Analysis of randomized algorithms via the probabilistic method

The probabilistic method is an extremely powerful tool in combinatorics that can be used to prove many surprising results. The idea is the following: to prove that an object with a certain property exists, we define a distribution of possible objects and use show that, among objects in the distribution, the property holds with non-zero probability. The key is that by using the tools and techniques of probability theory, we can vastly simplify proofs that would otherwise require very complicated combinatorial arguments.

As a technique, the probabilistic method developed rapidly during the latter half of the 20th century due to the efforts of mathematicians like Paul Erdős and increasing interest in the role of randomness in theoretical computer science. In essence, the probabilistic method allows us to determine how good a randomized algorithm's output is likely to be. Possibly applications range from graph property testing to computational geometry, circuit complexity theory, game theory, and even statistical physics.

In this talk, we will give a few examples that illustrate the basic method and show how it can be used to prove the existence of objects with desirable combinatorial properties as well as produce them in expected polynomial time via randomized algorithms. Our main goal will be to present a very slick proof from 1995 due to Spencer on the performance of a randomized greedy algorithm for a set-packing problem. Spencer, for seemingly no reason, introduces a time variable into his greedy algorithm and treats set-packing as a Poisson process. Then, like magic, he is able to show that his greedy algorithm is very likely to produce a good result using basic properties of expected value.

Properties of Poisson and Binomial distributions will be applied, but I'll remind everyone of the needed background for the benefit of those who might be a bit rusty. Stat 230 will be more than enough. Big O notation will be used, but not excessively.