Pseudorandomness and Combinatorial Constructions

TL;DR

This paper explores how the computational theory of pseudorandomness enables the derandomization of algorithms and the explicit construction of combinatorial objects like error-correcting codes and randomness extractors, linking theory and explicit methods.

Contribution

It reviews the connections between derandomization results and explicit combinatorial constructions, highlighting how pseudorandomness theory informs explicit object creation.

Findings

Derandomization results imply explicit constructions of combinatorial objects.

Connections between pseudorandomness and combinatorial constructions are elucidated.

The survey bridges theoretical pseudorandomness and practical explicit constructions.

Abstract

In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or unknown. In computer science, probabilistic algorithms are sometimes simpler and more efficient than the best known deterministic algorithms for the same problem. Despite this evidence for the power of random choices, the computational theory of pseudorandomness shows that, under certain complexity-theoretic assumptions, every probabilistic algorithm has an efficient deterministic simulation and a large class of applications of the the probabilistic method can be converted into explicit constructions. In this survey paper we describe connections between the conditional ``derandomization'' results of the computational theory of pseudorandomness and unconditional explicit constructions of certain combinatorial objects such as error-correcting codes and ``randomness extractors.''