A Quantization Framework for Smoothed Analysis of Euclidean Optimization Problems

We consider the smoothed analysis of Euclidean optimization problems. Here, input points are sampled according to density functions that are bounded by a sufficiently small smoothness parameter . For such inputs, we provide a general and systematic approach that allows designing linear-time approximation algorithms whose output is asymptotically optimal, both in expectation and with high probability. Applications of our framework include maximum matching, maximum TSP, and the classical problems of k-means clustering and bin packing. Apart from generalizing corresponding average-case analyses, our results extend and simplify a polynomial-time probable approximation scheme on multidimensional bin packing on -smooth instances, where is constant (Karger and Onak in Polynomial approximation schemes for smoothed and random instances of multidimensional packing problems, pp 1207-1216, 2007). Both techniques and applications of our rounding-based approach are orthogonal to the only other framework for smoothed analysis of Euclidean problems we are aware of (Blaser et al. in Algorithmica 66(2):397-418, 2013).