CSC 2411 - Linear Programming and Combinatorial Optimization ∗ Lecture 5 : Smoothed Analysis , Randomized Combinatorial Algorithms , and Linear Programming Duality

In this class, we discuss a few " post-simplex-algorithm " issues. We will first study the smoothed case analysis of Linear Programming problems. We then learn the Seidel's algorithm, a randomized com-binatorial algorithm that run in subexponential time, and its extensions. Last, we will be introduced to the duality theorem of Linear Programs. 1 Overview In the previous lecture, we learned the procedure of the simplex algorithm. The algorithm leads us to an optimum solution. The question now is how long it will take the algorithm to reach this final solution. Moreover, we may wonder how well the algorithm perform under different settings. In this lecture, we will first examine the complexity of the simplex algorithm. Following this analysis, we will learn three ran-domized combinatorial algorithms which improve the running time, at least for some sets of parameters. Last, we will start our discussion on Linear Program duality. The worst case analysis estimates the maximum running time T of an algorithm A on all possible inputs I n of a given length n, whereas the average case analysis provides the average running time of a program on a distribution of inputs of a given length. The smoothed analysis was invented a few years ago motivated by the strictness of the worst-case and average-case analyses. The smoothed analysis presents the maximum running time of cases within certain range of the average cases. The range is defined * Lecture Notes for a course given