Meetings

Click on the theme item for the meeting plan for that theme.
Click on the meeting item for references, exercises, and additional reading related to it.

Theme 1 : Counting vs Decision - 10 meetings

Meeting 01 : Mon, Jan 12, 10:00 am-10:50 am

Administrative Announcements. Counting Complexity. Computational Model. The class FP. Clique vs #Clique Problem. Cycle vs #Cycle Problem. If #CYCLE is in FP, then P = NP.

Meeting 02 : Tue, Jan 13, 09:00 am-09:50 am

The Class #P. #P = FP if and only if PP = P. The bits and value of the #P function. Counting as a generalization of NP, BPP, RP, PP and Parity P.

Meeting 03 : Wed, Jan 14, 08:00 am-08:50 am

Notion of #P-completeness.Attempt 1 : Parsimonious reductions. #SAT is #P-complete. #IND-SET is #P-complete. Shortcomings of the theory. General oracle query model of #P-completeness. Transitivity of reductions.

Meeting 04 : Tue, Jan 20, 09:00 am-09:50 am

The permanent and the determinant functions. Combinatorial interpretation of permanent of matrix with 0-1 entries. Counting perfect matchings.

Meeting 05 : Wed, Jan 21, 08:00 am-08:50 am

Permanent if integer matrices. Combinatorial interpretation in terms of cycle covers. The upper bound of FP^#P. Permanent over {-1,0,1} matrices is #P-complete. Proof idea, Gadget construction.

Meeting 06 : Fri, Jan 23, 12:00 pm-12:50 pm

Reduction from PERM(-1,0,1) to PERM(0,1,...,n), and then to PERM(0,1).

Meeting 07 : Tue, Jan 27, 09:00 am-09:50 am

Toda's Theorem. The Roadmap. U-SAT : Valiant-Vazirani Reduction.

Meeting 08 : Wed, Jan 28, 08:00 am-08:50 am

Hash families, Pairwise independent hashing. Proof of the uniqueness lemma. Reduction to Parity SAT. Amplification of success probability.

Meeting 09 : Mon, Feb 02, 10:00 am-10:50 am

Generalizations to show that PH BP-reduces to Parity-SAT.

Meeting 10 : Tue, Feb 03, 09:00 am-09:50 am

Any language that BP-reduces to Parity-SAT can be decided with a counting oracle. Proof idea. Toda's polynomials. Details of the proof.

Theme 2 : Interactive Proof Systems - 9 meetings

Meeting 11 : Wed, Feb 04, 08:00 am-08:50 am

Interactive Proof Systems. Deterministic, Randomized.
Power of Interaction 1 : Interactive Proof System for GNI.

Meeting 12 : Fri, Feb 06, 12:00 pm-12:50 pm

Power of Interaction 2: Making BPP one sided error using interaction.

Meeting 13 : Sat, Feb 07, 11:00 am-01:00 pm

(Compensatory class for Jan 16 & 19)
Public Coin Protocols. Classes AM and MA. Structural theorems:

MA is in Sigma_2.
Completeness for MA can be improved to 1.
MA is contained in AM.
Completeness for AM can be improved to 1.
AM is contained in Pi_2.
If CoNP is in AM, PH collapses to AM.

Meeting 14 : Mon, Feb 09, 10:00 am-10:50 am

Set lower bound protocol.Public Coin Protocol for GNI. GNI is in AM. Set lower bound protocol.
Hence if GI is NP-complete, PH = Sigma_2.

Meeting 15 : Tue, Feb 10, 09:00 am-09:50 am

Protocol for permanent. LFKN protocol and analysis. Protocol for #SAT. Generalization to prove IP = PSPACE.

Meeting 16 : Wed, Feb 11, 08:00 am-08:50 am

Arithmetization of QBF. Properties of Arithmetization. Mindblocks with respect to the protocol #SAT protocol.

Meeting 17 : Fri, Feb 13, 12:00 pm-12:50 pm

Degree Reduction using Quantifier introduction. Examples.

Meeting 18 : Mon, Feb 16, 10:00 am-10:50 am

Chinese remaindering. Coefficient size reduction. Completing Shamir's Proof of PSPACE is contained in IP
IP is contained in NEXP.
IP is contained in PSPACE.

Meeting 19 : Wed, Feb 18, 08:00 am-08:50 am

Multiprover interactive Proofs. Outline of MIP = NEXP proof. From MIP to PCP.

