Click on the theme item for the lecture plan for that section.
Click on the lecture item for suggested additional reading related to it.
The notes for the lectures are available at the moodle webpage of the course.
Theme 1: P vs NP : Time Complexity Classes, Reductions and Completeness. (9 meetings)
- Lecture 01 : Tue, Jan 4, 12-12:50pm
- Adminstrative annoucements. Plan for the course.
Recap of Turing machine model, Notion of a resource, Blum axioms, Tape reduction- Additiional Reading :
- The Computational Model and Why it Doesn't Matter - Chapter 1 of Arora-Barak Textbook.
- Chapter 1 of Book : Theory of Computational Complexity - Du and Ko
- Lecture 02 : Fri, Jan 7, 10-10:50am
- Time complexity. Tape reduction. UTMs, Enumeration of TMs,
Classes P and EXP.- Additiional Reading :
- Lecture 03 : Fri, Jan 7, 10-10:50am
- Time heirarchy theorem, Nondeterminism, P vs NP problem
- Additiional Reading :
- Lecture 04 : Thu, Jan 13, 11-11:50am
- Example of problems in NP, Reductions and Completeness.
Canonical complete problem.- Additiional Reading :
- Lecture 05 : Thu, Jan 13, 4:45-5:35pm (for Jan 3)
- Cook-Levin theorem, SAT is NP-complete
- Additiional Reading :
- Lecture 06 : Fri, Jan 14, 10-10:50am
- Examples of NP-complete reductions and examples
- Additiional Reading :
- Lecture 07 : Mon, Jan 17, 08-08:50am
- Padding techniques, Ladners theorem : outline of the proof
- Additiional Reading :
- Lecture 08 : Tue, Jan 18, 12-12:50pm
- Details of proof of Ladner's theorem.
- Additiional Reading :
- Lecture 09 : Tue, Jan 18, 4:45-5:35pm (for Jan 10)
- Oracle Turing Machines, Limits of diagonalisation proofs.
Baker-Gill-Solovoy : Oracles showing P eq NP and P ne NP.- Additiional Reading :
- Missed lectures : Jan 3, 10, 11
Compensatory lectures : Jan 13, 18 - Problem Set 1 : (Wed, Jan 19, 5pm - Wed, Feb 2, 5pm )
Theme 2: Below P and Above NP : Space Complexity Classes, Algorithms.(10 meetings)
- Lecture 10 : Thu, Jan 20, 11-11:50am
- Space Complexity : Space as a resource, Complexity classes.
constant space, o(loglog n) space.
Configuration graphs, basic complexity inclusions, reachability.- Additiional Reading :
- Lower Bounds using Crossing sequence arguments
http://www.cs.umd.edu/~jkatz/complexity/lower_bounds.pdf
- Lower Bounds using Crossing sequence arguments
- Lecture 11 : Fri, Jan 21, 10-10:50am
- Space hierarchy theorem. PSPACE, NL, L.
Savitch's theorem, PSPACE=NPSPACE.- Additiional Reading :
- Lecture 12 : Tue, Jan 25, 12-12:50pm
- Relook at PSPACE, TQBF is complete for PSPACE
Complexity of reductions, logspace reductions, composability.- Additiional Reading :
- Lecture 13 : Mon, Jan 31, 08-08:50am
- PATH is NL-complete,
Reachability in special graphs, Complementation & non-determinism- Additiional Reading :
- Lecture 14 : Tue, Feb 1, 12-12:50pm
- Immerman-Szelepcsenyi Theorem : NL = CoNL,
Reachability in Expander Graphs.- Additiional Reading :
- Lecture 15 : Thu, Feb 3, 11-11:50am
- Reachability in Bounded diameter regular graphs, Expander graphs,
Diameter of Expander Graphs. Algebraic & Combinatorial Expansion- Additiional Reading :
- Lecture 16 : Thu, Feb 3, 4:45-5:35pm (for Jan 11)
- Algebraic expansion => Combinatorial Expansion
- Additiional Reading :
- Lecture 17 : Fri, Feb 4, 10-10:50am
- Rotation Maps, Graph powering, graph tensoring and
replacement products.- Additiional Reading :
- Lecture 18 : Mon, Feb 7, 08-08:50am
- Reachability in Undirected graphs : Expanderization in Logspace
Reingold's algorithm.- Additiional Reading :
- Lecture 19 : Tue, Feb 8, 12-12:50pm
- Expansion properties of regular connected graphs,
remaining details of correctness proof of Reingold's algorithm- Additiional Reading :
- Missed Lectures : Jan 24
Compensatory Lectures : Feb 3 (for Jan 11) - Problem Set 2 : (Thu, Feb 10, 5pm - Fri, Feb 18, 5pm )
No lectures on Jan 27th and 28th - classes off for Saarang (IITM cultural festival).
