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Theory of Computing

2017-12-16T22:03:10Z

Passick: Add submission link for hw 4

== General Information ==

[http://www.mi.ras.ru/~podolskii/files/computability/grading.pdf Grading]

== Dates and Deadlines ==

'''Homework 1''' deadline: September 29, 2017, 23:59 AoE 
'''Homework 1, Extra Problems''' deadline: October 6, 2017, before seminar 
'''Homework 2 + Extra Problems''' deadline: November 3, 2017, before lecture 
'''Homework 3 + Extra Problems''' deadline: November 24, 2017, before lecture 
'''Homework 4 + Extra Problems''' deadline: December 17, 2017, 23:59 Moscow time, [https://www.dropbox.com/request/wYoVZeRnpDOZeyk3PuuM submit here]

== Colloquium ==

Date and time: December 11, 12:10 
Room: 505 
[http://www.mi.ras.ru/~podolskii/files/computability/col.pdf Program]

== Course Materials ==

{| class="wikitable"
|-
! Summary !! Problem list
|-
|| Complexity classes P, PSPACE, EXP. Time and space hierarchy theorems (see also Sipser Section 9.1) || [http://www.mi.ras.ru/~podolskii/files/computability/prob_1.pdf Problem list 1]
|-
|| Complexity class NP. Examples. Inclusions between P, NP and EXP. Non-deterministic TMs. Another definition of NP. Complexity class NEXP. Polynomial reductions, their properties. NP-hardness and NP-completeness, their properties. NP-completeness: CIRC-SAT, 3-SAT. || [http://www.mi.ras.ru/~podolskii/files/computability/prob_2.pdf Problem list 2]
|-
|| NP-completeness: NAE-3-SAT, Exactly-1-3-SAT, IND-SET, Subset-SUM, 3-COLORING. || [http://www.mi.ras.ru/~podolskii/files/computability/prob_3.pdf Problem list 3]
|-
|| Space complexity. Classes PSPACE and NPSPACE. Configuration graph. Inclusions between time and space classes. TQBF problem, its PSPACE-completeness. PSPACE = NPSPACE. NSPACE(s(n)) is in SPACE(s(n)^2) (additional material). Interpretation of PSPACE in terms of games. || [http://www.mi.ras.ru/~podolskii/files/computability/prob_4.pdf Problem list 4]
|-
|| Probabilistic computation. Probabilistic machines, the class BPP, prime testing and Carmichael numbers, invariance of the definition BPP for different thresholds, RP, coRP, PP, ZPP. BPP is in P/poly.
|| [http://www.mi.ras.ru/~podolskii/files/computability/prob_5.pdf Problem list 5]
|-
|| Definition of Sigma_2 and Pi_2. BPP is in Sigma_2 and Pi_2. Computations with oracles, its simple properties. There are oracles A and B such that P^A is equal to NP^A and P^B is not equal to NP^B.
|| [http://www.mi.ras.ru/~podolskii/files/computability/prob_6.pdf Problem list 6] Updated: 07.10.17
|-
|| Communication protocols. Functions EQ, GT, DISJ, IP. Fooling sets. Combinatorial rectangles. Rectangle size lower bound. Rank lower bound. Non-deterministic complexity. Communication complexity classes P, NP, coNP, intersection of NP and coNP.
|| [http://www.mi.ras.ru/~podolskii/files/computability/prob_7.pdf Problem list 7]
|-
|| D(f)=O(N^0(f)N^1(f)). Randomized communication complexity, definitions. R(EQ)=O(1). N^1(f) vs. R^1(f). Newman's theorem, formulation.
|| [http://www.mi.ras.ru/~podolskii/files/computability/prob_8.pdf Problem list 8]
|-
|| Proof of Newman's theorem, distributional complexity and the characterization of public coin communication complexity, the discrepancy method.
|| [http://www.mi.ras.ru/~podolskii/files/computability/prob_9.pdf Problem list 9]
|-
|| Randomized communication complexity of IP. Streaming algorithms. Finding the majority element. Deciding whether there is a most frequent element is hard. One-sided probabilistic complexity of disjointness.
|| [http://www.mi.ras.ru/~podolskii/files/computability/prob_10.pdf Problem list 10] Updated: 22.11.17
|-
|| Property testing: definitions, testing of halfplanes, sorted listed, connectedness of graphs, testing of linearity. [https://www.dropbox.com/s/r35jr22fb3lfo3k/propTest.pdf?dl=0 Lecture notes.] Version 25.11.17
|| [https://www.dropbox.com/s/uzh9dur6aiy88dl/prob_11.pdf?dl=0 Problem list 11] Updated: 22.11.17
|-
|| Property testing: connectedness of graphs (cont.) and testing of monotonicity. (See notes from the previous lecture.)
|| [https://www.dropbox.com/s/y2xujzeftsg6zlr/prob_12.pdf?dl=0 Problem list 12]
|-
|| Property testing: lower bounds for monotonicity (see Sect 4 [http://theory.stanford.edu/~tim/w15/l/l8.pdf here]) and k-linearity using communication complexity. Approximation algorithms for some NP-complete problems (see seminar).
|| [https://www.dropbox.com/s/5xsxqv098y5wmx0/prob_13.pdf?dl=0 Problem list 13]
|-
|| The class PCP: definition, basic properties, and relation to with the MAX-Clique approximation problem. Beginning of the proof that NP is a subset of PCP(poly(n), 1). See Dexter Kozen, "Introduction to the theory of computation", lectures 18-20 (or see the book of Arora and Barak, chapter 11).
|| [https://www.dropbox.com/s/gbg30ajwes45ne8/prob_14.pdf?dl=0 Problem list 14]
|-
|| Next week: we finish the proof that NP is a subset of PCP(poly(n),1), continue working on problem list 14, and present solutions of previous lists upon request.
|| No problem list.
|}

