­­­­­­­­CS395T: Approximability

51815 | ­Fall 2016 | T TH 11-12:30 | GDC 4.302  

 

Instructor: Dana Moshkovitz

 

This class is about approximation algorithms and their limitations. It covers: combinatorial approximation algorithms; approximation algorithms based on linear and semidefinite programming; hierarchies of linear and semidefinite programming; limitations of the aforementioned techniques with connections to communication complexity and proof complexity; hardness of approximation and connections to multi-prover games and probabilistic checking of proofs; the proof of the PCP theorem, including combinatorial and algebraic techniques; sum-check; linearity testing; low degree testing; locally testable and decodable codes; composition; parallel repetition; long code, Fourier analysis and optimal inapproximability results; the Unique Games Conjecture; dictator tests and integrality gaps.

 

Format: Some of the meetings will be regular lectures, and some will be “workshops” in which the students give mini-lectures. Before each workshop there will be a schedule with the topics of the mini-lectures. Usually there will be problems to solve and a mini-lecture will present a solution, but sometimes mini-lectures will be surveys or discussions of open problems. Students should register to give mini-lectures after the release of each workshop schedule. Workshops will give students ample opportunity to give lectures and receive feedback. There are no exams or problem sets apart from workshop participation and presentation.

 

Syllabus

 

 

Date

Topic

1

Thursday,

August 25

Overview; basic combinatorial approximation algorithms: vertex cover, set cover, 3SAT

2

Tuesday,

August 30

Workshop on approximation basics, including constraint satisfaction problems, label-cover, unique label-cover

3

Thursday, September  1

Linear programming (LP) and approximation algorithms

4

Tuesday, Septemb­­­er 6

Self-study on primal-dual method

5

Thursday, September 8

Problem solving in teams/LP-based approximation

6

Tuesday,

September 13,

Workshop on approximation algorithms based on linear programming

7

Thursday,

September 15,

Semidefinite programming (SDP), Max-Cut approximation and integrality gap

8

Tuesday,

September 20,

Workshop on SDP and their limitations

9

Thursday,

September 22

Hierarchies of linear and semidefinite programming and their limitations

10

Tuesday,

September 27

Workshop on hierarchies, connections to proof complexity and communication complexity

11

Thursday, September 29

Hardness of Approximation, Multi-prover games and probabilistic checking of proofs (PCP)

12

Tuesday, October 4

Workshop on PCP basics, including expanders and randomness efficient sequential repetition, transformation to two queries

13

Thursday, October 6

Linearity testing

14

Tuesday, October 11

Fourier analysis

15

Thursday, October 13

Hadamard-based PCP

16

Tuesday, October 18

Workshop on linearity testing and quadratic testing, Fourier, locally testable and decodable codes

17

Thursday, October 20

Sum-Check and PCP with polylog queries

18

Tuesday, October 25

Workshop on sum-check and its many applications, low degree testing,

19

Thursday, October 27

Composition, PCP of proximity, robust PCP, decoding PCP

20

Tuesday, November 1

Workshop on composition, connection to code concatenation

21

Thursday, November 3

Gap amplification (powering)

22

Tuesday, November 8

Workshop on gap amplification and its limitations

23

Thursday, November 10

Parallel repetition

24

Tuesday, November 15

Workshop on parallel repetition, including lower bounds

25

Thursday, November 17

The long code and optimal inapproximability results

26

Tuesday, November 22

Workshop on long code and optimal inapprox

Thursday, November 24

Thanksgiving

27

Tuesday, November 29

The Unique Games Conjecture

28

Thursday, December 1

Dictator tests and integrality gaps