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TR06-092 | 5th July 2006 00:00

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


Authors: Matthias Englert, Heiko Röglin, Berthold Vöcking
Publication: 2nd August 2006 23:51
Downloads: 3451


2-Opt is probably the most basic and widely used 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 Euclidean instances was
known so far. In this paper, we clarify this issue by presenting a
family of Euclidean 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 $\tilde{O}(n^{10})$ for Euclidean instances. We consider
a more advanced model of probabilistic instances in which the points
can be placed according to general distributions on $[0,1]^2$. In
particular, we allow different distributions for different points.
We study the expected running time in terms of the number $n$ of
points and the maximal density $\phi$ of the probability
distributions. We show an upper bound on the expected length of any
2-Opt improvement path of $\tilde{O}(n^{4+1/3}\cdot\phi^{8/3})$.
When starting with an initial tour computed by an insertion
heuristic, the upper bound on the expected number of steps improves
even to $\tilde{O}(n^{3+5/6}\cdot\phi^{8/3})$. If the distances are
measured according to the Manhattan metric, then the expected number
of steps is bounded by $\tilde{O}(n^{3.5}\cdot\phi)$. In addition,
we prove an upper bound of $O(\sqrt{\phi})$ on the expected
approximation factor w.r.t. both of these metrics.

Let us remark that our probabilistic analysis covers as special
cases the uniform input model with $\phi=1$ and a smoothed analysis
with Gaussian perturbations of standard deviation $\sigma$ with
$\phi\sim 1/\sigma^2$. Besides random metric instances, we also
consider an alternative random input model in which an adversary
specifies a graph and distributions for the edge lengths in this
graph. In this model, we achieve even better results on the expected
running time of 2-Opt.

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