Smoothed Analysis of Algorithms: Why the Simplex Algorithm Usually Takes Polynomial Time

Man, there was a lot of obtuse argument to wade through to get to this post. This is an interesting fact. Apparently this is the source? https://arxiv.org/abs/cs/0111050

The article discusses how the two mathematicians (Spielman and Teng) came up with Smoothed Analysis for Algorithms (as opposed to Worst case) which gives insight into why Simplex works well. We introduce the smoothed analysis of algorithms, which is a hybrid of the worst-case and average-case analysis of algorithms. In smoothed analysis, we measure the maximum over inputs of the expected performance of an algorithm under small random perturbations of that input. We measure this performance in terms of both the input size and the magnitude of the perturbations. We show that the simplex algorithm has polynomial smoothed complexity https://arxiv.org/abs/cs/0111050 The two authors won the Godel prize in 2008 for this work.

> If you have tens of thousands of variables, you definitely don't want to risk dealing with the simplex method's worst-case behavior. The risk is extremely low. In fact, the risk is so low that, for deacdes, it was an open question why the simplex algorithm was so good in practice on a wide variety of workloads despite having poor worst-case complexity (unlike, for example, pure quicksort, which is often problematic in practice). Spielman solved that problem in 2001 and showed that even very small perturbations from a contrived worst-case input have good performance: http://arxiv.org/pdf/cs/0111050.pdf

OTOH, simplex is very efficient already. Average-case complexity for random matrices is polynomial, and there are some interesting results that extend beyond that (see http://arxiv.org/abs/cs/0111050).