Theme 3: Between P and PSPACE : PH, #P, PP, ParityP, BPP, RP (15 meetings)
- Lecture 20 : Thu, Feb 10, 11-11:50am
- Back to Oracle Turing Machines, Polynomial Heirarchy :
between NP and PSPACE, Characterisations
- Additiional Reading :
- Lecture 21 : Fri, Feb 11, 10-10:50am
- Quantifier based characterisations of PH
- Additiional Reading :
- Lecture 22 : Mon, Feb 14, 08-08:50am
- Alternating Turing Machines, Characterisations for PH
- Additiional Reading :
- Lecture 23 : Tue, Feb 15, 12-12:50pm
- Alternating Space vs Deterministic Time, Alternating Time vs Deterministic Space
- Additiional Reading :
- Lecture 24 : Thu, Feb 17, 11-11:50am
- Number of certificates : Classes PP, ParityP. Containments.
Functional Problems, Class FP, #P- Additiional Reading :
- Lecture 25 : Fri, Feb 18, 10-10:50am
- Parsimonious reductions, complete problems for PP, ParityP
- Additiional Reading :
- Lecture 26 : Mon, Feb 21, 08-08:50am
- PP vs P and #P vs FP, #P-completeness.
Decision problems whose counting problems are hard
#P-completeness of #SAT and #CYCLE- Additiional Reading :
- Lecture 27 : Tue, Feb 22, 12-12:50pm
- Permanent, Perfect Matchings, Cycle Covers
- Additiional Reading :
- Lecture 28 : Thu, Feb 24, 11-11:50am
- Computing Permanent of integer matrices is #P-hard
- Additiional Reading :
- Lecture 29 : Fri, Feb 25, 10-10:50am
- #P-completeness of Permanent of Boolean matrices,
BPP, RP in terms of counting.- Additiional Reading :
- Lecture 30 : Mon, Feb 28, 08-08:50am
- Valiant Vazirani Theorem
- Additiional Reading :
- Lecture 31 : Tue, Mar 1, 12-12:50pm
- Toda's Theorem - I
- Additiional Reading :
- Lecture 32 : Thu, Mar 3, 11-11:50am
- Toda's Theorem - II
- Additiional Reading :
- Lecture 33 : Fri, Mar 4, 10-10:50am
- Error reduction for BPP, RP, BPP is in PH
- Additiional Reading :
- Lecture 34 : Tue, Mar 8, 12-12:50pm
- BPP is in PH, BPP is in P/poly. Advice classes
- Additiional Reading :
- Problem Set 3 : (Mon, Feb 28, 5pm - Wed, Mar 9, 5pm )
No lecture on Mar 7th, 2011.
Theme 4: Non-uniform Models and Lower Bounds (16 meetings)
- Lecture 35 : Thu, Mar 10, 11-11:50am
- Boolean Circuits, Families, Circuit Parameters,
Circuit family. Connections with parllel computation,
P/poly is the set of languages for which there is a polysized circuit family, Uniformity.- Additiional Reading :
- Lecture 36 : Fri, Mar 11, 10-10:50am
- Characterisation of P, CVP problem is P-complete under
log-space reductions.
- Additiional Reading :
- Lecture 37 : Sat, Mar 12, 2pm-3:30pm (for Mar 7)
- If NP is in P/poly then PH collapses.
Circuit Lower Bound Problem, Shannon Bound, Lypanov Bound(statement).
Restricted circuits, Characterisation of NL.- Additiional Reading :
- Lecture 38 : Mon, Mar 14, 08-08:50am
- Restrictions on circuits, review of basic parameters, NC,
AC
NC^1 \subseteq L, NL \subseteq AC^1- Additiional Reading :
- Lecture 39 : Tue, Mar 15, 12-12:50pm
- Basic circuit constructions : Adding 2 n-bit numbers in AC^0
Iterated Addition, BITCOUNT, Multiplication in NC^1- Additiional Reading :
- Lecture 40 : Thu, Mar 17, 11-11:50am
- Log iterated Addition in AC^0, Constant depth reductions,
Iterated Addition reduces to BITCOUNT- Additiional Reading :
- Lecture 41 : Sat, Mar 19, 2pm-4pm (for Mar 18)
- Majority reduces to multiplication and BITCOUNT, Threshold gates, TC
Computing symmetric functions using threshold gates, Power of MOD gates, ACC.
