## Tuesday, August 5, 2014

### Lecture on Recommender Systems

Great lecture on Recommender Systems by Xavier Amatriain, Researcher on Netflix.

Labels:
Machine Learning,
Recommender Systems

## Wednesday, July 16, 2014

### Genetic Algorithm for Knapsack using Hadoop

*Development of Genetic Algorithm using Apache Hadoop framework to solve optimization problems*

**Introduction**

This project I developed during a course on my Master intends to construct a Genetic algorithm to solve optimization
problems, focusing on the Knapsack Problem. It uses as base the
distributed framework Apache Hadoop.

The idea is to show
that the MapReduce paradigm implemented by Hadoop is a good fit for
several NP-Complete optimization problems. As knapsack, many problems
present a simple structure and converge to optimal solutions given a
proper amount of computation.

**Genetic Algorithm**

The algorithm was
developed based on a Genetic paradigm. It starts with a

*initial random population*(random instances to the problem). Then, the best*individuals are selected*among the population (instances that generate the best profits for the knapsack). A phase of*crossover*was then implemented to generate new instances as combination of the selected individuals. In this phase, parts of each selected individual are randomly selected to form a new instance.
The next step of
the process is the

*mutation*phase. In this phase, “genes” are selected also randomly, and changed. In the knapsack context this means that random items are substituted from the solutions found.
This cycle can be
repeated several times, until good individuals are generated.

The steps are
reproduced below:

**Initial Population → Selection → Crossover → Mutation**

**The Knapsack Problem**

The goal is to maximize the objective function, where vi represents the value of each element i, and variable xi represents whether element i is going to be taken or not. The maximization has to take into account the weight of each element represented by the first constraint.

This problem was
chosen because its solution only require enough amount of processing,
and this processing can be done in parallel. The space of possible
solutions can be seen as a set of local maximums that are going to be
tested by the genetic algorithm. Hopefully, with enough computation, the global
maximum can be achieved.

**Validation Results**

To demonstrate that
the algorithm find a correct result with high probability, it was
tested against two validation datasets. One was composed of a set of
10 items and another with 15 items.

With the validation
input P01, we have:

P01 is a set of 10
weights and profits for a knapsack of capacity 165.

The optimal result value is 309, by selecting items 0, 1, 2, 3 and 5 .

The algorithm finds
the correct answer:

Best 5 answers:

**Value**

**Items Selected**

270 0 1 2 9

276 0 2 3 6

277 0 1 3 4

284 0 1 3 6

309 0 1 2 3 5

The input P02 has
optimal profit value 1458, and the algorithm also finds the correct
answer.

These dataset were found on:

**Setting up the Hadoop Cluster on a Single Machine**

For testing purposes, here follows how to start on Hadoop, and set it up on a single machine.

Download hadoop 1.21
from:

http://hadoop.apache.org/releases.html

Install a Java 6:

$ sudo apt-get
install oracle-java6-installer

Inside the hadoop
home directory, update a configuration file in conf/ :

$ vim
conf/hadoop-env.sh

export
JAVA_HOME=/usr/lib/jvm/java-6-oracle/

export
HADOOP_HOME=/home/renata/hadoop-1.2.1

export
HADOOP_VERSION=1.2.1

Create an input
directory, to place your input file:

$ mkdir input

Put some content in:

$ cp conf/*.xml
input

To test if your
hadoop is working properly:

$ bin/hadoop jar
hadoop-examples*.jar grep input output 'dfs[a-z.]+'

Don't forget to
remove the output file after running a Job. Hadoop doesn't do this
for you.

$ rm -rf output

Now, to test the
Genetic jar file:

$ export
HADOOP_HOME=/home/renata/hadoop-1.2.1

$ export
HADOOP_VERSION=1.2.1

To compile the code
run :

$ javac -classpath
${HADOOP_HOME}/hadoop-core*.jar -d genetic_classes Genetic.java

And create the jar:

$ jar -cvf
genetic.jar -C genetic_classes/ .

Now, to run the code once you put the input data on "input" directory:

$ bin/hadoop jar genetic.jar org.myorg.Genetic input output

Now, to run the code once you put the input data on "input" directory:

$ bin/hadoop jar genetic.jar org.myorg.Genetic input output

*PS: A large large dataset consisting of 10.000 items, with optimal profit value of 7145 and a weight capacity of 431 can be found on: https://github.com/jaredtking/knapsack-lisp/blob/master/datasets/large01.*

Labels:
Algorithms,
Apache Hadoop,
Data Mining,
Map Reduce

## Friday, January 31, 2014

### Dealing with NP-Hard Problems: An Introduction to Approximation Algorithms

*This is just a quick overview on approximation algorithms. It is a broad topic to discuss. For more information go to References.*

The famous

**NP-Complete class is known for its possible intractability.**NP means

*non deterministic polynomial*and for a problem to be NP-Complete it has to be

- NP (verified in polynomial time) and

- NP-Hard (as hard as any other problem in the NP class).

