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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5
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48 views

SVM - running time for detecting if data is linearly separable?

If my understanding is correct, one way to check if a set of $m$ data points is linearly separable is to use support vector machines to find a maximum margin hyperlane for separating the data; the ...
2
votes
1answer
52 views

H-representation of convex hull

Consider a set of polytopes $P_j\;\;j=1,2,\dots,r$ with the same structure as follows: $P_j=\Big\{(x_{j1},\dots, x_{jt})\Big| \sum_{i=1}^t x_{ji}=1, x_{ji}\in [a_{ji},b_{ji}]\subseteq [0,1]\Big\}$ ...
-2
votes
0answers
18 views

Where can integrating the inverse of a function be used in programming application?

Just like [a link]https://en.wikipedia.org/wiki/Fast_inverse_square_root can be used in calculating angle of reflection and incidence in first person shooting games, Where might calculating the ...
4
votes
2answers
189 views

What are some of the most ingenious linear programs developed for tackling hard combinatorial problems?

What are some known ingenious linear programs that have been developed for tackling hard combinatorial optimization problems, especially any linear programs which had helped in getting good ...
1
vote
0answers
19 views

Minimum cost flow with Demand and Edge Capacity scaling

Consider a the splittable minimum cost flow problem on network $G(V,E,W,C)$ and a set of commodities $(s_i,t_i)$ with demand $d_i$ for $i=1,2,\dots,k$. Here, $w_e$ and $c_e$ is the weight of edge $e$ ...
0
votes
1answer
200 views

Vertices of a polytope

Consider the polytope $P=\{(x_1,x_2,...,x_n)\in \mathbb{R}^n| \sum_{i=1}^n x_i=1; 0\leq a_i\leq x_i\leq b_i, i=1,...,n\}$ where $a_i$ and $b_i$ are constant lower and upper bounds for $x_i$. Is it ...
7
votes
1answer
173 views

Embedding a graph in the euclidean space

Given a graph $G=(V,E)$, find a mapping $f\colon V \rightarrow \mathbb R^d$ such that for every edge $(u,v) \in E$ we have that $||f(u)-f(v)|| \leq r$; and for every $(u,v) \not \in E$, we have the ...
5
votes
1answer
85 views

Is computing the dual optimum of a degenerate LP equivalent in complexity to solving LP?

Given a primal LP and an optimum solution thereof, it is well-known that complementary slackness conditions [1] can be used in non-degenerate cases to produce a dual optimum solution. For instance, ...
3
votes
0answers
111 views

Exactly solvable but non-trivial integrality gap

Are there interesting polynomial time solvable problems that we know of for which the natural convex relaxation has a non-trivial integrality gap? Note: Maximum matching doesn't qualify because I ...
-1
votes
1answer
83 views

convertion into integer linear program for Ising spin state problem [closed]

I am trying to model the Ising spin state problem into Integer linear program and find the optimal ground state using lp_solve. (This is just a miniature version of Ising state problem) $$ maximise: ...
0
votes
0answers
29 views

Hoare program correctness

Which is the easiest way to find a valid Invariant for a While Program in the Hoare program correctness verification? Is there a "guided" way to do that?
6
votes
0answers
126 views

Are highly symmetric inequalities solvable over integers?

Suppose I have $n$ variables $x_1,\ldots,x_n$ that satisfy some inequalities that are highly symmetric, e.g., for all $S\subset [n], |S|=k$ we have $\sum_{i\in S} f(x_i,k)\le \sum_{i\in [n]} ...
0
votes
2answers
160 views

Logic with Linear Programming

Can first-order logic be modeled/simulated as linear programming or integer programming? What about other forms of logic (say second order)? Update: am actually not a theory person, but more on the ...
0
votes
0answers
155 views

Problems which are solvable using Linear Programming

Can anyone share a link to a good survey/book about the different problem types for which we have a linear programming based solution for, as well as the related techniques?
0
votes
0answers
58 views

LP solver for sparse, PSD and strictly diagonally dominant matrix

I have a linear problem with a sparse, psd and strictly diagonally dominant matrix. Can you please point me to some known best solvers (in terms of runtime, or easy to be practically optimized for ...
0
votes
0answers
109 views

Fastest Algorithms for Determining the Nullity of a Matrix

How exactly does one go about determining the Nullity of a Matrix quicker than simply running Gaussian Elimination on the matrix itself? To be perfectly honest I can't think of a method that doesn't ...
-1
votes
1answer
59 views

Combinatorial algorithm for load balancing

I have a problem that can be solved with linear programming, but I'm hoping there's a combinatorial algorithm for this (even approximation is fine). This is basically a load balancing problem using ...
4
votes
2answers
2k views

Intuitively, why is the complementary slackness condition true?