Formulas and NC^1- Additiional Reading :
- Lecture 42 : Mon, Mar 21, 08-08:50am
- Decision trees and lower bounds for parity
- Additiional Reading :
- Lecture 43 : Tue, Mar 22, 12-12:50pm
- Braching Programs, parameters, Simulations, Skew circuits
- Additiional Reading :
- Problem Set 4 : (Wed, Mar 23, 5pm - Fri, Apr 1, 5pm )
- Lecture 44 : Thu, Mar 24, 11-11:50am
- Bounded width branching programs, programs over groups and their connection to BPs
- Additiional Reading :
- Lecture 45 : Fri, Mar 25, 10-10:50am
- Proof of Barrington's theorem : Simulating NC^1 by programs over non-solvable groups
- Additiional Reading :
- Lecture 46 : Tue, Mar 29, 12-12:50pm
- Circuit Lower Bounds Problem, Strategy of Proofs, Parity and AND decision trees, DNF/CNFs
- Additiional Reading :
- Lecture 47 : Thu, Mar 31, 11-11:50am
- Method of Random restrictions : Circuit lower bounds against constant depth circuits
- Additiional Reading :
- Lecture 48 : Fri, Apr 1, 10-10:50am
- Proof of Switching Lemma
- Additiional Reading :
- Lecture 49 : Tue, Apr 5, 12-12:50pm
- Degree method, Representations of Boolean functions with polynomials, Role of the underlying field. Examples.
- Additiional Reading :
- Lecture 50 : Thu, Apr 7, 11-11:50am
- Razborov-Smolensky Theorem : PARITY cannot be computed by poly size constant depth circuits
even when they are allowed to use MOD gates
- Additiional Reading :
- Lecture 51 : Thu, Apr 7, 5:15pm-6:30pm (for Mar 28)
- Additiional Reading :
Theme 5: Deranodomization, Hardness vs Randomness (6 meetings)
- No lecture on Apr 4 - Institute Holiday - Ugadi.
- Lecture 52 : Fri, Apr 8, 10-10:50am
- Derandomization Problem, Pseudo random generators, Extractors, Deterministic simulations
- Additiional Reading :
- No lecture on Apr 11 - JEE 2011 travel.
- Lecture 53 : Tue, Apr 12, 12-12:50pm
- Proof techniques, deterministic simulations, high-end, low-end generators
- Additiional Reading :
- Lecture 54 : Fri, Apr 15, 10-10:50am
- Hard functions from PRGs, PRGs from hard functions, the toy case of one bit stretch generator using a hard function. Unpredictability.
- Additiional Reading :
- Lecture 55 : Sat, Apr 16, 9:30am - 12:30pm (for Apr 11+extra)
- Unpredictability is equivalent to Pseudorandomness. Proof of the Toy case. Attempts on
generalisation.
Nisan-Wigderson generator, Proof of pseudorandomness, Construction of NW-designs.- Additiional Reading :
- Lecture 56 : Mon, Apr 18, 08-08:50am
- From worst case hardness to average case hardness, high-level view of the proof, Error
correcting codes, Using locally list-decodable codes to prove the result.
- Additiional Reading :
- Lecture 57 : Tue, Apr 19, 12-12:50pm
- Proof details. High-level view of the construction of the required code : Walsh-Hadamard
code, Reed-Solomon code, Reed-Muller codes, Concatenation. Parameters.
- Additiional Reading :
- Problem Set 5 : (Tue, Apr 19, 5pm - Fri, Apr 29, 5pm )
OPTIONAL:
Theme 6: Probablistic Proof Systems (2 meetings)
- Lecture 58 : Thu, Apr 21, 11-11:50am
- Interactive Proofs, Sumcheck Protocol, CoNP contained in IP, Ideas for the IP protocol for
PSPACE
- Additiional Reading :
- Lecture 59 : Sat, Apr 23, 9am - 12pm ("extra lecture")
- Probablistically Checkable Proofs, Introduction,
Connection to Inapproximability, PCP Theorem (statement) and history, references.
- Additiional Reading :
- Statistics : 59 meetings, 67 hours.
- End-semester Examination: (in-hall open-book examination): Apr 25, 9am-12pm.
- Course Projects : Suggested Topics and Presentation Schedule.
Last updated on Fri May 13 21:08:49 IST 2011