Time: Tue 09:30-11:00, Thu: 11:30-13:00
Location: A-212
Instructor:
Homepage:
http://www.tcs.tifr.res.in/~prahladh/teaching/2012-13/complexity
Course Calendar (Subcribe to the calendar (ical format))
15 Jan | 1. Introduction to computational complexity Administrivia, efficient computation, Turing machine, Universal Turing machine (UTM), efficient simulation by UTMs. Ref: [ AB (Chap. 1) ] |
17 Jan | 2. Efficient computation Review of efficient UTM simulation, uncomputability; class P Ref: [ AB (Chap. 1) ] |
22 Jan | 3. NP and NP Completeness (part I) NP, non-determinisim, polynomial time reductions, NP-completeness. Ref: [ AB (Chap. 2) ] |
24 Jan | 4. NP and NP Completeness (part II) Cook-Levin Theorem, web of reductions, decision vs. search, downward self-reducibility of SAT. Ref: [ AB (Chap. 2) ] |
29 Jan | 5. Diagonalization (part I) coNP, NP vs. coNP problem; diagonalization: time hierarchy theorem (deterministic and non-deterministic). Ref: [ AB (Chap. 2-3) ] |
31 Jan | 6. Diagonalization (part II) limits of diagonalization (Baker-Gill-Solovay theorem), NP-intermediate problems (Ladner's Theorem). Ref: [ AB (Chap. 3), vMel (Lecture 5) ] |
5 Feb | 7. Space Complexity (part I) Ladner's them (complete), space as a resouce, configuration graphs, TQBF, PSPACE completeness. Ref: [ vMel (Lecture 5) , AB (Chap. 4) ] |
7 Feb | 8. Space Complexity (part II) PSPACE completeness, Savitch's theorem, logspace reductions, NL-completeness. Ref: [ AB (Chap. 4) ] |
12 Feb | 9. Polynomial hierarchy and alternation (part I) certificate complexity, NL=coNL (Immerman-Szelepcsyéni Theorem), Classes Σ^{P},Π^{P}, polynomial time hierarchy. Ref: [ AB (Chap. 5), Sud (Lec. 3) ] |
19 Feb | 10. Time-space tradeoffs for SAT polynomial hierarchy, alternating machines, alernating space vs. time, Fortnow's time-space tradeoff theorem. Ref : [ AB (Chap. 5,6), Sud (Lec. 5) ] |
21 Feb | 11. Polynomial hierarchy and alternation (part II) time-space tradeoff theorem, alternating time vs. space, polynomial hierarchy via oracles. Ref: [ AB (Chap. 5), Sud (Lec. 3) ] |
26 Feb | 12. Boolean circuits Boolean circuits, P/poly, complexity classes with advice, Karp-Lipton-Sipser Theorem, Meyer's Theorem. Ref: [ AB (Chap. 6) ] |
28 Feb | 13. Randomized computation (part I) bounded depth circuit classes, circuit lower bounds discussion, randomized algorithms for primality (Solovay-Strassen) Ref: [ AB (Chap. 6, 7) ] |
1 Mar | 14. Randomized computation (part II) polynomial identity testing, PIT based algorithm for matching, RP error reduction by amplification. Ref: [ AB (Chap. 7) ] |
12 Mar | 15. Randomized computation (part III) BPP error reduction by amplification, BPP ⊂ P/poly, BPP ⊂ PH, randomized space. Ref: [ AB (Chap. 7) ] |
14 Mar | 16. Complexity of counting (part I) promise problems, Unique-SAT (Valiant Vazirani) Ref: [ AB (Chap. 17) |
19 Mar | 17. Complexity of Counting (part II) #P, FP, PP, #P-completeness, Valiant's Theorem: #P-completeness of permanent Ref: [ AB (Chap. 17) ] |
21 Mar | 18. Complexity of Counting (part III) Valiant's theorem: perm is #P-complete, Toda's Theorem, operators. Ref: [ AB (Chap. 17), DHW (gadgets for Valiant's theorem taken from Figure 2 on page 8), Sud (Lec. 10) ] |