== Homework ==
Send homework assignments via Dropbox (the link is in the telegram group chat), or submit them in person to one of the teachers or the teaching assistant (Gleb Posobin) before the deadline.

[https://docs.google.com/spreadsheets/d/1mIkfx5VKmHy-ZFynVdRlVBpAqKNI1e5qyLLHt8yxJRY/edit?usp=sharing Homework results]

[https://docs.google.com/spreadsheets/d/1TyV3DDPgeH9OF0J-Axywhfv_8WC8Z0_Qdxwc9_bA4fg/edit?usp=sharing Results for extra problems]

== Office hours ==

{| class="wikitable"
|-
! !! Person !! Monday !! Tuesday !! Wednesday !! Thursday !! Friday
|-
| <center>1</center> || Vladimir Podolskii || || || || || 16:40–18:00, room 621
|-
| <center>2</center> || Bruno Bauwens || || 15:05–18:00, room 620 || || || 15:05–18:00, room 620
|}

Theory of Computing

2017-10-20T11:12:36Z

Passick: Changed section about homeworks

Theory of Computing

2017-09-22T11:55:16Z

Passick:

== General Information ==

[http://www.mi.ras.ru/~podolskii/files/computability/grading.pdf Grading]

== Course materials ==

{| class="wikitable"
|-
! Summary !! Problem list
|-
|| Complexity classes P, PSPACE, EXP. Time and space hierarchy theorems (see also Sipser Section 9.1) || [http://www.mi.ras.ru/~podolskii/files/computability/prob_1.pdf Problem list 1] Updated: 11.09.17
|-
|| Complexity class NP. Examples. Inclusions between P, NP and EXP. Non-deterministic TMs. Another definition of NP. Complexity class NEXP. Polynomial reductions, their properties. NP-hardness and NP-completeness, their properties. NP-completeness: CIRC-SAT, 3-SAT. || [http://www.mi.ras.ru/~podolskii/files/computability/prob_2.pdf Problem list 2]
|-
|| NP-completeness: NAE-3-SAT, Exactly-1-3-SAT, IND-SET, Subset-SUM, 3-COLORING. || [http://www.mi.ras.ru/~podolskii/files/computability/prob_3.pdf Problem list 3]
|}

== Homework ==
Send homework assignments by email to posobin+hw[at]gmail.com, or submit them in person to one of the teachers or the teaching assistant (Gleb Posobin) before the deadline.

== Office hours ==

{| class="wikitable"
|-
! !! Person !! Monday !! Tuesday !! Wednesday !! Thursday !! Friday
|-
| <center>1</center> || Vladimir Podolskii || || || || || 16:40–18:00, room 621
|-
| <center>2</center> || Bruno Bauwens || || 15:05–18:00, room 620 || || || 15:05–18:00, room 620
|}

Theory of Computing

2017-09-22T11:53:53Z

Passick: Added section about homeworks

== General Information ==

[http://www.mi.ras.ru/~podolskii/files/computability/grading.pdf Grading]

== Course materials ==

{| class="wikitable"
|-
! Summary !! Problem list
|-
|| Complexity classes P, PSPACE, EXP. Time and space hierarchy theorems (see also Sipser Section 9.1) || [http://www.mi.ras.ru/~podolskii/files/computability/prob_1.pdf Problem list 1] Updated: 11.09.17
|-
|| Complexity class NP. Examples. Inclusions between P, NP and EXP. Non-deterministic TMs. Another definition of NP. Complexity class NEXP. Polynomial reductions, their properties. NP-hardness and NP-completeness, their properties. NP-completeness: CIRC-SAT, 3-SAT. || [http://www.mi.ras.ru/~podolskii/files/computability/prob_2.pdf Problem list 2]
|-
|| NP-completeness: NAE-3-SAT, Exactly-1-3-SAT, IND-SET, Subset-SUM, 3-COLORING. || [http://www.mi.ras.ru/~podolskii/files/computability/prob_3.pdf Problem list 3]
|}

== Homework ==
Send homeworks by email to posobin+hw[at]gmail.com, or submit them in person to one of the teachers or the teaching assistant (Gleb Posobin) before the deadline.

== Office hours ==

{| class="wikitable"
|-
! !! Person !! Monday !! Tuesday !! Wednesday !! Thursday !! Friday
|-
| <center>1</center> || Vladimir Podolskii || || || || || 16:40–18:00, room 621
|-
| <center>2</center> || Bruno Bauwens || || 15:05–18:00, room 620 || || || 15:05–18:00, room 620
|}