Smoothed analysis of local search for the maximum-cut problem

Even though local search heuristics are the method of choice in practice for many well-studied optimization problems, most of them behave poorly in the worst case. This is in particular the case for the Maximum-Cut Problem, for which local search can take an exponential number of steps to terminate and the problem of computing a local optimum is PLS-complete. To narrow the gap between theory and practice, we study local search for the Maximum-Cut Problem in the framework of smoothed analysis in which inputs are subject to a small amount of random noise. We show that the smoothed number of iterations is quasi-polynomial, i.e., it is bounded from above by a polynomial in n^{log n} and ϕ where n denotes the number of nodes and ϕ denotes the perturbation parameter. This shows that worst-case instances are fragile and it is a first step in explaining why they are rarely observed in practice.