1,256 research outputs found
The Geometric Maximum Traveling Salesman Problem
We consider the traveling salesman problem when the cities are points in R^d
for some fixed d and distances are computed according to geometric distances,
determined by some norm. We show that for any polyhedral norm, the problem of
finding a tour of maximum length can be solved in polynomial time. If
arithmetic operations are assumed to take unit time, our algorithms run in time
O(n^{f-2} log n), where f is the number of facets of the polyhedron determining
the polyhedral norm. Thus for example we have O(n^2 log n) algorithms for the
cases of points in the plane under the Rectilinear and Sup norms. This is in
contrast to the fact that finding a minimum length tour in each case is
NP-hard. Our approach can be extended to the more general case of quasi-norms
with not necessarily symmetric unit ball, where we get a complexity of
O(n^{2f-2} log n).
For the special case of two-dimensional metrics with f=4 (which includes the
Rectilinear and Sup norms), we present a simple algorithm with O(n) running
time. The algorithm does not use any indirect addressing, so its running time
remains valid even in comparison based models in which sorting requires Omega(n
\log n) time. The basic mechanism of the algorithm provides some intuition on
why polyhedral norms allow fast algorithms.
Complementing the results on simplicity for polyhedral norms, we prove that
for the case of Euclidean distances in R^d for d>2, the Maximum TSP is NP-hard.
This sheds new light on the well-studied difficulties of Euclidean distances.Comment: 24 pages, 6 figures; revised to appear in Journal of the ACM.
(clarified some minor points, fixed typos
The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme
The Traveling Salesman Problem (TSP) is among the most famous NP-hard
optimization problems. We design for this problem a randomized polynomial-time
algorithm that computes a (1+eps)-approximation to the optimal tour, for any
fixed eps>0, in TSP instances that form an arbitrary metric space with bounded
intrinsic dimension.
The celebrated results of Arora (A-98) and Mitchell (M-99) prove that the
above result holds in the special case of TSP in a fixed-dimensional Euclidean
space. Thus, our algorithm demonstrates that the algorithmic tractability of
metric TSP depends on the dimensionality of the space and not on its specific
geometry. This result resolves a problem that has been open since the
quasi-polynomial time algorithm of Talwar (T-04)
Worst case and probabilistic analysis of the 2-Opt algorithm for the TSP
2-Opt is probably the most basic local search heuristic for the TSP. This heuristic achieves amazingly good results on âreal worldâ 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 pâN , 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 O~(n10) 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 dâ„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 Ï of the probability distributions. We show an upper bound on the expected length of any 2-Opt improvement path of O~(n4+1/3â
Ï8/3) . When starting with an initial tour computed by an insertion heuristic, the upper bound on the expected number of steps improves even to O~(n4+1/3â1/dâ
Ï8/3) . If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by O~(n4â1/dâ
Ï) . In addition, we prove an upper bound of O(Ïâd) 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 Ï=1 and a smoothed analysis with Gaussian perturbations of standard deviation Ï with ÏâŒ1/Ï d
Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane
Given a set of points with their pairwise distances, the traveling
salesman problem (TSP) asks for a shortest tour that visits each point exactly
once. A TSP instance is rectilinear when the points lie in the plane and the
distance considered between two points is the distance. In this paper, a
fixed-parameter algorithm for the Rectilinear TSP is presented and relies on
techniques for solving TSP on bounded-treewidth graphs. It proves that the
problem can be solved in where denotes the
number of horizontal lines containing the points of . The same technique can
be directly applied to the problem of finding a shortest rectilinear Steiner
tree that interconnects the points of providing a
time complexity. Both bounds improve over the best time bounds known for these
problems.Comment: 24 pages, 13 figures, 6 table
Quantum Annealing and Analog Quantum Computation
We review here the recent success in quantum annealing, i.e., optimization of
the cost or energy functions of complex systems utilizing quantum fluctuations.
The concept is introduced in successive steps through the studies of mapping of
such computationally hard problems to the classical spin glass problems. The
quantum spin glass problems arise with the introduction of quantum
fluctuations, and the annealing behavior of the systems as these fluctuations
are reduced slowly to zero. This provides a general framework for realizing
analog quantum computation.Comment: 22 pages, 7 figs (color online); new References Added. Reviews of
Modern Physics (in press
Comparing Mean Field and Euclidean Matching Problems
Combinatorial optimization is a fertile testing ground for statistical
physics methods developed in the context of disordered systems, allowing one to
confront theoretical mean field predictions with actual properties of finite
dimensional systems. Our focus here is on minimum matching problems, because
they are computationally tractable while both frustrated and disordered. We
first study a mean field model taking the link lengths between points to be
independent random variables. For this model we find perfect agreement with the
results of a replica calculation. Then we study the case where the points to be
matched are placed at random in a d-dimensional Euclidean space. Using the mean
field model as an approximation to the Euclidean case, we show numerically that
the mean field predictions are very accurate even at low dimension, and that
the error due to the approximation is O(1/d^2). Furthermore, it is possible to
improve upon this approximation by including the effects of Euclidean
correlations among k link lengths. Using k=3 (3-link correlations such as the
triangle inequality), the resulting errors in the energy density are already
less than 0.5% at d>=2. However, we argue that the Euclidean model's 1/d series
expansion is beyond all orders in k of the expansion in k-link correlations.Comment: 11 pages, 1 figur
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