Heiko Röglin

k-Means Has Polynomial Smoothed Complexity

The k-means method is one of the most widely used clustering algorithms, drawing its popularity from its speed in practice. Recently, however, it was shown to have exponential worst-case running time. In order to close the gap between practical performance and theoretical analysis, the k-means method has been studied in the model of smoothed analysis. But even the smoothed analyses so far are unsatisfactory as the bounds are still super-polynomial in the number n of data points. In this paper, we settle the smoothed running time of the k-means method. We show that the smoothed number of iterations is bounded by a polynomial in n and 1/sigma, where sigma is the standard deviation of the Gaussian perturbations. This means that if an arbitrary input data set is randomly perturbed, then the k-means method will run in expected polynomial time on that input set.

Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP

Abstract2-Opt is probably the most basic local search heuristic for the TSP. This heuristic achieves amazingly good results on “real world” Euclidean instances both with respect to running time and approximation ratio. There are numerous experimental studies on the performance of 2-Opt. However, the theoretical knowledge about this heuristic is still very limited. Not even its worst case running time on 2-dimensional Euclidean instances was known so far. We clarify this issue by presenting, for every

On the Impact of Combinatorial Structure on Congestion Games

p\in\mathbb{N}

Smoothed Analysis of the k-Means Method

, a family of Lp instances on which 2-Opt can take an exponential number of steps.Previous probabilistic analyses were restricted to instances in which n points are placed uniformly at random in the unit square [0,1]2, where it was shown that the expected number of steps is bounded by

The Smoothed Number of Pareto Optimal Solutions in Bicriteria Integer Optimization

\tilde{O}(n^{10})

Theoretical Analysis of Two ACO Approaches for the Traveling Salesman Problem

for Euclidean instances. We consider a more advanced model of probabilistic instances in which the points can be placed independently according to general distributions on [0,1]d, for an arbitrary d≥2. In particular, we allow different distributions for different points. We study the expected number of local improvements in terms of the number n of points and the maximal density ϕ of the probability distributions. We show an upper bound on the expected length of any 2-Opt improvement path of

Smoothed analysis of integer programming

\tilde{O}(n^{4+1/3}\cdot\phi^{8/3})

Uncoordinated Two-Sided Matching Markets

. When starting with an initial tour computed by an insertion heuristic, the upper bound on the expected number of steps improves even to

Smoothed Analysis of Multiobjective Optimization

\tilde{O}(n^{4+1/3-1/d}\cdot\phi^{8/3})

Decision-making based on approximate and smoothed Pareto curves

. If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by