Here is a real gift to the field from David Johnson: After a thirteen year intermission, David is restarting his NP-completeness column. His column will now appear about twice yearly in ACM Transactions on Algorithms. Welcome back David, and thanks! And for those for whom a diet of two per year won’t do, meals past can be found at http://www.research.att.com/ ̃dsj/columns.html. As to the Complex...

Foreword These are lectures notes for the introductory graduate courses on geometric complexity theory (GCT) in the computer science department, the university of Chicago. Part I consists of the lecture notes for the course given by the first author in the spring quarter, 2007. It gives introduction to the basic structure of GCT. Part II consists of the lecture notes for the course given by the...

This chapter provides an accessible and straightforward tour through some of the most basic notions and remarkable phenomena in machine-independent computational complexity. It includes versions of the recursion, compression, speedup, gap, union, and honesty theorems. Where possible, the results are derived from machine-dependent ones, in the particularly convenient concrete setting of space-bo...

Complexity theory is the part of theoretical computer science that attempts to prove that certain transformations from input to output are impossible to compute using a reasonable amount of resources. Theorem 1 below illustrates the type of ‘‘impossibility’’ proof that can sometimes be obtained (1); it talks about the problem of determining whether a logic formula in a certain formalism (abbrev...

In this article, we shall describe some of the most interesting topics in the subject of Complexity Theory for a general audience. Anyone with a solid foundation in high school mathematics (with some calculus) and an elementary understanding of computer programming will be able to follow this article. First, we shall describe the P versus NP problem and its significance. Next, we shall describe...

Complexity distortion theory (CDT) is a mathematical framework providing a unifying perspective on media representation. The key component of this theory is the substitution of the decoder in Shannon’s classical communication model with a universal Turing machine. Using this model, the mathematical framework for examining the efficiency of coding schemes is the algorithmic or Kolmogorov complex...

This paper furthers the study of quasi-linear time complexity initiated by Schnorr [Sch76] and Gurevich and Shelah [GS89]. We show that the fundamental properties of the polynomial-time hierarchy carry over to the quasilinear-time hierarchy. Whereas all previously known versions of the Valiant-Vazirani reduction from NP to parity run in quadratic time, we give a new construction using error-cor...

Most existing theory re nement systems are not incremental. However, any theory re nement system whose input and output theories are compatible can be used to incrementally assimilate data into an evolving theory. This is done by continually feeding its revised theory back in as its input theory. An incremental batch approach, in which the system assimilates a batch of examples at each step, se...

Autoreducibility and mitoticity express weak forms of redundancy of information that a set might possess. We describe many results known about these concepts. Mitoticity always implies autoreducibility, but the converse holds in some situations and fails in others. Among the results we describe are that NP-complete sets are many-one autoreducible and that every many-one autoreducible set is man...

1 Recall We state what was proven in Multiparty Communication Complexity I. has all of the inputs except x i. Let d(f) be the total number of bits broadcast in the optimal deterministic protocol for f. At the end of the protocol all parties must know the answer. This is called the multiparty communication complexity of f. The scenario is called the forehead model.