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Computational Complexity
CPSC 468/568, Fall 2007Time: Tu & Th, 1:00-2:15 pmRoom: AKW 200
Dist. Group: IV (not Natural Sci.)Satisfies “QR”
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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Schedule
Sept. 20: First HW Assignment DueOct. 2: Second HW Assignment DueOct. 11: Third HW Assignment DueOct. 18: First In-Class ExamOct. 23: Grades on HW1, HW2, HW3, and Ex1 will
be returned to students today or earlier.Oct. 26: Drop DateOct. 31: Fourth HW Assignment DueNov. 13: Fifth HW Assignment DueNov. 29: Sixth HW Assignment Due Dec. 6: Second In-Class Exam
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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
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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: Tuesday 2:30 - 3:30 pm
Thursday 11 am - noonPhone: 203-432-6432Assistant: Judi Paige([email protected], 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.
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TA: Nicholas RuozziOffice: AKW 305Office Hours: TBAEmail: [email protected]
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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)
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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?
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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