Theme 3 : PCPs and Inapproximability - 12 meetings

Meeting 20 : Fri, Feb 20, 12:00 pm-12:50 pm

The setting of Inapproximability. Independent Set Problem. GapIS. Connection between GapIS problem and inapproximability of IS. GapSAT and q-GapCSP.
From PCP theorem to hardness of q-GapCSP.

Meeting 21 : Mon, Feb 23, 10:00 am-10:50 am

L is in PCP(O(log n),q) if and only if L reduces to q-GapCSP. Statement of PCP theorem.

Meeting 22 : Tue, Feb 24, 09:00 am-09:50 am

Hardness of GapSAT. Hardness of GapIS. Constant factor inapproximability for Independent Set problem.
History of PCP theorem. A weaker version first. PCP for LIN.

Meeting 23 : Wed, Feb 25, 08:00 am-08:50 am

Weak PCP for LIN. Random vector testing. Allowing exponential sized proofs. Need for linearity testing. BLR Test for linearity.

Meeting 24 : Fri, Feb 27, 12:00 pm-12:50 pm

BLR test. Closeness to a Linear Function. Proof of Linearity testing.

Meeting 25 : Mon, Mar 02, 12:00 pm-12:50 pm

Proof of correctness of linearity testing using combinatorial arguments. Soundness for PCP for LIN.
Generalizing to setting of Qudratic Equations. Proof of consistency.

Meeting 26 : Tue, Mar 03, 09:00 am-09:50 am

Soundness Analysis of the weak PCP for QE.
Dinur's proof of PCP theorem. The strategy. Constraint graph of a 2-query CSP. Outline of the proof. Query reduction.

Meeting 27 : Wed, Mar 04, 08:00 am-08:50 am

Proof of Query reduction at the expense of alphabet size and soundness.

Meeting 28 : Mon, Mar 09, 10:00 am-10:50 am

Degree reduction and detailed proof of upper bound for increase in the soundness. Role of expanders.

Meeting 29 : Tue, Mar 10, 09:00 am-09:50 am

Expanderization. Review of second largest eigen value. Soundness Amplification (construction and overview).

Meeting 30 : Wed, Mar 11, 08:00 am-08:50 am

Alphabet Reduction construction and proof sketch.

Meeting 31 : Fri, Mar 13, 12:00 pm-12:50 pm

Intution and proof sketch for soundness amplification.

Theme 4 : Boolean Circuit Complexity - 25 meetings

Meeting 32 : Mon, Mar 16, 10:00 am-10:50 am

Boolean Circuit Model of Computation. Why circuits? Connections to parallel algorithm design. The relevant parameters of the circuit model - Size, depth, fanin. What is allowed as a gate? Basis and completeness. Post's characterization of a complete basis (statement).

Meeting 33 : Tue, Mar 17, 09:00 am-09:50 am

PSIZE = P/poly. Circuit Lower Bound Problem. Shannon's counting argument. Lupanov's construction (sketch)

Meeting 34 : Fri, Mar 20, 12:00 pm-12:50 pm

Gate Elimination Argument for Circuit Lower Bounds. Lower bounds for Parity and Threshold.

Meeting 35 : Sat, Mar 21, 10:00 am-11:30 am

Uniformity, Log-space Uniformity. P is computed by uniform polysize circuits.
Parity and Addition Functions. NC^1 and AC^0. Uniform NC^1 is contained in L.

Meeting 36 : Mon, Mar 23, 10:00 am-10:50 am

The class AC-hierarchy. Interleaving with NC hierarchy. NL is contained in uniform-AC^1.

Meeting 37 : Tue, Mar 24, 09:00 am-09:50 am

Back to NC^1, Adding n, n-bit numbers. Trivial AC^1 upper bound. Offman's technique and the NC^1 upper bound. Majority in NC^1. Constant depth reductions. Zooming in to NC^1. Review of constant depth reductions. Six Problems : ADD(n,n). MULT(2,n), Th(n,k), BCOUNT, MAJ. NC^1 upper bounds. Constant depth reductions among these problems. Motivation and definition of the class TC^0.

Meeting 38 : Wed, Mar 25, 08:00 am-08:50 am

Symmetric Functions are in TC^0. Motivation and definition of the class ACC^0. The hierarchy interleaving with NC and AC hierarchies.

Meeting 39 : Fri, Mar 27, 12:00 pm-12:50 pm

Formulas, BF = NC^1. Formula size lower bounds,

Meeting 40 : Sat, Mar 28, 10:00 am-11:30 am

Nechiporuk's Method. Lower bound for Indirect Access function. Proof of Nechiporuk's Theorem. Nechiporuk's "hardness embedding" technique. Sabbotavskaya's Lower bound using Restrictions. Lower Bounds for Parity.

