Lecture Notes on Computational Complexity

Foreword These are scribed notes from a graduate courses on Computational Complexity offered at the University of California at Berkeley in the Fall of 2002, based on notes scribed by students in Spring 2001 and on additional notes scribed in Fall 2002. I added notes and references in May 2004. The first 15 lectures cover fundamental material. The remaining lectures cover more advanced material, that is different each time I teach the class. In these Fall 2002 notes, there are lectures on Håstad's optimal inapproximability results, lower bounds for parity in bounded depth-circuits, lower bounds in proof-complexity, and pseudorandom generators and extractors. The notes have been only minimally edited, and there may be several errors and impre-cisions. I will be happy to receive comments, criticism and corrections about these notes. The syllabus for the course was developed jointly with Sanjeev Arora. Sanjeev wrote the notes on Yao's XOR Lemma (Lecture 11). Many people, and especially Avi Wigderson, were kind enough to answer my questions about various topics covered in these notes. I wish to thank the scribes This course assumes CS170, or equivalent, as a prerequisite. We will assume that the reader is familiar with the notions of algorithm and running time, as well as with basic notions of discrete math and probability. A main objective of theoretical computer science is to understand the amount of resources (time, memory, communication, randomness ,. . .) needed to solve computational problems that we care about. While the design and analysis of algorithms puts upper bounds on such amounts, computational complexity theory is mostly concerned with lower bounds; that is we look for negative results showing that certain problems require a lot of time, memory, etc., to be solved. In particular, we are interested in infeasible problems, that is computational problems that require impossibly large resources to be solved, even on instances of moderate size. It is very hard to show that a particular problem is infeasible, and in fact for a lot of interesting problems the question of their feasibility is still open. Another major line of work in complexity is in understanding the relations between different computational problems and between different " modes " of computation. For example what is the relative power of algorithms using randomness and deterministic algorithms, what is the relation between worst-case and average-case complexity, how easier can we make an optimization problem if we only look …