# CS 278 Computational Complexity Theory (Spring 2024)

Course Description

Computational Complexity studies the power and limitations of efficient computation. What can be computed with bounded resources such as time, memory, randomness, communication, and parallel cores? In this course, we will explore these beautiful questions. While most of them are widely open (e.g., Is verifying easier than proving? Is parallelism always helpful? Does randomness help in computation?), we will see many surprising connections between them. The course will be based on selected chapters from the book “Computational Complexity” by Sanjeev Arora and Boaz Barak.

Among the highlights, we will discuss Randomized Algorithms, Bounded-Space Algorithms, Savitch's Theorem, Immerman-Szelepcsényi's Theorem, the PCP Theorem and its connections to Hardness of Approximation, Interactive Proofs and IP = PSPACE, Hardness vs. Randomness, Pseudorandomness and Derandomization, Hardness Amplification, Introduction to Communication Complexity, Karchmer-Wigderson games, Circuit Complexity, Hardness within P.

General Information

Time and Place: Tuesday, Thursday 2:00 - 3:30 PM, Soda 405.

Instructor: Avishay Tal, email: atal "at" berkeley.edu, Office Hours: Thursday 3:30-4:30 PM.

TA: Hongxun Wu, email: wuhx "at" berkeley.edu, Office Hours: Thursday 9:00 - 10:00 AM at Soda 326.

Grading: Homework assignments - 50% (5 assignments), scribe - mandatory + replaces the lowest-grade hw assignment, Final Project & Presentation - 50%.

Prerequisite: CS170 or equivalent is required. CS172 or equivalent is recommended.

Pre-Reading: For those of you who want a refresher on the general setting, or those who haven't taken 172, please see Chapter 3 in Sipser's book ("Introduction to the Theory Of Computation" by Michael Sipser) or Chapters 1 & 2 Arora-Barak.

Textbooks & Lecture Notes:

Sanjeev Arora, Boaz Barak - Computational Complexity, A Modern Approach + Web Addendum

Oded Goldreich - Computational Complexity, A Conceptual Perspective

Avi Wigderson - Mathematics and Computation

Gil Cohen - A Taste of Circuit Complexity Pivoted at NEXP not in ACC (and more)

Topics

Time-Complexity (Chapter 3 AB)

Hierarchy Theorems

Barriers to diagonalization

Space-Complexity (Chapter 4 AB)

PSPACE, L (log-SPACE)

st-conn is NL complete

Savitch’s Theorem (NSPACE[s] in DSPACE[s^2])

Immerman-Szelepcsényi's Theorem (NSPACE is closed under complement)

Randomness in Computation (Chapters 5,7 AB)

Relation to the Polynomial Hierarchy

Relation to Circuits

Derandomization

Circuit Complexity (Chapters 6, 14, 23 AB)

The circuit complexity approach to P!=NP

The Karp-Lipton Theorem

Unconditional lower bounds for restrictive circuit classes like bounded-depth circuits

KW-Games

NEXP not in ACC^0

Barriers: Natural Proofs

Hardness vs Randomness (Chapters 19, 20 AB)

Pseudorandom Generators from Hard functions

Derandomization implies Circuit Lower Bounds

Hardness Amplification

Interactive Proofs (Chapter 8 AB)

Arthur-Merlin Protocols

IP = PSPACE

The PCP theorem and its connections to Hardness of Approximation (Chapter 11 AB)

Communication Complexity -

Karchmer-Wigderson Games (the connection between communication complexity and circuit complexity)

Lower Bounds on Randomized Communication Complexity

Lifting Theorems

Hardness within P