Among the several important problems that are NP-Complete or NP-Hard (on its optimization form) we can name the Knapsack, the Travel Salesmen, and the Set Cover problem.

Even though no efficient optimal solution might exist for NP-Complete problems we still need to address this issue due to the amount of practical problems existent in the NP-Complete class.

Considering that even for medium volumes of data exponential

*brute-force*is impractical, the option is abdicating the optimum solution as minimum as possible and pursuing an efficient algorithm.

**Approximation algorithms are a way to deal with NP-Hard problems**. They are

**polynomial-time algorithms, that return guaranteed near optimal solutions**for any instance of the problem. The objective is to make the output as close to the optimal value as possible. We say that an

*α-approximation*, is an algorithm that is within a α distance from the optimal value.

### Measures of Approximation

Among the measures of approximation it is possible to highlight:

- Absolute approximation
- Approximation factor

**absolute approximation**for an instance I if it stays under a absolute distance k of the optimum algorithm OPT:

|A(I) − OPT(I)| ≤ k ∀ instance I.

An algorithm has an

**approximation factor**of α if:

A(I) ≤ α · OPT(I) ∀I, for minimization

A(I) ≥ α · OPT(I) ∀I, for maximization

being that α>1 for minimization and α<1 for maximization problems.

### Approximation Techniques

**Greedy approaches**are a highly intuitive and usually easy to implement approximation algorithm. It basically consists of making the "best" local choice at each step. For example, for the Set Cover problem, a greedy algorithm would be:

"At each round choose the set that minimizes the ratio weight of its weight to the number of currently uncovered elements it contains."

You can check a proof of approximation here [1].

Another way to create an approximation is by making a

**linear relaxation**of a integral program. Remembering that many problems can be written in a more mathematical form, were we have a

*objective function*to maximize (or minimize),

*constraints*and

*variables*.

The Knapsack is a good example:

The goal here is to maximize the objective function, where

*vi*represents the value of each element

*i*, and variable

*xi*represents whether element

*i*is going to be taken or not. The maximization has to take into account the weight of each element represented by the first constraint.

Approximation through linear relaxation can be done in a very simple way. While Integral programming is NP-Hard, linear programming is not. If we relax (substitute) the constraint

to

Here is what is would look like:

The linear relaxation serves as a

*lowerbound*of the optimal solution in minimization problems and

*upperbound*in maximization problems.

There is also a very powerful method called

**Primal-dual schema**.

It is well known that, in some cases, optimal solutions satisfy the complementary slackness conditions.

The Primal-Dual schema works with a relaxed version of the complementary slackness conditions. It starts with a integral feasible solution for the primal and a feasible solution for the dual, for example. Iteratively, it satisfies the slackness conditions so that no constraint problem is broken in the process. When all conditions are satisfied the program ends.

It can be a little complex, so I am going to do a post about it on the future. For now, to understand it in details please read [2].

### Classes of Approximation

Algorithms as the mentioned above can be classified by their approximation "degree", or how much it is possible to approximate them of the optimal solution.

Following there are listed the classifications:

- PO - problems for which there are exact polynomial algorithms. This an extension of the P class to optimization problems.

- PTAS (Polynomial Time Approximation Scheme) - It presents an approximation scheme which is a (1 +
*e*)OPT approximation for minimization, and, (1 -*e*)OPT for maximization, where*e*is a relative error (rational number).

- FPTAS (Fully Polynomial Time Approximation Scheme) - as PTAS, FPTAS is also a polynomial scheme, but FPTAS running time is polynomial in the inverse of the error (1/
*e)*.

- APX - there is at least one polynomial α-approximation (for some constant α).

- NPO - is an extension of the NP class to optimization problems. This means that an algorithm
*A*is in the NP Optimization class if its instances are polynomial in size, and solution for the problem can be verified in polynomial time and the size of the objective function (solution) can be calculated in polynomial time.

PO ⊆ FPTAS ⊆ PTAS ⊆ APX ⊆ NPO

If an algorithm is in the FTPAS class, this usually means that it can have better approximations than one in the APX class.

The knapsack problem, for example, has been proved to be in FPTAS [3]. If by any chance someone proved it to be also in any other class of the above, than P=NP.

### References

*[1] D. P. Williamson and D. B. Shmoys. The Design of Approximation Algorithms. 2010. Cambridge University Press*

*[2] V. Vazirani. Approximation Algorithms. 2001. Springer-Verlag.*

*[3]*

**(PORTUGUESE)**M.H. Carvalho, M.R. Cerioli, R. Dahab, P. Feofiloff, C.G. Fernandes, C.E. Ferreira, K.S. Guimarães, F.K. Miyazawa, J.C. Pina Jr., J. Soares, Y. Wakabayashi. Uma introdução sucinta a algoritmos de aproximação. 23o Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro.*[4] Giorgio Ausiello, Pierluigi Crescenzi, and Marco Protasi. Approximate Solution of NP Optimization Problems.*

*Theor. Comput. Sci.**150(1):1-55*(*1995*)
Subscribe to:
Posts (Atom)