CPS 240: Computational Complexity

Department of Computer Science

Duke University

John H. Reif
   Spring Semester, 2011

Classes: Tues, Thurs: 11:40 AM – 12:55 PM

Schedule:

Class	Date	Topics	Primary Lecture Notes	Secondary Lecture Notes	Primary Textbook Readings	Secondary Textbook Readings	Handouts
1	Thurs Jan 13	Introduction: Turing Machines, Non-Determinism, Church Turing Hypothesis Overview of Course	[T, Ch1] [T, Ch2] [T, Ch3] [T, Ch4]	[Arora] [Beame] [Spielman] [Trevisan]	[AB, Intro] [AB, Ch 1] [G, Ch1]	[HMU, Ch 8]
2	Tues Jan 18	Diagonalization Universal TMs Diagonalization of Real Numbers Proofs of Undecidability via Diagonalization of TMs Time-Space Hierarchy Theorems	[AB, Ch 1] [AB, Ch 4] [G, Ch4] [Arora]	[Beame]	[Pap,Ch7,8] [Pap, Ch14]
3	*Thurs Jan 20	Review of Formal Language Theory Finite State Automata & Regular Expressions Push Down Automata & Context Free Languages Non-Determinism in Automata Complexity Definions of P, NP, Polytime Reductions, Completness	Complexity Definions: [Arora] [Spielman]
4	Tues Jan 25	Further Review of Formal Language Theory Relativized Proof Techniques Oracle Turing Machines Relativized Proofs	Relativization: [AB, Ch 4] [Arora] [Spielman]
5	Thurs Jan 27	P and NP-Completeness: P, NP, NP-Completeness Cook-Lewin Theorem and 3SAT NP-Completeness via Reductions coNP, EXPTIME, NEXPTIME	[AB, Ch 2] [T, Ch5] [T, Ch7] [T, Ch8]	[Arora] [Spielman]	[G, Ch2] [Pap, Ch9]	[G, AppA] [GJ, Ch3,4] [HMU, Ch10]
6	Tues Feb 1	Polynomial Hierarchy (PH): Alternation Properties of PH Oracle Characterization of PH	[Arora]	[Beame] [Spielman]	[AB, Ch 5] [Pap, Sec16.2] [CKS]	[T, Ch9]
7	*Thurs Feb 3	Quiz on Formal Language Theoery and Recursion Theory: - Finite State Automata - Context Free Languages -Diagonalization - Undecidability
8	Tues Feb 8	Space Complexity: PSPACE Completeness and TQBF Examples of PSPACE Complete Problems Savitch’s proof PSPACE=NPSPACE NSPACE=coNSPACE  L, NL, Logspace Reductions PATH is NL-Complete	[Arora]	[Trevisan]	[AB, Ch 3] [G, Ch5] [Pap, Ch19]	[T, Ch6] [T, Ch10] [HMU, Ch11]
9	Thurs Feb 10	Space Complexity, Cont: PSPACE Complete Problems:Quantified Boolean Logic Complexity of 2 Player Games Optional: Reingold's deterministic logspace algorithm for undirected connectivity	[AB, Ch 3]	Optional: [Spielman] [Trevisan] [Trevisan]	[AB, Ch 16] [G, Ch5] [Pap, Ch7,16]
10	Tues Feb 15	Review of Classical Computation Theory
11	*Thurs Feb 17	Quiz on Classical Complexity Theory: -Speedup & Slowdown Thms -Complexity Classes -Reductions -Completeness Proofs
12	Tues Feb 22	Kolmogorov Complexity Definitions Incompressibility for lowerbounds	[ppt:Li, Li] [Li] [ppt:Li, Li] [ppt:Li, Li]
13	Thurs Feb 24	Circuit Complexity: P/poly, Logspace uniform classes Can P/poly resolve P versus NP? P-Completeness and Linear Programming Log-depth Circuits, NC Relation of NC to Parallel Computing Examples of NC parallel algorithms	[Arora]	[Beame] [Spielman] [Spielman] [Trevisan]	[AB, Ch 6] [G, Ch3] [Pap, Ch11]
14	Tues March 1	Circuit Complexity, Cont: Circuit Lowerbounds: Parity is not in AC0 proof with polynomials Hastad ‘s proof via switching lemma Applications to learning Lowerbounds for monotone circuits	[Arora]	[Arora] [Beame] [Beame] [Trevisan] [Trevisan] [Trevisan]	[AB, Ch 13] [G, AppB] [Pap, Ch11]
15	Thurs March 3	Randomized Complexity: RP, BPP, ZPP Examples of Probabilistic TMs Proof of the Schwartz-Zippel Lemma Error amplification Relation of BPP to PH and P/poly	[Arora]	[Beame] [Spielman]	[AB, Ch 7] [AB, Ch 11] [G, Ch6]	[HMU, Ch11] [G, AppD]
	March 5-13	No Class - SPRING BREAK
16	Tues March 14	Randomized Complexity, Cont: Probabilistic decision trees: Yao's lemma on decision trees Unique Sat: The Valiant-Vazirani Theorem	[Arora] [Spielman] [Trevisan]		[AB, Ch 11] [G, Ch6]	[HMU, Ch11] [G, AppD]
17	Tues March 17	Counting Complexity Classes: #P defined Permanent is #P complete	[Arora] [Beame] [Spielman]	[Beame] [Beame] [Spielman] [Trevisan]	[AB, Ch 8] [G, Ch6]
18	*Tues March 22	Mid-Term Exam - Kolmogorov Complexity, - Circuit Complexity
19	Thurs March 24	Hardness amplification and error correcting codes: - Worst-case to average case reduction. Error-Correcting Codes - List decoding and its use for hardness amplification	[AB, Ch 18] [Arora] [Arora]		[G, Ch7] [G, AppE]
20	Tues March 29	CrytoComplexity: Overview of Cryptography: Trapdoor Functions, Cryptography RSA and Discrete Log Cryptosystems	[AB, Ch 10]		[Pap, Ch18] [G, AppC]
21	Thurs March 31	CrytoComplexity, Cont: Pseudorandom Number Generation: One Way Functions: Discrete Log Pseudorandom Number Generators: Goldreich-Levin Hardcore Bit Pseudorandom generation using average-case hardness	[Trevisan] [AB, Ch 10]		[AB, Ch 16] [G, Ch8] [Pap, Ch11]	[G, AppC]
22	*Tues April 5	Introduction to Communication complexity	[Arora]		[AB, Ch 12] [Pap, Ch11]
23	Thurs April 7	Interactive Proofs (IP): Brief Overview of Zero Knowledge Proofs -Interactive proofs of graph non-isomorphism -Applications to Program Checking -Author-Merlin Games Proof IP = PSPACE	[Arora]	[Beame] [Beame] [Beame] [Spielman] [Spielman] [Spielman]	[AB, Ch 9] [G, Ch9] [Pap, Ch19]
24	*Tues April 12	Intro to Quantum Comp Qubits Unitary Operations & Projections Quantum Circuits Example Quantum Computations Further topics to be Overviewed: Qunatum Factoring Quantum Search Quantum Information Theory Quantum Compression	[Arora]	[Spielman] [Spielman] [Spielman] [Spielman] Extra reading Shor’s Quantum Algorithm for integer factorization [Spielman] Grover's Quantum Algorithm for NP Search [Spielman]	[AB, Ch 21]
25	*Thurs April 14	Exam - Randomized Complexity - CrytoComplexity - Interactive Proofs
26	Tues April 19 (last class)	Logic & Proof in complexity theory: Why are complexity lower bounds so difficult? Natural proofs Proof complexity Independence in Logics	[Arora] [A]		[AB, Ch 23]	[AB, Ch 22]
(extra topic)		Hardness of Approximation: Overview of Direct Proof of PCP Theorem: Hardness of Approximating MAX-SAT NP in PCP(poly(n),1)	[Beame] [Beame] [Beame] [Beame]	Spielman] [Spielman] [Spielman] [Spielman]	[AB, Ch 19] [AB, Ch 20]
		(FINAL PROJECTS DUE APRIL 26)

