3,241 research outputs found
A time- and space-optimal algorithm for the many-visits TSP
The many-visits traveling salesperson problem (MV-TSP) asks for an optimal
tour of cities that visits each city a prescribed number of
times. Travel costs may be asymmetric, and visiting a city twice in a row may
incur a non-zero cost. The MV-TSP problem finds applications in scheduling,
geometric approximation, and Hamiltonicity of certain graph families.
The fastest known algorithm for MV-TSP is due to Cosmadakis and Papadimitriou
(SICOMP, 1984). It runs in time and
requires space. An interesting feature of the
Cosmadakis-Papadimitriou algorithm is its \emph{logarithmic} dependence on the
total length of the tour, allowing the algorithm to handle
instances with very long tours. The \emph{superexponential} dependence on the
number of cities in both the time and space complexity, however, renders the
algorithm impractical for all but the narrowest range of this parameter.
In this paper we improve upon the Cosmadakis-Papadimitriou algorithm, giving
an MV-TSP algorithm that runs in time , i.e.\
\emph{single-exponential} in the number of cities, using \emph{polynomial}
space. Our algorithm is deterministic, and arguably both simpler and easier to
analyse than the original approach of Cosmadakis and Papadimitriou. It involves
an optimization over directed spanning trees and a recursive, centroid-based
decomposition of trees.Comment: Small fixes, journal versio
RoboTSP - A Fast Solution to the Robotic Task Sequencing Problem
In many industrial robotics applications, such as spot-welding,
spray-painting or drilling, the robot is required to visit successively
multiple targets. The robot travel time among the targets is a significant
component of the overall execution time. This travel time is in turn greatly
affected by the order of visit of the targets, and by the robot configurations
used to reach each target. Therefore, it is crucial to optimize these two
elements, a problem known in the literature as the Robotic Task Sequencing
Problem (RTSP). Our contribution in this paper is two-fold. First, we propose a
fast, near-optimal, algorithm to solve RTSP. The key to our approach is to
exploit the classical distinction between task space and configuration space,
which, surprisingly, has been so far overlooked in the RTSP literature. Second,
we provide an open-source implementation of the above algorithm, which has been
carefully benchmarked to yield an efficient, ready-to-use, software solution.
We discuss the relationship between RTSP and other Traveling Salesman Problem
(TSP) variants, such as the Generalized Traveling Salesman Problem (GTSP), and
show experimentally that our method finds motion sequences of the same quality
but using several orders of magnitude less computation time than existing
approaches.Comment: 6 pages, 7 figures, 1 tabl
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)
Maximum Scatter TSP in Doubling Metrics
We study the problem of finding a tour of points in which every edge is
long. More precisely, we wish to find a tour that visits every point exactly
once, maximizing the length of the shortest edge in the tour. The problem is
known as Maximum Scatter TSP, and was introduced by Arkin et al. (SODA 1997),
motivated by applications in manufacturing and medical imaging. Arkin et al.
gave a -approximation for the metric version of the problem and showed
that this is the best possible ratio achievable in polynomial time (assuming ). Arkin et al. raised the question of whether a better approximation
ratio can be obtained in the Euclidean plane.
We answer this question in the affirmative in a more general setting, by
giving a -approximation algorithm for -dimensional doubling
metrics, with running time , where . As a corollary we obtain (i) an
efficient polynomial-time approximation scheme (EPTAS) for all constant
dimensions , (ii) a polynomial-time approximation scheme (PTAS) for
dimension , for a sufficiently large constant , and (iii)
a PTAS for constant and . Furthermore, we
show the dependence on in our approximation scheme to be essentially
optimal, unless Satisfiability can be solved in subexponential time
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