What's an intuitive proof that shows that the conditions of complementary slackness are indeed true: If $x^*_j > 0$ then the $j$-th constraint in the dual is binding. If the $j$-th constraint in ...
7
votes
1answer
262 views

Why is complementary slackness important?

Complementary slackness (CS) is commonly taught when talking about duality. It establishes a nice relation between the primal and the dual constraint/variables from a mathematical viewpoint. The two ...
2
votes
1answer
176 views

Literature for Generalized Load Balancing

i am looking for literature on this kind of problem. $$ \begin{align} \min_x \max_k &\quad \sum_{i,j} x_{ij}c_{ijk}\\ \text{subject to}&\\ &\sum_j x_{ij}=1,&& \forall i\in\mathcal ...
5
votes
0answers
176 views

Practical algorithm for testing whether an edge is Delaunay

I have a set of vertices $V\subset\mathbb R^3$ and a set of edges $S=\{(a,b)|a,b \in V\}$. I want to know whether an edge in the set $S$ is Delaunay against the vertices in $V$. My assumed ...
2
votes
1answer
170 views

Complexity of linear programming

It is known that Linear Programming (LP) is P-complete. I am interested in approximation algorithms for LP. There are numerous inapproximability results for NP optimization problems, e.g. it is ...
0
votes
0answers
93 views

Definition of convex optimization problem by Stephen Boyd and Lieven Vandenberghe

Boyd and Vandenberghe say that a convex optimization problem is one of the form: minimize $f_0(x)$ subject to $$f_i(x)\le 0, i=1,\ldots m$$ $$a_i^\top x=b_i, i=1,\ldots p$$ ...
3
votes
1answer
170 views

What is known about this binary representation polytope?

Consider the following set. $S_n =\{ (x,y) ~|~x \in \mathbb{Z}_+~\wedge~ y \in \{0,1\}^n ~\wedge~x=\sum_{i=0}^{n-1} 2^i y_i \}$ $S_n$ is a collection of pairs $(x,y)$, where $x$ is an integer ...
1
vote
0answers
102 views

Optimization as LP on the convex hull of its solution space

I have heard claims that for a certain class of optimization problems, one can re-write the problem as a linear-programming problem on the convex hull of the solution space of the original problem ...
-1
votes
1answer
248 views

equivalent way(s) of expressing P=?NP problem in linear programming?

the paper "In defense of the Simplex Algorithm's worst-case behavior" Disser/Skutella [1] was recently cited on this tcs.se site by saeed on another interesting question. the paper introduces the idea ...
2
votes
1answer
143 views

Proof-techniques for the hardness of optimization problems (esp. Polynomial time)

I've given an optimization problem for which I want to show that it is solvable in polynomial time. Now, I have two questions: Can this be done by formulating a mixed-integer linear program such ...
23
votes
3answers
722 views

Optimization problems with good characterization, but no polynomial-time algorithm

Consider optimization problems of the following form. Let $f(x)$ be a polynomial-time computable function that maps a string $x$ into a rational number. The optimization problem is this: what is the ...
1
vote
1answer
114 views

Feasibility of equations with max constraints

Relating to a question I posted here, I have formulated the following question: Notation: $\max\{ x_1, \cdots, x_n \}$ denotes the maximal number among $x_1, \cdots, x_n$. How to check the following ...
12
votes
1answer
321 views

Equivalence of feasibility checking and optimization for linear systems

One way to show that checking the feasibility of a linear system of inequalities is as hard as linear programming is via the reduction given by the ellipsoid method. An even easier way is to guess the ...
2
votes
2answers
262 views

A variant of linear programming

Consider this "variant" of linear programming: Notation: $\max\{ x_1, \cdots, x_n \}$ denotes the maximal number among $x_1, \cdots, x_n$; minimize $\sum a_i x_i$ such that $\max\{x_i\mid ...
2
votes
1answer
104 views

Testing feasibility of special linear programs

We have a linear program in the standard form $Ax=b,x\geq 0$ with $A(i,j)\in\{-1,0,1\}\:\forall\,i,j$ and $b(i)\in\{0,1\}\:\forall\,i$. $A$ is not full rank. Is the time complexity of testing the ...
24
votes
1answer
525 views

Is cubic complexity still the state of the art for LP?