Required Class Text Books: (digital text books [AB] and [G] used by permission)

• [Pap]  Christos Papadimitriou. Computational Complexity. Addison-Wesley Longman, 1994. Errata.
• [AB] Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach, digital preprint of yet to be published book, Princeton University.
• [G] Oded Goldreich, Computational Complexity: A Conceptual Perspective, digital preprint of yet to be published book, Weizmann Institute.
• Other Relevant References for Further Reading

Additional Readings in Complexity Theory: ([Tompa] used by permission)

• [GJ] M. Garey and D. Johnson.  Computers and intractability: A guide to NP-Completeness. W. H. Freeman & Co., 1979.
• [HMU]  J. Hopcroft, R. Motwani, and J. Ullman. Introduction to automata theory, language, and computation. Addison Wesley, 2001
• [T] Martin Tompa, Introduction to Computational Complexity, unpublished digital notes, University of Washington
• [A] Scott Aaronson, Is P Versus NP Formally Independent? Bulletin of the EATCS 81: 109-136 (2003).
• Other Relevant References for Further Reading

Lecture Notes Credits: (all used by permission)

• [Arora] Sanjeev Arora, Princeton University
• [Beame] Paul Beame, University of Washington
• [Li] Ming Li, University of Watterloo
• [Spielman] Daniel A. Spielman, MIT
• [Trevisan] Luca Trevisan, U.C. Berkeley

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