CS 151: Complexity Theory (Spring 2007)

Instructor: Chris Umans

Office: Jorgensen 286

Times: Tu/Th 1:00-2:25 in Jorgensen 74

TAs:

• Xiaojie Gao
• Kevin Ko

Office hours:

• Wednesdays 3-4 in Jorgensen 286 (Chris)
• by appointment (Xiaojie)
• by appointment (Kevin)

Announcements:

• Packets of graded material (including the final) can be picked up from me this week.
• Kevin has prepared a LaTeX template (tex, pdf) that you can use for your writeups if you wish.

Handouts:

• Syllabus (pdf)

Lecture Slides:

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

Problem Sets:

• Problem Set 1: (pdf) Solutions: (pdf) (pts: 9; mean 7.6)
• Problem Set 2: (pdf) Solutions: (pdf) (pts: 21; mean 16.5)
• Problem Set 3: (pdf) Solutions: (pdf) (pts: 33; mean 29.2)
• Problem Set 4: (pdf) Solutions: (pdf) (pts: 15; mean 11.6)
• Midterm: (pdf) Solutions: (pdf) (pts:21; mean 16.75)
• Problem Set 5: (pdf) Solutions: (pdf) (pts:12; mean 10.4)
• Problem Set 6: (pdf) Solutions: (pdf) (pts:12; mean 11.0)
• Problem Set 7: (pdf) (LaTeX source) Solutions: (pdf) (pts: 15; mean 14.2)
• Final: (pdf) (LaTeX source) Solutions: (pdf) (pts: 24; mean 19.4)