CS599: Linear and Semi-Definite Programming in Approximation Algorithms(Autumn 2014)

Konstantin Makarychev Email: komakary at microsoft(dot)com

Mohit Singh Email: mohits at microsoft(dot)com

Time: Monday/Friday 10:30-11:50am
Location: CSE 403.

Course description

The area of approximation algorithms deals with finding provable guarantees on the performance of heuristic solutions to hard combinatorial optimization problems. Linear and semi-definite programming have played a central role in development of approximation algorithms which will be the focus of this course. The course will introduce the rich class of algorithmic techniques developed to analyze linear and semi-definite programing relaxations. The course will start from basics and also cover some recent advances in the field. Topics include: LP and SDP Duality, Randomized rounding, Graph Coloring, Interior Point Algorithms, Graph Partitioning, Discrepancy and Unique Games. The course assumes background in basic probability theory. Key mathematical concepts will be reviewed before they are used, but a certain level of mathematical maturity is expected. The course does not involve any programming and can be taken either for a grade or for credit/no-credit.

Grading:

There will be 3-4 homeworks and no exams. Grading will be based on the homeworks.

Lecture Readings.

Lecture 1. Introduction. Set Cover. Chapter 1 from Book WS.
Lecture 2. Min-Cut, LP-Duality, Multiway Cut.
Lecture 3. Multiway Cut. 8.1 and 8.2 from Book WS.
Lecture 4. Sparsest Cut and Metric Embeddings. Notes.
Lecture 5. Primal Dual Method. 7.3 and 7.4 from Book WS.
Lecture 6. Iterative Rounding. 11.2 from Book WS.
Lecture 7. SDP Introduction. Max-Cut. Chapter 6 from Book WS.
Lecture 8. Graph Coloring. 13.2 from Book WS.
Lecture 9. Grothendieck Inequality.
Lecture 10. Ellipsoid Method. 11.2 from Lecture Notes due to Michel Goemans.
Lecture 11. Interior Point Method. See Paper by Monteiro and Adler. and also see Chapter 6 from Gartner and Matousek.
Lecture 12. Sparset Cut and SDPs.
Lecture 13. ARV algorithm.
Lecture 14. ARV contd and metric embeddings. Section 4 from James Lee's paper
Lecture 15. Discrepancy and SDPs Bansal's paper.
Lecture 16. Unique Games. Notes.
Lecture 17. Unique Games and Orthogonal Separators. Notes.
Lecture 18. Hierarchies. Notes by Thomas Rothvoss.

Homeworks.

Homework 1. Due 10/24. Homework 1 Solutions.
Homework 2. Due 12/5. Homework 2 Solutions.
Homework 3. Due 12/12.