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Computational Complexity

CPSC 468/568, Fall 2010Time: Tu & Th, 1:00-2:15 pmRoom: AKW 500

Satisfies the QR requirement.

http://zoo.cs.yale.edu/classes/cs468/

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Partial Topic Outline

• Complexity classes (P, NP, L, NL, etc.)• Reductions and completeness• The roles of, e.g.,

– Randomness– Interaction– Approximation

? Communication complexity

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Requirements

• Modest reading assignments, mostly in Arora and Barak, Computational Complexity: A Modern Approach, Cambridge Univ. Press, 2009.

• 6 Written HW Assignments, each worth 10% of the course grade

• 2 In-Class Exams, each worth 20% of the course grade

• No final exam during exam week

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Tentative Schedule

Sept. 21: First HW Assignment DueSept. 30: Second HW Assignment DueOct. 12: Third HW Assignment DueOct. 14: First In-Class ExamOct. 22: Fall Semester Drop DateNov. 02: Fourth HW Assignment DueNov. 16: Fifth HW Assignment DueNov. 30: Sixth HW Assignment Due Dec. 2: Second In-Class Exam

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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.

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Instructor: Joan FeigenbaumOffice: AKW 512Office Hours: Thursdays 11:30 am - 12:30

pm and by appointment

Phone: 203-432-6432Assistant: Judi Paige(judi.paige@yale.edu, 203-436-1267,

AKW 507a, 8:30 am – 4:30 pm M-F)

Note: Do not send email to Professor Feigenbaum, who suffers from RSI. Contact her through Ms. Paige or the TA.

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TA: David CostanzoOffice: AKW 301 (or open zoo)Email: David.Costanzo@yale.eduOffice Hours: Tuesdays, 4 – 5 pm

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If you’re undecided …

Check out:• zoo.cs.yale.edu/classes/cs468/spr07/,

…/fall07/, …/fall08, and …/fall09• www.cs.princeton.edu/theory/complexity/ (draft of textbook 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)

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Questions?

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Introduction to Complexity Classes

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Computational Complexity Themes

•“Easy” vs. “Hard”•Reductions (Equivalence)

•Provability•Randomness

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Poly-Time Solvable

• Nontrivial Example : Matching

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Poly-Time Solvable• Nontrivial Example : Matching

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Poly-Time Verifiable• Trivial Example : Hamiltonian Cycle

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Poly-Time Verifiable• Trivial Ex. : Hamiltonian Cycle

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•Is it Easier to Verify a Proof than to Find one?

•Fundamental Conjecture of Computational Complexity:

PNP

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• Matching:

• HC:

Fundamentally Different

Distinctions

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Reduction of B to A

•If A is “Easy”, then B is, too.

BAlgorith

m

A “oracle”

“black box”

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•NP-completeness•P-time reduction•Cook’s theorem

If B ε NP, thenB ≤ P-time SAT

•HC is NP-complete

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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…

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NP coNP

P

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Random poly-time Solvable

x ε L?

poly-time Algorithm

xr

YES

NO

x ε {0,1}n

r ε {0,1}poly(n)

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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 ResultPRIMES ε ZPP ( = RP ∩ coRP)

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Two-sided Error

BPPx L “yes” w.p. ¾x L “no” w.p. ¾

Question to Audience: BPP set not known to be in RP or coRP?

Note: QR is in NP and coNP but not known to be in RP or coRP.

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RP coRPZPP

NP coNP

P

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Interactive Provability

P V [PPT, ¢]

x

yes/no

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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

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PSPACE

RP coRPZPP

NP coNP

P

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PSPACE

EXP

P#P

PH

iP

2P

NP

P