Author:
Englert Matthias,Röglin Heiko,Vöcking Berthold
Abstract
Abstract
2-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
$p\in\mathbb{N}$
, a family of L
p
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 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
$\tilde{O}(n^{4+1/3}\cdot\phi^{8/3})$
. When starting with an initial tour computed by an insertion heuristic, the upper bound on the expected number of steps improves even to
$\tilde{O}(n^{4+1/3-1/d}\cdot\phi^{8/3})$
. If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by
$\tilde{O}(n^{4-1/d}\cdot\phi)$
. In addition, we prove an upper bound of
$O(\sqrt[d]{\phi})$
on the expected approximation factor with respect to all L
p
metrics.
Let us remark that our probabilistic analysis covers as special cases the uniform input model with ϕ=1 and a smoothed analysis with Gaussian perturbations of standard deviation σ with ϕ∼1/σ
d
.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computer Science Applications,General Computer Science
Reference20 articles.
1. Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM 45(5), 753–782 (1998)
2. Bronshtein, I.N., Semendyayev, K.A., Musiol, G., Mühlig, H.: Handbook of Mathematics. Springer, Berlin (2007)
3. Chandra, B., Karloff, H.J., Tovey, C.A.: New results on the old k-Opt algorithm for the traveling salesman problem. SIAM J. Comput. 28(6), 1998–2029 (1999)
4. Dubhashi, D.P., Ranjan, D.: Balls and bins: a study in negative dependence. Random Struct. Algorithms 13(2), 99–124 (1998)
5. Englert, M., Röglin, H., Vöcking, B.: Worst case and probabilistic analysis of the 2-Opt algorithm for the TSP. In: Proc. of the 18th ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 1295–1304 (2007)
Cited by
64 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献