28 Mar | 19. Complexity of Counting (part IV) Toda's Theorem (part 1: PH ⊂ BP.⊕P) Toda's Theorem (part 2: BP.⊕P ⊂ P^{#P}). Ref: [ AB (Chap. 17), Sud (Lec. 10-11) ] |
2 Apr | 20. Approximate counting Approximate counting is in BPP^{NP}, average case complexity: permanent reconstruction Ref: [ Tre2 (Chap. 8-9) ] |
4 Apr | 21. Interactive proofs (part I) permanent reconstruction (contd); what is a proof?, interactive proofs, graph non-isomorphism. Ref: [ Tre2 (Chap. 9), Vad (Lec. 17) ] |
11 Apr | 22. Interactive proofs (part I) interactive proofs, P^{#P} ⊂ IP. Ref: [ Vad (Lec. 17) ] |
12 Apr | 23. Interactive proofs (part III) PSPACE ⊂ IP, Arthur-Merlin games. Ref: [ AB (Chap. 8), Vad (Lec. 18 & 19) ] |
16 Apr | 24. Interactive proofs (part IV) Arthur-Merlin games, Public Coins = Private Coins, Round reduction for AM, GI - NP-complete? Ref: [ AB (Chap. 8), Vad (Lec. 18 & 19) ] |
19 Apr | 25. Cryptography (part I) Introduction, private key encryption, one-time pad, pseudorandomness, one-way functions. Ref: [ AB (Chap. 9), Tre1 (Lec. 11 & 12) ] |
23 Apr | 26. Cryptography (part II) examples of OWF/OWP, hardcore predicate, OWP with hardcore predicate implies PRG (with 1-bit stretch). Ref: [ BG (Sec. 2.3-2.4), Tre1 (Lec. 12) ] |
25 Apr | 27. Cryptography (part III) PRGs with polynomial stretch, PRGs imply derandomization, hardcore bit theorem (Goldreich-Levin). Ref: [ Han (Lec. 21, Sec. 1-2), Tre1 (Lec. 13) ] |
30 Apr | 28. Cryptography (part IV) hardcore bit theorem (Goldreich-Levin), pseudorandom functions Ref: [ Han (Lec. 21, Sec. 1-2), Tre1 (Lec. 14 & 13.2) ] |
2 May | 29. Cryptography (part V) pseudorandom function family, GGM construction, application to private key cryptography. Ref: [ Tre3 (Lec. 14), Tre1 (Lec. 13.3) ] |
Students taking the course for credit will be expected to:
[AB] | Sanjeev Arora and Boaz Barak. Computational Complexity: A Modern Approach. Cambridge University Press, 2009. [ .html ] |
[BG] | Mihir Bellare and Shafi Goldwasser. Lecture Notes on Cryptography. Summer course offered at MIT (1996-2008). [ .html ] |
[DHW] | Holger Dell, Thore Husfeldt and Martin Wahlen. Exponential Time Complexity of the Permanent and the Tutte Polynomial. In Proc. 37th ICALP (Part I), pages 426-437, 2011. [ DOI | eccc ] |
[Han] | Kristoffer Arnsfelt Hansen. CT10: Computational Complexity, Fall 2010. A course offered at Aarhys University (Fall 2010). [ .html ] |
[vMel] | Dieter van Melkebeek. CS710: Computational Complexity, 2010. A course offered at Univ. Wisconsin-Madison (Fall 2010). [ .html ] |
[Sud] | Madhu Sudan. 6.841/18.405J Advanced Complexity Theory, 2010. A course offered at MIT (Spring 2003). [ .html ] |
[Tre1] | Luca Trevisan. CS 278 Computational Complexity, 2001. A course offered at UC, Berkeley (Spring 2001). [ .pdf ] |
[Tre2] | Luca Trevisan. CS 278 Computational Complexity, 2002. A course offered at UC, Berkeley (Fall 2002). [ .pdf ] |
[Tre3] | Luca Trevisan. CS 276 Cryptography, 2009. A course offered at UC, Berkeley (Spring 2009). [ .html ] |
[Vad] | Salil Vadhan. CS221: Computational Complexity Theory, 2010 A course offered at Harvard (Spring 2010). [ .html ] |
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Prahladh Harsha |