15-855: Graduate Computational Complexity Theory, Fall 2017

Prerequisite: An undergraduate course in computational complexity theory, covering most of "Part III" of Sipser and/or most of Carnegie Mellon's 15-455.

Potential topics: Models and Time Hierarchy Theorem. Nondeterminism, padding, Hopcroft-Paul-Valiant Theorem. Circuits and advice. Randomized classes. Cook-Levin Theorem and SAT. Nondeterministic Time Hierarchy Theorem, and nondeterministic models. Oracles, alternation, and the Polynomial Time Hierarchy. Kannan's Theorem, Karp-Lipton, and PH vs. constant-depth circuits. Time-Space tradeoffs for SAT. Randomized classes vs. PH. Interactive proofs and the AM hierarchy. NP in BPP implies PH in BPP, and Boppana-Hastad-Zachos. BCGKT Theorem and Cai's Theorem. Counting classes and the permanent. Valiant's Theorem. Algebraic Complexity. IP = PSPACE and interactive proofs. Instance checkers and Santhanam's Theorem. Random restrictions and AC0 lower bounds for parity. Monotone circuit lower bounds. Razborov-Smolensky lower bounds for AC0[p]. Valiant-Vazirani and Toda Theorems. Beigel-Tarui Theorem. Hardness vs. Randomness and Nisan-Wigderson. Hardness amplification and derandomization. Williams's Theorem. Natural proofs and barriers.

There will be 11 homeworks, and two take-home "tests".

• Homeworks will come out on Tuesdays and be due on Tuesdays.
• Homeworks will have 3 problems, each worth 10 points. Due to TA time limitations, only 2 out of the 3 problems will be graded.

• The two take-home "tests" will be Oct. 10 -- Oct. 17 and Nov. 30 -- Dec. 7.
• The tests will have 4 problems, each worth 20 points. All 4 problems will be graded.

Your well-being and happiness is very important to us at CMU, and there are many resources to help you with it. Please contact me directly if you need assistance or would like to talk about any such issues.

• All homework must be typeset, with PDFLaTeX highly encouraged.
• For LaTeX help, I recommend the book More Math Into LaTeX by Grätzer and also the website TeX Stack Exchange.
• Homework must be turned in on Gradescope; it will generally be due Tuesdays at 10am.
• You have six (6) "late days" for use on homework throughout the semester.
• No more than 3 late days can be used on any one assigment.
• Any amount of time between 1 minute and 24 hours counts as one late day.
• You are responsible for knowing how many late days you have left. If you are unsure, you could ask a TA.
• Except through the use of late days described above, no late homework will be accepted. If you are out of late days and your homework is late, you will get 0 points.
• You may discuss homework problems with others in the class. However, you should not share any written notes, and of course all your writeups should be done individually.
• Homework will be graded both for correctness and for clarity of exposition.

• Same instructions as for the homework, except: no late days may be used, and no collaboration of any sort allowed.

Let me know if your favorite book or set of lecture notes does not appear here!

Textbooks:

• Computational Complexity: A Modern Approach, by Arora and Barak
• Computational Complexity, by Papadimitriou
• Theory of Computational Complexity, by Du and Ko
• Complexity Theory, by Wegener
• Computational Complexity: A Conceptual Perspective, by Goldreich
• The Complexity Theory Companion, by Hemaspaandra and Ogihara
• Theory of Computation, by Kozen
• Computability and Complexity Theory, by Homer and Selman
• Structural Complexity I and II, by Balcázar, Díaz, and Gabarró
• Boolean Function Complexity: Advances and Frontiers, by Jukna
• The Nature of Computation, by Moore and Mertens
• Introduction to the Theory of Computation, by Sipser

• Van Melkebeek: 2007, 2010, 2011, 2015, 2016
• Harsha: '11/'12, '12/'13, '13/'14
• Trevisan: 2001, 2002, 2004, 2008, 2010, 2012, 2014
• Sudan: 2002, 2003, 2007, 2009
• Spielman: 1998, 1999, 2000, 2001
• Moshkovitz: 2012, 2016
• Katz: 2005, 2011
• Umans
• Hansen 2010
• Vadhan 2002
• Cai
• Gács and Lovász
• Beame 2008
• Arora 2001
• Miltersen 2006
• Håstad
• M. Naor '04/'05
• Guruswami--O'Donnell 2009

• Regan 2015