2 . 2 A 78 - Approximation Algorithm for MAX 3 SAT

In CS109 and CS161, you learned some tricks of the trade in the analysis of randomized algorithms, with applications to the analysis of QuickSort and hashing. There’s also CS265, where you’ll learn more than you ever wanted to know about randomized algorithms (but a great class, you should take it). In CS261, we build a bridge between what’s covered in CS161 and CS265. Specifically, this lecture covers five essential tools for the analysis of randomized algorithms. Some you’ve probably seen before (like linearity of expectation and the union bound) while others may be new (like Chernoff bounds). You will need these tools in most 200and 300-level theory courses that you may take in the future, and in other courses (like in machine learning) as well. We’ll point out some applications in approximation algorithms, but keep in mind that these tools are used constantly across all of theoretical computer science. Recall the standard probability setup. There is a state space Ω; for our purposes, Ω is always finite, for example corresponding to the coin flip outcomes of a randomized algorithm. A random variable is a real-valued function X : Ω → R defined on Ω. For example, for a fixed instance of a problem, we might be interested in the running time or solution quality produced by a randomized algorithm (as a function of the algorithm’s coin flips). The expectation of a random variable is just its average value, with the averaging weights given by a specified probability distribution on Ω: