Randomness and Pseudo-Randomness in Discrete Mathematics

The discovery, demonstrated in the early work of Paley, Zygmund, Erdős, Kac, Turan, Shannon, Szele and others, that deterministic statements can be proved by probabilistic reasoning, led already in the first half of the century to several striking results in Analysis, Number Theory, Combinatorics and Information Theory. It soon became clear that the method, which is now called the probabilistic method, is a very powerful tool for proving results in Discrete Mathematics. The early results combined combinatorial arguments with fairly elementary probabilistic techniques, whereas the development of the method in recent years required the application of more sophisticated tools from probability. The books [10], [54] are two recent texts dealing with the subject. Most probabilistic proofs are existence, non-constructive arguments. The rapid development of theoretical Computer Science, and its tight connection to Combinatorics, stimulated the study of the algorithmic aspects of these proofs. In a typical probabilistic proof, one establishes the existence of a combinatorial structure satisfying certain properties by considering an appropriate probability space of structures, and by showing that a randomly chosen point of this space is, with positive probability, a structure satisfying the required properties. Can we find such a structure efficiently, that is, by a (deterministic or randomized) polynomial time algorithm ? In several cases the probabilistic proof provides such a randomized efficient algorithm, and in other cases the task of finding such an algorithm requires additional ideas. Once an efficient randomized algorithm is found, it is sometimes possible to derandomize it and convert it into an efficient deterministic one. To this end, certain explicit pseudo-random structures are needed, and their construction often requires tools from a wide variety of mathematical areas including Group Theory, Number Theory and Algebraic Geometry. The application of probabilistic techniques for proving deterministic theorems, and the application of deterministic theorems for derandomizing probabilistic existence proofs, form an interesting combination of mathematical ideas from various areas, whose intensive study in recent years led to the development of fascinating techniques. In this paper I survey some of these developments and mention several related open problems. ∗Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel. Email: noga@math.tau.ac.il. Research supported in part by a USA Israeli BSF grant and by the Fund for Basic Research administered by the Israel Academy of Sciences.