CS 151: Complexity Theory (Spring 2021)

Instructor: Chris Umans

Office: Annenberg 311

Times: Tu/Th 1:00-2:25 via Zoom

TAs: Vinayak Kumar and Mayank Pandey

Office hours: Zoom links are on the Canvas course page. All times are Pacific Time.

• Wednesdays 2-3pm (Chris)
• Wednesdays 6-7pm (Vinayak)
• Wednesdays 8-9pm (Mayank)

Announcements:

• All solutions are posted. You should have received your final grades. Have a wonderful summer everyone!

Handouts:

• Syllabus (pdf)
• Collaboration table (pdf)

Lecture slides:

• Lecture 1: intro; languages, complexity classes, Turing Machines (pptx, pdf)
• Lecture 2: reductions and completeness, time and space classes, hierarchy theorems, relationships between classes (pptx, pdf)
• Lecture 3: a P-complete problem, padding and succinctness, nondeterminism, NP- and NEXP- complete problems (pptx, pdf)
• Lecture 4: NTIME hierarchy theorem, Ladner's Theorem, unary languages and NP (pptx, pdf)
• Lecture 5: nondeterministic space classes, STCONN, Savitch's Theorem, I-S Theorem, nonuniformity and advice (pptx, pdf)
• Lecture 6: NC hierarchy, formula lower bound on Andreev function (pptx, pdf)
• Lecture 7: Razborov's lower bound on monotone circuits for clique (pptx, pdf)
• Lecture 8: Schwartz-Zippel, Valiant-Vazirani Theorem, randomized complexity classes, error reduction, BPP in P/poly, (pptx, pdf)
• Lecture 9: Goldreich-Levin hard bit, Yao's Lemma, BMY generator (pptx, pdf)
• Lecture 10: Nisan-Wigderson generator, error-correcting codes (pptx, pdf)
• Lecture 11: transforming worst-case hardness into average-case hardness (pptx, pdf)
• Lecture 12: extractors, RL, oracles, the PH and alternating quantifiers (pptx, pdf)
• Lecture 12.5: the PH and alternating quantifiers, complete problems for levels of the PH and PSPACE, Karp-Lipton Theorem, BPP in PH, the class #P, complete problems for #P (pptx, pdf)
• Lecture 13: #Matching is #P-completes, interactive proof systems, graph non-isomorphism, the power of IP (pptx, pdf)
• Lecture 14: IP = PSPACE, Arthur-Merlin games, the classes MA and AM (pptx, pdf)
• Lecture 15: derandomization of MA and AM, optimization, approximation, and PCPs (pptx, pdf)
• Lecture 16: elements of the proof of the PCP Theorem (pptx, pdf)
• Lecture 17: finishing up PCPs (pptx, pdf)
• Lecture 18: relativization and natural proofs; course summary (pptx, pdf)

Problem sets:

• Problem Set 1: (pdf, LaTeX source) Solutions (pdf)
• Problem Set 2: (pdf, LaTeX source) Solutions (pdf)
• Problem Set 3: (pdf, LaTeX source) Solutions (pdf)
• Problem Set 4: (pdf, LaTeX source) Solutions (pdf)
• Midterm: (pdf, LaTeX source) Solutions (pdf)
• Problem Set 5: (pdf, LaTeX source) Solutions (pdf)
• Problem Set 6: (pdf, LaTeX source) Solutions (pdf)
• Problem Set 7: (pdf, LaTeX source) Solutions (pdf)
• Final: (pdf, LaTeX source) Solutions (pdf)

Resources:

• Here is a LaTeX template (tex, pdf) that you can use for your writeups if you wish.
• all 2017 lectures as a .tar.gz archive.
• Videos of 2019 lectures (Caltech only) here.