Meeting 41 : Mon, Mar 30, 10:00 am-10:50 am

Extensions of Sabotavskaya's argument to random restrictions.

Meeting 42 : Tue, Mar 31, 09:00 am-09:50 am

Andreev's Lower Bound.

Meeting 43 : Mon, Apr 06, 10:00 am-10:50 am

Constant depth circuits cannot compute parity in polysize. Overview of the proof. Trivial cases like AND, OR. First nontrivial case - DNF.

Meeting 44 : Tue, Apr 07, 09:00 am-09:50 am

Inductive Argument. Exponential size lower bounds for constant depth circuits computing parity.

Meeting 45 : Wed, Apr 08, 08:00 am-08:50 am

Review of the proof. Proof of Switching Lemma.

Meeting 46 : Fri, Apr 10, 12:00 pm-12:50 pm

Details of Proof of switching lemma.

Meeting 47 : Mon, Apr 13, 10:00 am-10:50 am

Can Majority be computed by Constant depth circuits of poly-size using PARITY, AND and OR.
Main Technique : Approximation of Boolean functions by polynomials. Outline of the argument.

Meeting 48 : Tue, Apr 14, 09:00 am-09:50 am

OR-approximation Lemma over F_q. Lower Bounds for the degree of polynomials approximating Threshold functions well. Majority is not in AC^o[2].
Degree upper bounds for the polynomials approximating circuits over MOD-3, AND, OR using OR-approximation lemma for q=3. Lower bounds for the degree for approximating parity using polynomials over F_3. Razborov Smolensky theorem - PARITY is not in AC^0[3].

Meeting 49 : Wed, Apr 15, 08:00 am-08:50 am

Monotone Circuits. Razoborov's lower bound for monotone circuits. Proof strategy. Clique Approximators. Inductively building clique approximators for a monotone circuit.

Meeting 50 : Fri, Apr 17, 12:00 pm-12:50 pm

Lower bound on the error estimate for the approximate circuit.The lower bound argument.

Meeting 51 : Mon, Apr 20, 10:00 am-10:50 am

Upper bound on the error by counting that for each gate. Proof of the Sunflower Lemma.

Meeting 52 : Tue, Apr 21, 09:00 am-09:50 am

Branching Programs and Skew circuits. From space bounded algorithms to Branching Programs. Width of the Branching Programs.

Meeting 53 : Wed, Apr 22, 08:00 am-08:50 am

Programs over groups. Permutations. Conjugates, Cycle conjugacy lemma. Barrington's theorem for width-5. Role of Non-solvability of the group.

Meeting 54 : Fri, Apr 24, 12:00 pm-12:50 pm

Derandomization Problem, Hardness assumptions. Pseudo random generators.

Meeting 55 : Mon, Apr 27, 10:00 am-10:50 am

Nisan-Wigderson generator, NW Design. Proof of pseudorandomness. Hybrid argument.

Meeting 56 : Tue, Apr 28, 09:00 am-09:50 am

Arithmetic Circuits. Identity Testing Problem. From Derandomization to Hard Functions - Impagliazzo-Kabanets Theorem.

Evaluation Meetings
- 9 meetings
Meeting 57 : Mon, Apr 13, 04:00 pm-06:00 pm
Interim Meeting : Ameya, Aounon, Meenakshi, Vishvajeet
References
Exercises
Reading
Interim Meeting : Ameya, Aounon, Meenakshi, Vishvajeet
References : None
Meeting 58 : Thu, Apr 16, 02:00 pm-05:00 pm
Interim Meetings : Akshay Gadre, Purnata, Samir, Mitali
References
Exercises
Reading
Interim Meetings : Akshay Gadre, Purnata, Samir, Mitali
References : None
Meeting 59 : Sat, Apr 18, 11:00 am-01:00 pm
Interim Meetings : Ranjan, Aditi, Akshayram, Madhuri
References
Exercises
Reading
Interim Meetings : Ranjan, Aditi, Akshayram, Madhuri
References : None
Meeting 60 : Mon, Apr 20, 04:30 pm-05:30 pm
Interim Meetings : Vidhya, Srinivasan
References
Exercises
Reading
Interim Meetings : Vidhya, Srinivasan
References : None
Meeting 61 : Thu, Apr 23, 01:30 pm-04:00 pm
Presentations : Ameya, Meenakshi
References
Exercises
Reading
Presentations : Ameya, Meenakshi
References : None
Meeting 62 : Sat, Apr 25, 08:00 am-10:00 am
Presentations : Purnata, Akshay
References
Exercises
Reading
Presentations : Purnata, Akshay
References : None
Meeting 63 : Sat, Apr 25, 11:00 am-01:00 pm
Presentations : Samir, Ranjan
References
Exercises
Reading
Presentations : Samir, Ranjan
References : None
Meeting 64 : Sat, Apr 25, 01:30 pm-06:30 pm
Presentations : Mitali, Vishvajeet, Akshayram, Madhuri.
References
Exercises
Reading
Presentations : Mitali, Vishvajeet, Akshayram, Madhuri.
References : None
Meeting 65 : Mon, Apr 27, 01:30 pm-05:30 pm
Presentations : Vidhya, Srinivasan, Aditi, Aounon.
References
Exercises
Reading
Presentations : Vidhya, Srinivasan, Aditi, Aounon.
References : None

