CS 151: Complexity Theory (Spring 2011)

Instructor: Chris Umans

Office: Annenberg 311

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

TAs:

• Arda Antikacioglu
• Patrick Xia

Office hours:

• Wednesdays 3-4 in Annenberg 311 (Chris)
• Wednesdays 8-9 in Annenberg 106 (Arda or Patrick)

Announcements:

• All solutions are now posted. Graded material and final grades may be picked up from me (Annenberg 311). Have a great summer!

Handouts:

• Syllabus (pdf)

Lecture slides:

• Lecture 1: intro; languages, complexity classes, Turing Machines (ppt, pdf)
• Lecture 2: reductions and completeness, time and space classes, hierarchy theorems, relationships between classes, a P-complete problem, padding and succinctness (ppt, pdf)
• Lecture 3: nondeterminism, NP- and NEXP- complete problems, NTIME hierarchy theorem, Ladner's Theorem (ppt, pdf)
• Lecture 4: unary languages and NP, nondeterministic space classes, STCONN, Savitch's Theorem, I-S Theorem, circuits and uniformity (ppt, pdf)
• Lecture 5: uniformity and advice, NC hierarchy, formula lower bound on Andreev function (ppt, pdf)
• Lecture 6: 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, error reduction, BPP in P/poly (ppt, pdf)
• Lecture 8: Goldreich-Levin hard bit, Yao's Lemma, BMY generator (ppt, pdf)
• Lecture 9: Nisan-Wigderson generator, error-correcting codes (ppt, pdf)
• Lecture 10: transforming worst-case hardness into average-case hardness, extractors (ppt, pdf)
• Lecture 11: Trevisan's extractor, strong error reduction, oracles, the Polynomial-Time Hierarchy (ppt, pdf)
• Lecture 12: the PH and alternating quantifiers, complete problems for levels of the PH and PSPACE, Karp-Lipton Theorem (ppt, pdf)
• Lecture 13: BPP in PH, the class #P, complete problems for #P, interactive proof systems, graph non-isomorphism (ppt, pdf)
• Lecture 14: the power of IP, graph non-isomorphism, IP = PSPACE, Arthur-Merlin games, the classes MA and AM (ppt, pdf)
• Lecture 15: derandomization of the classes MA and AM, optimization problems and approximation algorithms (ppt, pdf)
• Lecture 16: elements of the proof of the PCP Theorem (ppt, pdf)
• Lecture 17: finishing up PCPs, relativization, natural proofs (ppt, pdf)
• Lecture 18: course summary and worked through PS7 problem 3 (ppt, pdf)

Problem sets:

• Problem Set 1: (pdf, LaTeX source) Solutions (pdf) (pts: 15; mean: 12.0; median: 13)
• Problem Set 2: (pdf, LaTeX source) Solutions (pdf) (pts:24; mean: 21.4; median: 22)
• Problem Set 3: (pdf, LaTeX source) Solutions (pdf) (pts: 36; mean: 32.3; median: 34.5)
• Problem Set 4: (pdf, LaTeX source) Solutions (pdf) (pts: 24; mean: 19.1; median: 23)
• Midterm: (pdf, LaTeX source) Solutions (pdf) (pts: 15; mean: 10.7; median: 12.5)
• Problem Set 5: (pdf, LaTeX source) Solutions (pdf) (pts: 24; mean: 18.8; median 20)
• Problem Set 6: (pdf, LaTeX source) Solutions (pdf) (pts: 21; mean: 18.0; median 18.0)
• Problem Set 7: (pdf, LaTeX source) Solutions (pdf) (pts: 18; mean: 14.3; median 17.0)
• Final: (pdf, LaTeX source) Solutions (pdf) (pts: 36; mean: 28.4; median 34.0)

Resources:

• Here is a LaTeX template (tex, pdf) that you can use for your writeups if you wish.