Smoothed Analysis of Local Search

Local search is a powerful paradigm for finding good solutions to intractable combinatorial optimization problems. However, for many local search heuristics there exist worst-case instances on which they are extremely slow or provide solutionsthat are far from optimal. Smoothed analysis is a semi-random input model that has been invented to bridge the gap between poor worst-case and good empirical performance of algorithms. In smoothed analysis, an adversary picks an arbitrary input, which is then slightly randomly perturbed. In particular, smoothed analysis has been applied successfullyto local search algorithms in a variety of cases. We use the 2-opt heuristic for the traveling salesman problem and the k-means method for clustering as examples to explain how local search heuristics can be analyzed in the framework of smoothed analysis. For both algorithm, as for many other local search algorithms, the worst-case running time is exponential in the input size, but polynomial in the framework of smoothed analysis.