CS6840 - Advanced Complexity Theory

Today : Fri, Dec 6, 2024

Announcements

Meetings

References	Arora-Barak Textbook chapter on Complexity of Counting - section 17.1
Exercises	<ul> <li>Review the reduction from SAT to HAMCYCLE Problem.</li> <li>Calculate the exact gadget length that you need for the reduction to #CYCLE problem. </li> </ul>
Reading

References	Kristoffer Hansen's <a href="http://www.cs.au.dk/~arnsfelt/CT10/scribenotes/lecture10.pdf">Lecture Notes</a>
Exercises
Reading

References	<a href="http://www.cs.umd.edu/~jkatz/complexity/f11/lecture24.pdf">Jonathan Katz's Lecture notes</a>
Exercises
Reading

References	Arora Barak Textbook Chapter on Interactive Proofs. First few sections.
Exercises
Reading

References	Jonathan Katz's <a href="http://www.cs.umd.edu/~jkatz/complexity/f11/lecture16.pdf">lecture notes</a>.
Exercises
Reading

(Compensatory class for Jan 16 & 19) <br> Public Coin Protocols. Classes AM and MA. Structural theorems:<ul> <li>MA is in Sigma_2.</li> <li>Completeness for MA can be improved to 1.</li> <li>MA is contained in AM.</li> <li>Completeness for AM can be improved to 1.</li> <li>AM is contained in Pi_2.</li> <li>If CoNP is in AM, PH collapses to AM.</li> </ul>

References	:	Jonathan Katz's lecture notes.
Reading	:	Lowerbounds for switching Arthur and Merlin.

References	:	Jonathan Katz's Lecture Notes
Reading	:	IP = AM : Paper by Goldwasser and Sipser

References	:	None
Reading	:	Course by Prahladh Harsha : Limits of approximation algorithms : PCPs and Unique Games - Spring Semester (2009-10)

References	<ul> <li>Class Notes</li> <li>Zwick's Notes (sent to the mailing list) is the reference for Post's theorem.</li> </ul>
Exercises
Reading

References	Arora-Barak Textbook chapter on Complexity of Counting - section 17.2
Exercises	Question to think about : how do we develop a theory for integer valued functions which produces negative values too? <br> A reading related to the same : <a href="http://dx.doi.org/10.1006/inco.1996.0080">Gap Definable Counting Classes</a>
Reading	<ul> <li><a href="www.cse.buffalo.edu/~regan/papers/pdf/MPJrev.pdf">Power of the middle bit of a #P function.</a></li> </ul>

References	<ul> <li>Previous Year's Class Notes</li> <li>Jukna's Textbook Section 1.4</li> </ul>
Exercises
Reading

References	<ul> <li>Jukna's Textbook Section 1.6</li> </ul>
Exercises
Reading

References	<ul> <li>Lecture Notes from V.Vinay (sent to the mailing list)</li> </ul>
Exercises
Reading

References	<li>Class Notes</li>
Exercises
Reading	<li>Jukna's Textbook Section 6.1</li>

References	<li>Jukna's Textbook Section 6.2, 6.3</li>
Exercises
Reading

References	<li>Jukna's Textbook Section 6.2, 6.3, 6.4</li>
Exercises
Reading

References	<li><a href="http://homes.cs.washington.edu/~beame/papers/primer.ps">Switching Lemma Primer</a> - Paul Beame.</li>
Exercises
Reading

References	<li>Jukna's Textbook section 12.3</li>
Exercises
Reading

References	<li>Jukna's Textbook section 2.6, 12.3 and 12.4</li>
Exercises
Reading

References	<li><a href="http://www.cmi.ac.in/~ramprasad/lecturenotes/raz_clique.pdf">Lectures Notes</a> by Ramprasad Saptarishi in Arvind's course</li>
Exercises
Reading