CAREER: Randomized Computations and Probabilistically Checkable Proofs

Columbia University, New York NY

Investigators

Abstract

Summary: The use of randomness affects computations in dramatic and not yet fully understood ways: in algorithm design it yields simpler and more efficient ways to solve computational problems; in complexity theory it suggests new concepts and models that lead sometimes to unexpected (and far-reaching) results. This career development project involves a collection of research and educational activities related to computational randomness. The research component of this project deals with two main themes. One goal is the development of general tools that can be used to make randomized algorithms more robust, so that they can work even if they are implemented using biased, and limited, sources of randomness. Such tools are randomness extractors, procedures that convert biased distributions into almost uniform ones, and pseudorandom generators, procedures that stretch a short random input into a much longer output that has the property of being indistinguishable (a term that is given a precise technical meaning) from the uniform distribution. The other theme of the research component is the study of probabilistically checkable proofs (PCP), a model of computation that gives a surprising characterization of NP in terms of efficient randomized proof-checking. The PCP model is the best known tool to prove results about the complexity of finding approximate solutions for NP-hard combinatorial optimization problems. The goal of this project is to look for stronger characterizations of NP in the PCP model, for more applications to the study of the approximability of optimization problems and, with special emphasis, for a simplified proof of the PCP characterization of NP, a result that currently has an exceedingly complicated proof. The educational component of this project will integrate material on randomized algorithms, pseudorandomness, and probabilistic proof-systems into existing courses on algorithms and complexity and into a new course on cryptography that the principal investigator is developing. A main goal of the educational component is to give elementary presentations of some results that have so far been confined to research-oriented graduate courses. This is unfortunate because they are relevant and entertaining, not particularly hard to explain, and can have a strong motivational influence. An extensive set of lecture notes will be developed on this material.