  The 2-opt heuristic is a very simple local search heuristic for the traveling
salesperson problem. In practice it usually converges quickly to solutions
within a few percentages of optimality. In contrast to this, its running-time
is exponential and its approximation performance is poor in the worst case.
  Englert, R\"oglin, and V\"ocking (Algorithmica, 2014) provided a smoothed
analysis in the so-called one-step model in order to explain the performance of
2-opt on d-dimensional Euclidean instances, both in terms of running-time and
in terms of approximation ratio. However, translating their results to the
classical model of smoothed analysis, where points are perturbed by Gaussian
distributions with standard deviation sigma, yields only weak bounds.
  We prove bounds that are polynomial in n and 1/sigma for the smoothed
running-time with Gaussian perturbations. In addition, our analysis for
Euclidean distances is much simpler than the existing smoothed analysis.
  Furthermore, we prove a smoothed approximation ratio of O(log(1/sigma)). This
bound is almost tight, as we also provide a lower bound of Omega(log n/ loglog
n) for sigma = O(1/sqrt n). Our main technical novelty here is that, different
from existing smoothed analyses, we do not separately analyze objective values
of the global and local optimum on all inputs (which only allows for a bound of
O(1/sigma)), but simultaneously bound them on the same input.
