CS 254: Computational Complexity
General Information
Instructor: Li-Yang Tan (liyang@cs.stanford.edu)
CAs: Tom Knowles (tknowles@stanford.edu)
Can Liu (canliu@stanford.edu)
Time: Mondays and Wednesdays, 4:00-5:20pm
Tom's OH: Saturdays 12-1pm
Can's OH: Tuesdays 7-9pm
Homework parties: Mondays 6-8pm
CAs: Tom Knowles (tknowles@stanford.edu)
Can Liu (canliu@stanford.edu)
Time: Mondays and Wednesdays, 4:00-5:20pm
Tom's OH: Saturdays 12-1pm
Can's OH: Tuesdays 7-9pm
Homework parties: Mondays 6-8pm
Textbooks
Computational Complexity: A Modern Approach, by Sanjeev Arora and Boaz Barak.
Mathematics and Computation, by Avi Wigderson.
Mathematics and Computation, by Avi Wigderson.
List of Topics
- Space Complexity
ST-connectivity and its role in space complexity
Non-determinism in space complexity: Savitch's theorem
NL = coNL: Immerman-Szelepcsényi theorem
- Polynomial Hierarchy
P, NP, coNP, and friends
NP ∩ coNP ≈ having a good characterization
Efficient computation in a world where P = NP - Randomized Complexity
Randomness as a resource. Does P = BPP?
Randomness versus non-determinism
Unique-SAT: Valiant-Vazirani theorem - Non-Uniform Computation
Circuit complexity
Randomness versus non-uniformity: Adelman's theorem
Small circuits for NP? Karp-Lipton theorem
- Interactive Proofs
Arthur and Merlin, and generalizations of NP
Approximate counting: Goldwasser-Sipser theorem
IP = PSPACE - If time permits:
Beyond worst-case complexity
Hardness within P
Circuit lower bounds
Hardness versus randomness
Barriers to P versus NP
...
Lecture schedule
(Will be updated as the quarter progresses. Supplementary material listed in gray.)
- Jan 11: Course overview; the grand challenges of complexity theory
- Jan 13: CS154 recap
- Jan 20: Space complexity; Savitch’s theorem (AB §4.1-4.3)
- Jan 25: Nondeterministic space and NL-completeness of STCONN (AB §4.4)
The complexity of graph connectivity, Avi Wigderson
Undirected connectivity in log-space, Omer Reingold - Jan 27: Immerman-Szelepcsényi theorem (AB §4.4)
- Feb 1: NP, coNP, and NP ∩ coNP (AB §2.6-2.7)
Chapter 3.5 of Wigderson's book: The class coNP, the NP versus coNP question, and efficient characterization
Chapter 6 of Wigderson's book: Proof complexity
Propositional proof complexity: past, present, and future, Paul Beame and Toniann Pitassi
The limits of proof, video of a talk by Paul Beame
Proof complexity 2020, video of a talk by Paul Beame
- Feb 3: The polynomial hierarchy (AB §5)
Completeness in the polynomial-time hierarchy, Marcus Schaefer and Chris Umans
- Feb 8: PSAT and oracle Turing machines (AB §5)
- Feb 10: The power of randomness in computation (AB §7)
Chapter 7 of Wigderson's book: Randomness in computation
Finding hay in haystacks: the power and limits of randomness, video of a talk by Avi Wigderson
Pseudorandomness, monograph by Salil Vadhan - Feb 17: Randomized complexity. P versus BPP; NP versus BPP (AB §7)
Pure randomness extracted from two poor sources, Don Monroe
How random is your randomness, and why does it matter, Eshan Chattopadhyay and David Zuckerman
Research Vignette: Ramsey graphs and the error of explicit 2-source extractors, Amnon Ta-Shma
- Feb 22: Non-uniform computation and circuit complexity (AB §6)
Chapter 5 of Wigderson's book: Lower bounds, boolean circuits, and attacks on P vs. NP
P =? NP, Scott Aaronson
Some estimated likelihoods for computational complexity, Ryan Williams
- Feb 24: Relating P/poly to BPP and NP: Adelman's theorem and the Karp-Lipton theorem (AB §6)
- March 1: Interactive proofs (AB §8)
Chapter 10 of Wigderson's book: Randomness in proofs
Proofs, Knowledge, and Computation, video of a talk by Silvio Micali
A history of the PCP theorem, Ryan O'Donnell
E-mail and the unexpected power of interaction, László Babai
1993 Gödel prize citation, for Babai-Moran and Goldwasser-Micali-Rackoff - March 3: Interactive proof for #3SAT (AB §8)
Evaluation
• 4 problem sets (70%, weighted by total score per set)
• Course project (30%)
☐ Interim progress report (5%)
☐ Final written report (15%)
☐ Your peer evaluation report (10%)
• CR: ≥ 70% on 2 psets or ≥ 70% on Course project
Problem set policies:
• Course project (30%)
☐ Interim progress report (5%)
☐ Final written report (15%)
☐ Your peer evaluation report (10%)
• CR: ≥ 70% on 2 psets or ≥ 70% on Course project
• 4 late days, at most 2 per pset
• Late days can only be used for psets, not the project
• Regrade requests must be submitted within 1 week
• Late days can only be used for psets, not the project
• Regrade requests must be submitted within 1 week
Coursework schedule
(Tentative; subject to change.)
• Pset 1: Jan 27
• Pset 2: Feb 10
• Pset 3: Feb 24
• Pset 4: Mar 8
• Course project
☐ Decide on project topic: Feb 17
☐ Interim progress report: Mar 3
☐ Final written report: Mar 15
☐ Your peer evaluation report: Mar 19
• Pset 2: Feb 10
• Pset 3: Feb 24
• Pset 4: Mar 8
• Course project
☐ Decide on project topic: Feb 17
☐ Interim progress report: Mar 3
☐ Final written report: Mar 15
☐ Your peer evaluation report: Mar 19