable/3689168. I have no clue what complexity he provides on Mixed Integer Programming (it is too terse and it is not a stand alone paper as he assumes ...
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1answer
205 views

Cases of Linear programming known to be in $NC$?

Linear programming is $P$-complete. However are there special situations where we know an $NC$ algorithm?
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1answer
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About the sign-rank of the Minsky-Pappert function

Apologies this might be a very trivial thing I am getting confused by! Firstly in corollary 1.1 (page 3) in this paper, https://eccc.weizmann.ac.il/report/2016/075/ the authors claim that they have ...
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1answer
217 views

Max weight travel on a graph with deadline

Given a deadline $D>0$ and a complete graph $K_n$ (with loops) in which each edge $e_{ij}$ has a weight $w(e_{ij}) \ge 0$ and a travel time $l(e_{ij}) > 0$. Starting from one of the nodes, we ...
17
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1answer
511 views

How not to compute the smallest circle enclosing a finite set of circles

Suppose we have a finite set $L$ of disks in $\mathbb{R}^2$, and we wish to compute the smallest disk $D$ for which $\bigcup L\subseteq D$. A standard way to do this is to use the algorithm of ...
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1answer
56 views

How to check whether graph of n vertex contains n/k disjoint k - complete graph by linear programming? [closed]

Edges are given in form of $X_{ij}$, which denotes whether there is edge in between $i^{th}$ and $j^{th}$ vertex. I am solving integer optimization problem and want to add this constraint to it.
2
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1answer
287 views

Are there specific examples of integral polyhedra that are neither Totally Unimodular nor Total Dual Integral?

It is well known that if a constraint matrix $A$ is total dual integral or totally unimodular, then this is a sufficient condition of integrality of the polyhedron defined by the system $Ax \leq \beta$...
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1answer
267 views

Solving linear equations involving min

Solve for $\alpha$ if solution exists: $\min(r_1, s_1 \alpha_1) + \min(r_2, s_2\alpha_1) = d_1$, where $r_1,r_2,s_1,s_2,d_1$ are integer constants. One way seems to be enumerate all possible outcomes,...
3
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1answer
178 views

Min cost set of edges to connect 2 subgraphs s.t dist of nodes between subgraphs <= K

I find myself with another graph problem that I can't find the name of. I was wondering if anyone was able to identify if this problem and any efficient algorithms to solve it are known. The ...
6
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1answer
149 views

Brute force search algorithm for semidefinite programming (representation of spectrahedron)

I was wondering if there exists a brute force search algorithm for semidefinite programming problems. Specifically, can we find finite number of points in the positive semidefinite cone such that for ...
3
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1answer
353 views

List of Pivot rules for simplex methods

Any implementation of the simplex method depends on the choice of pivot rule, which determines how the corners of the search space polyhedron are traversed. Many different have been proposed ...