According to D. den Hertog, Interior Point Approach to Linear, Quadratic and Convex Programming, 1994, a linear program with $n$ variables, $n$ constraints and precision $L$ is solvable in $O(n^3L)$ ...
3
votes
2answers
342 views

Polynomial algorithm for correlated equilibrium

I searched through the web for a polynomial algorithm for correlated equilibrium. I found a lot of papers by C.H. Papadimitriou that proposes a solution using the ellipsoid algorithm. Is there a ...
8
votes
0answers
154 views

Approximating a convex polyhedron, with fewer inequalities

I have a convex polyhedron $\mathcal{P}$, given by $n$ linear inequalities $a_i \cdot x \le c_i$ where $x$ is a $d$-dimensional vector over the non-negative real numbers. In other words, ...
7
votes
2answers
303 views

what can be solved with semidefinite programming that can't be solved with linear programs?

I'm familiar with linear programs in that they can solve problems with linear objective functions and linear constraints. But what can semidefinite programming solve that linear programs can't? I ...
21
votes
1answer
400 views

Is there a polynomial time algorithm to determine if the span of a set of matrices contains a permutation matrix?

Precisely what the title says, I'm looking to find a polynomial time algorithm that given some set of matrices, will find if their span contains a permutation matrix. If any one knows if this ...
2
votes
1answer
278 views

Approximate Maximum Weight Matching

I am looking for an approximated (or randomized) maximum weight matching algorithm. Do you have any suggestion for me? In my problem, I have a bipartite graph with N abound 1000 (#vertices on each ...
1
vote
2answers
298 views

Applications of duality

We have recently covered linear programming and I am comfortable with the weak and strong duality theorems. However, I don't understand what the applications of duality are that are specific to TCS. ...
4
votes
1answer
122 views

Batch membership testing for convex polyhedron specified in vertex representation

I have a convex shape defined by a set of vertices (the so-called vertex representation of a convex polyhedron). I also have a large set of points and I would like to test which are contained in the ...
3
votes
0answers
327 views

Linear programming optimization problems using parallel algorithms

I'm looking for methods and algorithms for solving linear programming algorithms, characterized by up to 20 variables but up to thousands of constraints in a parallel way. There are several approaches ...
3
votes
1answer
181 views

Efficient flow problem for a complex integer program

I have a bunch of marbles each with some weight (they can also have negative weights). I want to pick the nodes such that the weight is maximized. The only rule is that if I pick X1 and X2 I have to ...
2
votes
1answer
247 views

Linear programming, a non standard handling of absolute value

This is really a basic (undergrad) question in LP but still i would like to clarify it for myself to be sure. I have a minimization problem from the sort $\min \sum_i |x_i|$ s.t $Ax \le b$. I've seen ...
0
votes
1answer
721 views

Difference between Multiple Knapsack problem and Multidimensional Knapsack Problem

What is the difference between Multiple Knapsack problem and Multidimensional Knapsack Problem? (http://en.wikipedia.org/wiki/List_of_knapsack_problems#Multiple_constraints) According to the ...
6
votes
1answer
156 views

Research about geometry negotiation in widget layout

It's very common to view GUI layout as an optimization problem. I've uncovered a lot of research around linear constraint systems, but a lot of real world layouts can not be represented linearly. For ...
2
votes
0answers
76 views

General covering approximation

Consider the following integer program (general covering): $\min c \cdot x$ subject to $Ax \ge b$, where all entries in $A, b, c$ are nonnegative and $x$ is required to be nonnegative and integral. ...
3
votes
2answers
232 views

Generalization of independent set

I know the definition of the independent set problem in graph theory. An independent set cannot contain any two adjacent vertices. How about if you allow no more than $k$ pairs of adjacent vertices? ...
1
vote
0answers
135 views

What does “no integrality gap” implies?

I'm currently working on a linear time heuristic for the rectangle decomposition of binary matrix. This problem has a polynomial time solution, which in our case it too slow for large scale ...
0
votes
1answer
260 views

Approximation algorithm for graph problem

In the process of trying to create an approximation algorithm for the following problem. Let $G = (V,E)$ be a graph, $c_e, c_{iv} \ge 0$, for $e \in E$, $i \in L$, and $v \in V$, where $L$ is a ...
1
vote
1answer
99 views

Expressing a set of 0-1 strings by Extended Formulation

Extended Formulation of a polytope $P \subseteq \mathbb{R}$ is a system of linear constraints which satisfies the following condition: $x \in P \iff \exists y\in \mathbb{R}^{d} \mbox{ such that } Ax ...