CS6840 Modern Complexity Theory

Feb - May 2021

The lecture videos and the notes written during the lectures are available below.

Week 1
Administrative details about the course; Recap from last semester - probabilistic complexity classes; Complexity of counting - the classes PP and #P; Notion of #P-completeness and parsimonious reductions; Perfect matchings, cycle covers and permanent of matrices.

Week 2
Valiant’s theorem - counting perfect matchings in bipartite graphs is #P-complete. Toda’s theorem - BP and parity operators and their properties; Isolation lemma and its proof.

Week 3
Toda’s theorem - Randomized reduction from NP to \(\oplus\) P; relativizing the reduction to PH; Derandomizing the reduction - Modulus amplifying polynomials; Boolean circuits and lower bounds; Shannon’s lower bound and Lupanov’s upper bound.

Week 4
Uniform circuit families; P-uniform P/poly is P; NC and AC hierarchy; Examples of functions in NC and AC; P vs NC question and P-completeness; Lower bounds: Restrictions and Gate elimination - Schnorr’s lower bound; Random restrictions and shrinkage - Subbotovskaya’s lower bound; Lower bounds for parity - decision trees and the switching lemma.

Week 5
Lower bounds for parity - decision trees and the switching lemma; Proof of the switching lemma.

Week 6
Majority in \(NC^1\); Parity constant-depth reduces to Majority; Lower bound for \(AC^0(\oplus)\) - Razborov-Smolensky technique.

Week 7
Lower bound for \(AC^0(MOD_{p^k})\) - Razborov-Smolensky technique; Hardness vs Randomness - Definition of PRGs; Derandomization using PRGs; PRGs and unpredictability.

Week 8
PRGs and unpredictability; Hardness vs Randomness - the Nisan-Wigderson generator; Hardness amplification: Yao’s XOR lemma - from mild to strong hardness; Impagliazzo’s hardcore lemma; Brief outline of locally decodable codes - from worst-case hardness to mildly average-case hardness.
  • Videos: (will be updated)
  • References: [AB] - Chapters 19 and 20
  • Scribbles: (will be uploaded soon)

Week 9
Mid-semester break

Week 10
Interactive proofs - private coins and public coins; Arthur-Merlin games; Interactive proof for graph non-isomorphism; Set lower bound protocol - AM protocol for graph non-isomorphism; properties of the classes AM and MA.
  • Videos: (will be updated)
  • Reference: [AB] - Chapters 8; Lecture notes - Katz here and here; Kabanets here
  • Scribbles: (will be uploaded soon)

Week 11
Simulating the optimal prover in PSPACE; Permanent in IP: LFKN protocol; IP = PSPACE: Using self-reducibility, arithmetization and linearization.
  • Videos: (link sent on the course mailing list)
  • References: [AB] - Chapter 8, Babai’s exposition
  • Scribbles: (will be uploaded soon)

Week 12
IP=PSPACE: final protocol; Proof verification view of interactive proofs; MIP = PCP(\(n^c, n^c\)); Statement of the PCP theorem; PCP theorem and inapproximability.
  • References: [AB] - Chapters 8, 11

Week 13
Example of a PCP for NP - NP \(\subseteq\) PCP\((n^c, O(1))\); Derandomization implies circuit lower bounds - Kabanets-Impagliazzo theorem; IKW theorem - hard witnesses and derandomization; From SAT algorithms to circuit lower bounds - statement of Williams’ theorem.

Week 14
From SAT algorithms to circuit lower bounds - statement and proof of Williams’ theorem; Succinct certificates and verification; Wrap-up of the course.