CS 151: Complexity Theory (Spring 2009)

Instructor: Chris Umans

Office: Jorgensen 286

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

TAs:

• Dave Buchfuhrer
• Domenic Denicola
• Cheng William Hong

Office hours:

• Wednesdays 3-4 in Jorgensen 286 (Chris)
• Wednesdays 4-5 in Jorgensen 166 (Dave)
• Wednesdays 7-8 in VLSI Lab (=JRG 154) (Cheng)
• Wednesdays 10-11 in VLSI Lab (=JRG 154) (Domenic)

Announcements:

• Packets of graded material (with final grades for the course) are available for pickup from Diane Goodfellow in JRG 266. Have a great summer!
• Solutions for all problem sets and the midterm are posted.

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 (ppt, pdf)
• Lecture 3: a P-complete problem, padding and succinctness, nondeterminism, NTIME hierarchy theorem (ppt, pdf)
• Lecture 4: Ladner's Theorem, unary languages and NP, nondeterministic space classes, STCONN, Savitch's Theorem (ppt, pdf)
• Lecture 5: I-S Theorem, 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: finishing up Razborov's clique lower bound, randomness in communication complexity, Polynomial Identity Testing + Schwartz-Zippel,(ppt, pdf)
• Lecture 8: Valiant-Vazirani Theorem, randomized complexity classes, error reduction, BPP in P/poly, PRGs (ppt, pdf)
• Lecture 9: Goldreich-Levin hard bit, Yao's Lemma, BMY generator (ppt, pdf)
• Lecture 10: Nisan-Wigderson generator, error-correcting codes (ppt, pdf)
• Lecture 11: transforming worst-case hardness into average-case hardness, extractors (ppt, pdf)
• Lecture 12: Trevisan's extractor, strong error reduction, oracles, the Polynomial-Time Hierarchy (ppt, pdf)
• Lecture 13: the PH and alternating quantifiers, complete problems for levels of the PH and PSPACE (ppt, pdf)
• Lecture 14: Karp-Lipton Theorem, BPP in PH, the class #P, complete problems for #P, interactive proof systems (ppt, pdf)
• Lecture 15: the power of IP, graph non-isomorphism, IP = PSPACE (ppt, pdf)
• Lecture 16: Arthur-Merlin games, the classes MA and AM, optimization problems and approximation algorithms (ppt, pdf)
• Lecture 17: gap-producing reductions and PCPs, elements of the proof of the PCP Theorem (ppt, pdf)
• Lecture 18: relativization, natural proofs, course summary (ppt, pdf)

Problem sets:

• Problem Set 1: (pdf, LaTeX source) Solutions (pdf) (pts: 15; mean: 13)
• Problem Set 2: (pdf, LaTeX source) Solutions (pdf) (pts: 24; mean 18)
• Problem Set 3: (pdf, LaTeX source) Solutions (pdf) (pts: 36; mean 33.2)
• Problem Set 4: (pdf, LaTeX source) Solutions (pdf) (pts: 15; mean 12.7)
• Midterm: (pdf, LaTeX source) Solutions (pdf) (pts: 21; mean 16.3; median 19)
• Problem Set 5: (pdf, LaTeX source) Solutions (pdf) (pts: 18; mean 15.6)
• Problem Set 6: (pdf, LaTeX source) Solutions (pdf) (pts: 21; mean 18.3)
• Problem Set 7: (pdf, LaTeX source) Solutions (pdf) (pts: 18; mean: 17.0)
• Final: (pdf, LaTeX source) Solutions (pdf) (pts: 30; mean: 25.07)

Resources:

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