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Computational Complexity

Computational Complexity. CPSC 468b, Spring 2007 Time: Tu & Th, 1:00-2:15 pm Room: AKW 500 Dist. Group: IV (not Natural Sci.) Satisfies “QR” http://zoo.cs.yale.edu/classes/cs468. Partial Topic Outline. Complexity classes (P, NP, L, NL, etc .) Reductions and completeness

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Computational Complexity

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  1. Computational Complexity CPSC 468b, Spring 2007 Time: Tu & Th, 1:00-2:15 pm Room: AKW 500 Dist. Group: IV (not Natural Sci.) Satisfies “QR” http://zoo.cs.yale.edu/classes/cs468

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

  3. Schedule Feb. 1: First HW Assignment Due Feb. 13: Second HW Assignment Due Feb. 22: Third HW Assignment Due Mar. 1: First In-Class Exam Mar. 6: Grades on HW1, HW2, HW3, and Ex1 will be returned to students today or earlier. Mar. 9: Drop Date Apr. 10: Fourth HW Assignment Due Apr. 19: Fifth HW Assignment Due Apr. 26: Second In-Class Exam

  4. Requirements • Modest reading assignments • 5 Written HW Assignments, 4 of which will each be worth 15% of the course grade and one of which will be worth 10% of the course grade • 2 In-Class Exams, each worth 15% of the course grade • No final exam during exam week

  5. Rules and Guidelines • Deadlines are firm. • 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. Instructor: Joan Feigenbaum Office: AKW 512 Office Hours: Tuesday 4-5 pm Thursday 9-10 am 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. TA: Nikhil Srivastava Office: AKW 413 Office Hours: Tuesday 10am-11am Friday 10am-11am Email: nikhil.srivastava@yale.edu Phone: 203-432-1245

  8. If you’re undecided … Check out: • 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. Questions?

  10. Computational Complexity

  11. The Big Picture • Analysis of Algorithmsupper bounds on resources to solve a computational problem • Complexitylower bounds… OR • Computability Theory What can be computed? • Complexity What can be computed, given limited resources?

  12. “Computational Problem” input 2 {0,1}* output 2 {0,1}* R

  13. Decision Problem • Does the input satisfy <property>? input 2 {0,1}* output 2 {YES/NO} R

  14. Decision Problem ~ Language • L = {YES instances}µ {0,1}* {0,1}* YES

  15. Examples • Is n composite? • Is there a path of length at most k between s and t in G? (input <G,s,t,k>) • Does G contain a clique of size ¸k? • Is G 3-colorable?

  16. Example -- SAT • Is  satisfiable? • Boolean formula on x1… xn (x1Ç x3Ç x5)Æ(x3Ç x4)Æ(x4Çx5Çx1) • Is there a satisfying assignment? • YES: (1Ç 0Ç 0)Æ (0Ç 0)Æ (0Ç0Ç 1)

  17. Search Problem • Given x, find y such that R(x,y), or output “no such y”. x 2 {0,1}* y 2 {0,1}*/NO R

  18. Examples • Find a factor f of n such that 1<f<n. • Find an s-t path of length at most k. • Find a clique of size at least k in G. • Find a 3-coloring of G. • Find an assignment to x1… xn that satisfies .

  19. Optimization Problem • Given x, find y that optimizes some criterion. y 2 {0,1}*, that achieves the optimizing value x 2 {0,1}* R

  20. Examples • Find the largest factor f of n s.t. 1<f<n. • Find the shortest path between s and t. • Find the largest clique in G. • Find a coloring of G that uses as few colors as possible.

  21. Optimization reduces to Decision • Optimization problems can be reduced to decision problems with small overhead. • For example, to find the size of the biggest clique in G, binary search on: Does G contain a clique of size k?

  22. Counting Problem • Given x, find #y satisfying R(x,y) x 2 {0,1}* |{y:R(x,y)}| R

  23. Self-Reducibility of SAT •  can be reduced to two smaller instances • Each variable must be 0 or 1. • (x1… xn)2 SAT iff (0,x2…xn)2 SAT or(1,x2…xn)2 SAT

  24. An Algorithm • Given (x1… xn) • If (0,x2…xn)2 SAT, solve recursively on (0,x2…xn) • If (1,x2…xn)2 SAT, solve recursively on (1,x2…xn) • Else return UNSAT • At most n recursive calls

  25. Why is this important? • R is an NP-relation if 9 poly. p s.t. • For all (x,y)2 R, |y|· p(|x|) • Given x,y, we can check R(x,y) in time · p(|x|) • A search problem over such an R is an NP-search problem.

  26. So… • NP-decision problem: • Given x, does 9y : R(x,y)? • Cook’s theorem: • If there is a poly. time alg. For SAT, there are poly. time algs for all NP decision problems • By self-reducibility, this extends to search problems!

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