1. 1 Computational Complexity CPSC 468/568, Fall 2007 Time: Tu & Th, 1:00-2:15 pm Room: AKW 200 Dist. Group: IV (not Natural Sci.) Satisfies “QR” http://zoo.cs.yale.edu/classes/cs468

2. 2 Partial Topic Outline Complexity classes (P, NP, L, NL, etc.) Reductions and completeness The roles of, e.g., –Randomness –Interaction –Approximation Communication complexity

3. 3 Schedule Sept. 20: First HW Assignment Due Oct. 2: Second HW Assignment Due Oct. 11: Third HW Assignment Due Oct. 18: First In-Class Exam Oct. 23: Grades on HW1, HW2, HW3, and Ex1 will be returned to students today or earlier. Oct. 26: Drop Date Oct. 31: Fourth HW Assignment Due Nov. 13: Fifth HW Assignment Due Nov. 29: Sixth HW Assignment Due Dec. 6: Second In-Class Exam

4. 4 Requirements Modest reading assignments 6 Written HW Assignments, each of will each be worth 10% of the course grade 2 In-Class Exams, each worth 20% of the course grade No final exam during exam week

5. 5 Rules and Guidelines Deadlines are firm. Late penalty: 5% per day. Announcements and assignments will be posted on the class webpage (as well as conveyed in class). No “collaboration” on homeworks unless you are told otherwise. Pick up your graded homeworks and exams promptly, and tell the TA promptly if one is missing.

6. 6 Instructor: Joan Feigenbaum Office: AKW 512 Office Hours: Tuesday 2:30 - 3:30 pm Thursday 11 am - noon Phone: 203-432-6432 Assistant: Judi Paige (judi.paige@yale.edu, 203-436-1267, AKW 507a, 9am-2pm M-F) Note: Do not send email to Professor Feigenbaum, who suffers from RSI. Contact her through Ms. Paige or the TA.

7. 7 TA: Nicholas Ruozzi Office: AKW 305 Office Hours: TBA Email: nicholas.ruozzi@yale.edu

8. 8 If you’re undecided … Check out: zoo.cs.yale.edu/classes/cs468/spr07 www.cs.princeton.edu/theory/complexity/ (a new complexity-theory text by Sanjeev Arora and Boaz Barak of Princeton) www.cs.berkeley.edu/~luca/cs278-02/ (a complexity-theory course taught by Luca Trevisan at Berkeley in 2002) www.cs.lth.se/home/Rolf_Karlsson/bk/retro.pdf (“NP-Completeness: A Retrospective,” by Christos Papadimitriou, 1997 International Colloquium on Automata, Languages, and Programming)

9. 9 Questions?

10. 10 Introduction to Complexity Classes

11. 11 Computational Complexity Themes “Easy” vs. “Hard” Reductions (Equivalence) Provability Randomness

12. 12 Poly-Time Solvable Nontrivial Example : Matching

13. 13 Poly-Time Solvable Nontrivial Example : Matching

14. 14 Poly-Time Verifiable Trivial Example : Hamiltonian Cycle

15. 15 Poly-Time Verifiable Trivial Ex. : Hamiltonian Cycle

16. 16 Is it Easier to Verify a Proof than to Find one? Fundamental Conjecture of Computational Complexity: P  NP

17. 17 Matching: HC: Fundamentally Different Distinctions

18. 18 Reduction of B to A If A is “Easy”, then B is, too. B Algorithm A “oracle” “black box”

19. 19 NP-completeness P-time reduction Cook’s theorem If B ε NP, then B ≤ P-time SAT HC is NP-complete

20. 20 Equivalence NP-complete problems are an equivalence Class under polynomial-time reductions. 10k’s problems Diverse fields Math, CS, Engineering, Economics, Physical Sci., Geography, Politics…

21. 21 NPcoNP P

22. 22 Random poly-time Solvable x ε L? poly-time Algorithm x r YES NO x ε {0,1} n r ε {0,1} poly(n)

23. 23 Probabilistic Classes x ε L  “yes” w.p. ¾ x  L  “no” w.p. 1 x ε L  “yes” w.p. 1 x  L  “no” w.p. ¾ RP coRP (Outdated) Nontrivial Result PRIMES ε ZPP ( = RP ∩ coRP)

24. 24 Two-sided Error BPP x  L  “yes”w.p. ¾ x  L  “no” w.p. ¾ Question to Audience: BPP set not known to be in RP or coRP?

25. 25 RPcoRP ZPP NPcoNP P

26. 26 Interactive Provability P V [PPT, ¢ ] x yes/no

27. 27 L ε IP x ε L   P: “yes” w.p. ¾ x  L   P*: “no” w.p. ¾ Nontrivial Result Interactively Provable Poly-Space Solvable

28. 28 PSPACE RPcoRP ZPP NPcoNP P

29. 29 PSPACE EXP P #P PH iPiP 2P2P NP P