CS153: Current Topics in Theoretical Computer Science (Spring 2010)

Instructor: Chris Umans

Office: Annenberg 311

Times: Tu/Th 1:00-2:25 in Annenberg 314

Office hours: email me

Announcements:

• Final grades have been entered. Please stop by if you'd like to pick up graded material. Have a great summer!
• Solutions for PS2 have been posted.
• Presentations for Tuesday June 1: Dave (Khot-Regev); Bill (Aharanov-Arad-Landau-Vazirani); Arda (low-degree testing)
• Presentations for Thursday June 3: Jeremy (Khot-Kindler-Mossel-O'Donnell); Ben (Raghavendra-Steurer 2010); Chris (Kolla); Emma (O'Donnell-Wu); Brian (Raz 2008 "A counterexample...")

Handouts:

• Syllabus (pdf)

Lectures:

• Lecture 1: introduction; approximation algorithms
• Lecture 2: GW algorithm for MAX-CUT; hardness of approximation: gap problems, connection to locally/probabilistically checkable proof systems; Label Cover
• Lecture 3: the basic algebraic PCP
• Lecture 4: robust soundness of basic PCP, equivalence between robust PCPs and label cover
• Lecture 5: decodable PCPs, composition theorem, beginning of proof [from DH09]
• Lecture 6: end of proof of composition theorem, the Long code, Fourier analysis, the Long code test
• Lecture 7: analysis of the Long code test, Hastad's 3-bit PCP
• Lecture 8: hardness of MAX-3-LIN, MAX-3-SAT; first steps toward hardness of MAX-CUT: the Unique Games Conjecture, influence of variables
• Lecture 9: hardness of MAX-CUT
• Lecture 10: hardness of MAX-CUT; SDP relaxations again: for MAX-CUT with large OPT and for Unique Label Cover
• Lecture 11: rounding SDP for ULC; Raghavendra's paper
• Lecture 12: portions of the proof from Raghavendra's paper
• Lecture 13: Raghavendra continued
• Lecture 14: algorithms for Unique Games: ABS 2010
• Lecture 15: ABS continued

Problem sets:

• Problem Set 1: (LaTeX source, pdf) Solutions (pdf)
• Problem Set 2: (LaTeX source, pdf) Solutions (pdf)

Papers:

• P. Harsha and I. Dinur. Composition of low-error 2-query PCPs using decodable PCPs. 2009.
• P. Harsha. Robust PCPs of Proximity and Shorter PCPs. 2004.
• S. Khot, G. Kindler. E. Mossel, R. O'Donnell. Optimal inapproximability results for MAX-CUT and other two-variable CSPs? 2004.
• S. Arora and M. Sudan. Improved low-degree testing and its applications. 2003.
• P. Raghavedra. Optimal Algorithms and Inapproximability Results for every CSP? 2008.
• S. Arora, R. Impagliazzo, W. Matthews, D. Steurer. Improved Algorithms for Unique Games via Divide and Conquer. 2010.
• S. Arora, B. Barak, D. Steurer. Subexponential Algorithms for Unique Games and Related Problems 2010.

Surveys:

• I. Dinur. PCPs with small soundness error. 2008
• S. Khot. Inapproximability results via Long Code based PCPs. 2005.

Course notes:

• Prahladh Harsha: Limits of approximation algorithms: PCPs and Unique Games. 2010.
• Prahladh Harsha: PCPs, codes and inapproximability. 2007.
• DIMACS tutorial: Limits of Approximation Algorithms: PCPs and Unique Games. 2010.
• Venkat Guruswami and Ryan O'Donnell: The PCP Theorem and Hardness of Approximation. 2005.
• Madhu Sudan: Approximability of Optimization Problems. 1999.

Possible papers for presentation:

• Irit Dinur's relevant papers, available here.
• Prahladh Harsha's relevant papers, available here.
• Subhash Khot's relevant papers, available here.
• Ryan O'Donnell's relevant papers, available here.
• Prasad Raghavendra's relevant papers, available here.
• David Steurer's relevant papers, available here.
• G. Kol and R. Raz. Bounds on 2-Query Locally Testable Codes with Affine Tests. 2009.
• See also Locally Testable Codes Analogues to the Unique Games Conjecture Do Not Exist by the same authors.