18 . S 096 : Approximation Algorithms and Max - Cut Topics in Mathematics of Data Science ( Fall 2015 ) Afonso

Unless the widely believed P 6= NP conjecture is false, there is no polynomial algorithm that can solve all instances of an NP-hard problem. Thus, when faced with an NP-hard problem (such as the Max-Cut problem discussed below) one has three options: to use an exponential type algorithm that solves exactly the problem in all instances, to design polynomial time algorithms that only work for some of the instances (hopefully the typical ones!), or to design polynomial algorithms that, in any instance, produce guaranteed approximate solutions. This section is about the third option. The second is discussed in later in the course, in the context of community detection. The Max-Cut problem is the following: Given a graph G = (V,E) with non-negative weights wij on the edges, find a set S ⊂ V for which cut(S) is maximal. Goemans and Williamson [GW95] introduced an approximation algorithm that runs in polynomial time and has a randomized component to it, and is able to obtain a cut whose expected value is guaranteed to be no smaller than a particular constant αGW times the optimum cut. The constant αGW is referred to as the approximation ratio. Let V = {1, . . . , n}. One can restate Max-Cut as