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3
votes
2answers
39 views

Restricted Integer Programming

The integer feasibility problem is NP-complete: $Ax=b, x \geq 0, x \mbox{ integer}$ $A$ contains elements in $\mathbb{R}$ If we restrict this: $A$ contains only elements in: $\{1,0\}$ ...
3
votes
1answer
51 views

Find perfect matching whose weight is minimal, in polynomial time

Given a bipartite graph $G=(A,B,E)$ and a weight function $w: E \rightarrow\mathbb{R}^+$, I'd like to find a perfect matching $M\subseteq E$ with min. weight. I'm assuming $|A| \leq |B|$, and WLOG $G$ ...
0
votes
1answer
70 views

Polynominal reduction from unbounded knapsack problem to general integer programming

Given an oracle that can solve in polynominal time: $$a^Tx=b$$ $$x \geq 0$$ So it can solve the feasibility problem with one equality-constraint(a is here a vector and b is a constant, x is required ...
0
votes
1answer
52 views

Integer Programming Traveling Salesman - Checking if Tour is Complete

I'm reading a Wikipedia article on the Traveling Salesman problem as an Integer Programming formulation (http://en.wikipedia.org/wiki/Travelling_salesman_problem). The authors have all their ...
1
vote
1answer
30 views

Integer Programming - packing wolves and sheep

I'm new to linear/integer programming and I'm trying to solve a little problem I made up. I want to "pack" animals into a minimum number of bins where some of the animals cannot co-exist (wolves and ...
0
votes
1answer
29 views

Scheduling Problem Optimisation using LP solvers

I am currently working on a scheduling system which schedules individuals based on timeslot that have the following attributes : (1) Day of Week (2) Month (3) Session(AM/PM). The individuals ...
4
votes
0answers
22 views

Fastest known complexity for combinatorial ILP algorithm?

I'm wondering, what is the best known algorithm, in terms of Big-$O$ notation, to solve Integer Linear Programming? I know that the problem is $NP$-complete, so I'm not expecting anything polynomial. ...
1
vote
1answer
31 views

ILP and number of variables in constraints

This question is about the time impact of the length (i.e. number of variables) of the constraints in an Integer Linear Programming formulation. Most people try to reach the minimum number of ...
2
votes
0answers
26 views

How to convert a rank constraint into integer programming?

Consider the low-rank matrix completion problem: given an integer $k$ and a subset of entries of some matrix, can you fill in the rest of the entries so that the resulting matrix has rank at most $k$? ...
-1
votes
2answers
33 views

What is decision version of integer programming

I dont know what is meant by decision version of Integer Programming. I know ILP, but this terminology has me confused. There are no good resources on Google.
1
vote
1answer
39 views

Integer Linear Programs: An instance or not?

Given a set of integers $\{x_0, x_1, ... , x_{n-1}, x_n\} \subseteq \mathbb{Z}$, a set of integer variables $\{y_0, y_1, ... ,y_{n-1}, y_n\} \subseteq \mathbb{Z}$ and an integer $m \in \mathbb{Z}$ is ...
2
votes
0answers
31 views

Difficulty of Integer Linear Programming vs. Mixed Integer Linear Programming

I have recently been working on an applied project where I have to solve optimization problems. I have found that it is much easier to solve an integer linear program (ILP) as opposed to a ...
2
votes
0answers
44 views

How can k-means be reduced to Integer Programming

The k-means algorithm reduces to computing the objective function: $ \underset{\textbf{S}}{\operatorname{argmax}} \sum_{i=1}^k \sum_{\textbf{x}_j\in\textbf{S}_i} \lVert \textbf{x}_j - \mathbf{\mu}_i ...
3
votes
0answers
73 views

Formulating Integer Program for passing packages on a cycle

Can't seem to figure out the IP formulation for this. Question Suppose there are $n$ people connected in a circular fashion as demonstrated by the diagram. Individuals need to send packages to each ...
2
votes
1answer
203 views

Advantage of MTZ problem formulation of TSP

In class, we saw the Miller-Tucker-Zemlin formulation of the Travelling Salesmen Problem (TSP). MTZ is a way of formulating the TSP as an integer linear programming instance. I understand how MTZ ...
0
votes
1answer
50 views

How to reformulate my problem as a mixed-integer quadratic problem

I have an unknown $n$-dimensional vector $x$ whose analytical expression depends on the following sum $x = z + Ba$ where the vector $z$ and the matrix $B\in \mathbb{R}^{n\times s}$ are given. So the ...
1
vote
0answers
92 views

Is this NP-Hard problem? [closed]

I have the following problem. maximize $\sum\limits_{k=1}^Lx_k$ subject to: $\mathbf{x}^T\mathbf{A}~ \tilde{\mathbf{x}_i} \geq 0,~~ \forall~ i\in\{1, 2, \cdots, L\}.$ where, $~\mathbf{x}^T = (x_1, ...
5
votes
2answers
240 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 $\mathbb{Z}$). ...
8
votes
1answer
497 views

Poly-time reduction from ILP to SAT?

So, as is known, ILP's 0-1 decision problem is NP-complete. Showing it's in NP is easy, and the original reduction was from SAT; since then, many other NP-Complete problems have been shown to have ILP ...
16
votes
0answers
774 views

Is it NP-hard to fill up bins with minimum moves?

There are $n$ bins and $m$ type of balls. The $i$th bin has labels $a_{i,j}$ for $1\leq j\leq m$, it is the expected number of balls of type $j$. You start with $b_j$ balls of type $j$. Each ball of ...
8
votes
1answer
2k 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 ...
3
votes
3answers
2k 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 ...
2
votes
0answers
34 views

Why are optimization problems always NP-hard and not NP-complete and what does this mean for other levels of the polynomial time hierarchy? [duplicate]

I have read that optimization problems cannot be $\mathcal{NP}$-complete, but are always classified as $\mathcal{NP}$-hard. When a problem is NP-complete, I know it is contained in $\mathcal{NP}$P. ...
8
votes
1answer
121 views

Hardness of Approximating 0-1 Integer Programs

Given a $0,1$ (binary) integer program of the form: $$ \begin{array}{lll} \text{min} & f(x) & \\ \text{s.t.} &A\vec{x} = \vec{b} & \quad \forall i\\ &x_i\ge 0 & \quad \forall ...