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École normale supérieure
Approximation Algorithms Part II
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École normale supérieure

Approximation Algorithms Part II

Claire Mathieu

Instructor: Claire Mathieu

12,227 already enrolled

4 modules
Gain insight into a topic and learn the fundamentals.
4.8

(44 reviews)

35 hours to complete
3 weeks at 11 hours a week
Flexible schedule
Learn at your own pace

4 modules
Gain insight into a topic and learn the fundamentals.
4.8

(44 reviews)

35 hours to complete
3 weeks at 11 hours a week
Flexible schedule
Learn at your own pace
  • About
  • Modules
  • Recommendations
  • Testimonials
  • Reviews

Skills you'll gain

  • Algorithms
  • Solution Design
  • Combinatorics
  • Mathematical Modeling
  • Linear Algebra
  • Probability
  • Operations Research
  • Theoretical Computer Science

Details to know

Assessments

33 assignments

Taught in English

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There are 4 modules in this course

Approximation algorithms, Part 2

This is the continuation of Approximation algorithms, Part 1. Here you will learn linear programming duality applied to the design of some approximation algorithms, and semidefinite programming applied to Maxcut. By taking the two parts of this course, you will be exposed to a range of problems at the foundations of theoretical computer science, and to powerful design and analysis techniques. Upon completion, you will be able to recognize, when faced with a new combinatorial optimization problem, whether it is close to one of a few known basic problems, and will be able to design linear programming relaxations and use randomized rounding to attempt to solve your own problem. The course content and in particular the homework is of a theoretical nature without any programming assignments. This is the second of a two-part course on Approximation Algorithms.

This module does not study any specific combinatorial optimization problem. Instead, it introduces a central feature of linear programming, duality.

What's included

9 videos11 readings8 assignments1 peer review

9 videos•Total 87 minutes
  • Linear programming duality - example•15 minutes•Preview module
  • Properties of LP duality•6 minutes
  • Geometry of LP duality•10 minutes
  • Proof of weak duality theorem•6 minutes
  • Changing the form of the LP•10 minutes
  • Complementary slackness•5 minutes
  • Primal-dual algorithms•5 minutes
  • Vertex cover by primal-dual•23 minutes
  • Conclusion•3 minutes
11 readings•Total 110 minutes
  • Slides•10 minutes
  • Comment•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides-all•10 minutes
8 assignments•Total 240 minutes
  • Quiz 1•30 minutes
  • Quiz 2•30 minutes
  • Quiz 3•30 minutes
  • Quiz 4•30 minutes
  • Quiz 5•30 minutes
  • Quiz 6•30 minutes
  • Quiz 7•30 minutes
  • Quiz 8•30 minutes
1 peer review•Total 120 minutes
  • Assignment 1•120 minutes

This module uses linear programming duality to design an algorithm for another basic problem, the Steiner forest problem.

What's included

8 videos9 readings8 assignments1 peer review

8 videos•Total 72 minutes
  • Problem definition•3 minutes•Preview module
  • A special case: Steiner tree•12 minutes
  • LP relaxation for Steiner forest•6 minutes
  • ... and its dual•4 minutes
  • Primal-dual algorithm, Part1•10 minutes
  • Primal-dual algorithm,Part 2•12 minutes
  • Analysis•13 minutes
  • Proof of the main lemma•9 minutes
9 readings•Total 90 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides-all•10 minutes
8 assignments•Total 240 minutes
  • Quiz 1•30 minutes
  • Quiz 2•30 minutes
  • Quiz 3•30 minutes
  • Quiz 4•30 minutes
  • Quiz 5•30 minutes
  • Quiz 6•30 minutes
  • Quiz 7•30 minutes
  • Quiz 8•30 minutes
1 peer review•Total 120 minutes
  • Assignment 2•120 minutes

This module continues teaching algorithmic applications of linear programming duality by applying it to another basic problem, the facility location problem.

What's included

9 videos10 readings8 assignments1 peer review

9 videos•Total 63 minutes
  • Problem definition•5 minutes•Preview module
  • A linear programming relaxation•4 minutes
  • ...and its dual•8 minutes
  • A primal-dual algorithm•7 minutes
  • Analyzing the service cost•7 minutes
  • Analyzing the facility opening cost•7 minutes
  • A better algorithm•11 minutes
  • Analysis•7 minutes
  • Conclusion•4 minutes
10 readings•Total 100 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides-all•10 minutes
8 assignments•Total 240 minutes
  • Quiz 1•30 minutes
  • Quiz 2•30 minutes
  • Quiz 3•30 minutes
  • Quiz 4•30 minutes
  • Quiz 5•30 minutes
  • Quiz 6•30 minutes
  • Quiz 7•30 minutes
  • Quiz 8•30 minutes
1 peer review•Total 60 minutes
  • Assignment 3•60 minutes

We introduce a generalization of linear programming, semi-definite programming.This module uses semi-definite programming to design an approximation algorithm for another basic problem, the maximum cut problem.

What's included

11 videos12 readings9 assignments1 peer review

11 videos•Total 76 minutes
  • Definition•5 minutes•Preview module
  • A 2-approximation•5 minutes
  • A linear programming relaxation...•11 minutes
  • ...with an integrality gap of almost 2•10 minutes
  • Proof of Lemma•7 minutes
  • A quadratic programming relaxation•4 minutes
  • General facts about semidefinite programming•7 minutes
  • A rounding algorithm•7 minutes
  • Analysis•6 minutes
  • General facts about MaxCut•6 minutes
  • The end!•3 minutes
12 readings•Total 120 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Slides•10 minutes
  • Sldies•10 minutes
  • Slides•10 minutes
  • Slides-all•10 minutes
  • Comment•10 minutes
9 assignments•Total 270 minutes
  • Quiz 1•30 minutes
  • Quiz 2•30 minutes
  • Quiz 3•30 minutes
  • Quiz 4•30 minutes
  • Quiz 5•30 minutes
  • Quiz 6•30 minutes
  • Quiz 7•30 minutes
  • Quiz 8•30 minutes
  • Quiz 9•30 minutes
1 peer review•Total 120 minutes
  • Assignment 4•120 minutes

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Instructor

Claire Mathieu
Claire Mathieu
École normale supérieure
2 Courses•32,178 learners

Offered by

École normale supérieure

Offered by

École normale supérieure

L’École normale supérieure (ENS) est un établissement d'enseignement supérieur pour les études prédoctorales et doctorales (graduate school) et un haut lieu de la recherche française. L'ENS offre à 300 nouveaux étudiants et 200 doctorants chaque année une formation de haut niveau, largement pluridisciplinaire, des humanités et sciences sociales aux sciences dures. Régulièrement distinguée au niveau international, l'ENS a formé 10 médailles Fields et 13 prix Nobel.

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4.8

44 reviews

  • 5 stars

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  • 4 stars

    6.81%

  • 3 stars

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Showing 3 of 44

A
AP
5

Reviewed on Oct 28, 2016

Demanding course with lots of great algorithm concepts based on Linear Programming.

P
PV
5

Reviewed on Feb 16, 2017

Even better than the first! Very good classes (except for the two first of week 3 ...)

R
RA
5

Reviewed on Mar 14, 2016

It is remarkable to note that Professor Claire Mathieu explains such a complex subject in such a elegant and understandable manner.

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