54 research outputs found
Constant-Factor Approximation for TSP with Disks
We revisit the traveling salesman problem with neighborhoods (TSPN) and
present the first constant-ratio approximation for disks in the plane: Given a
set of disks in the plane, a TSP tour whose length is at most times
the optimal can be computed in time that is polynomial in . Our result is
the first constant-ratio approximation for a class of planar convex bodies of
arbitrary size and arbitrary intersections. In order to achieve a
-approximation, we reduce the traveling salesman problem with disks, up
to constant factors, to a minimum weight hitting set problem in a geometric
hypergraph. The connection between TSPN and hitting sets in geometric
hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure
A PTAS for Euclidean TSP with Hyperplane Neighborhoods
In the Traveling Salesperson Problem with Neighborhoods (TSPN), we are given
a collection of geometric regions in some space. The goal is to output a tour
of minimum length that visits at least one point in each region. Even in the
Euclidean plane, TSPN is known to be APX-hard, which gives rise to studying
more tractable special cases of the problem. In this paper, we focus on the
fundamental special case of regions that are hyperplanes in the -dimensional
Euclidean space. This case contrasts the much-better understood case of
so-called fat regions.
While for an exact algorithm with running time is known,
settling the exact approximability of the problem for has been repeatedly
posed as an open question. To date, only an approximation algorithm with
guarantee exponential in is known, and NP-hardness remains open.
For arbitrary fixed , we develop a Polynomial Time Approximation Scheme
(PTAS) that works for both the tour and path version of the problem. Our
algorithm is based on approximating the convex hull of the optimal tour by a
convex polytope of bounded complexity. Such polytopes are represented as
solutions of a sophisticated LP formulation, which we combine with the
enumeration of crucial properties of the tour. As the approximation guarantee
approaches , our scheme adjusts the complexity of the considered polytopes
accordingly.
In the analysis of our approximation scheme, we show that our search space
includes a sufficiently good approximation of the optimum. To do so, we develop
a novel and general sparsification technique to transform an arbitrary convex
polytope into one with a constant number of vertices and, in turn, into one of
bounded complexity in the above sense. Hereby, we maintain important properties
of the polytope
The traveling salesman problem for lines, balls and planes
We revisit the traveling salesman problem with neighborhoods (TSPN) and
propose several new approximation algorithms. These constitute either first
approximations (for hyperplanes, lines, and balls in , for ) or improvements over previous approximations achievable in comparable times
(for unit disks in the plane).
\smallskip (I) Given a set of hyperplanes in , a TSP tour
whose length is at most times the optimal can be computed in
time, when is constant.
\smallskip (II) Given a set of lines in , a TSP tour whose
length is at most times the optimal can be computed in polynomial
time for all .
\smallskip (III) Given a set of unit balls in , a TSP tour
whose length is at most times the optimal can be computed in polynomial
time, when is constant.Comment: 30 pages, 9 figures; final version to appear in ACM Transactions on
Algorithm
TSP With Locational Uncertainty: The Adversarial Model
In this paper we study a natural special case of the Traveling Salesman Problem (TSP) with point-locational-uncertainty which we will call the adversarial TSP problem (ATSP). Given a metric space (X, d) and a set of subsets R = {R_1, R_2, ...R_n} : R_i subseteq X, the goal is to devise an ordering of the regions, sigma_R, that the tour will visit such that when a single point is chosen from each region, the induced tour over those points in the ordering prescribed by sigma_R is as short as possible. Unlike the classical locational-uncertainty-TSP problem, which focuses on minimizing the expected length of such a tour when the point within each region is chosen according to some probability distribution, here, we focus on the adversarial model in which once the choice of sigma_R is announced, an adversary selects a point from each region in order to make the resulting tour as long as possible. In other words, we consider an offline problem in which the goal is to determine an ordering of the regions R that is optimal with respect to the ``worst\u27\u27 point possible within each region being chosen by an adversary, who knows the chosen ordering. We give a 3-approximation when R is a set of arbitrary regions/sets of points in a metric space. We show how geometry leads to improved constant factor approximations when regions are parallel line segments of the same lengths, and a polynomial-time approximation scheme (PTAS) for the important special case in which R is a set of disjoint unit disks in the plane
Dynamic Vehicle Routing for Data Gathering in Wireless Networks
We consider a dynamic vehicle routing problem in wireless networks where
messages arriving randomly in time and space are collected by a mobile receiver
(vehicle or a collector). The collector is responsible for receiving these
messages via wireless communication by dynamically adjusting its position in
the network. Our goal is to utilize a combination of wireless transmission and
controlled mobility to improve the delay performance in such networks. We show
that the necessary and sufficient condition for the stability of such a system
(in the bounded average number of messages sense) is given by {\rho}<1 where
{\rho} is the average system load. We derive fundamental lower bounds for the
delay in the system and develop policies that are stable for all loads {\rho}<1
and that have asymptotically optimal delay scaling. Furthermore, we extend our
analysis to the case of multiple collectors in the network. We show that the
combination of mobility and wireless transmission results in a delay scaling of
{\Theta}(1/(1- {\rho})) with the system load {\rho} that is a factor of
{\Theta}(1/(1- {\rho})) smaller than the delay scaling in the corresponding
system where the collector visits each message location.Comment: 19 pages, 7 figure
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)
On the Approximability of the Traveling Salesman Problem with Line Neighborhoods
We study the variant of the Euclidean Traveling Salesman problem where instead of a set of points, we are given a set of lines as input, and the goal is to find the shortest tour that visits each line. The best known upper and lower bounds for the problem in , with , are -hardness and an -approximation algorithm which is based on a reduction to the group Steiner tree problem. We show that TSP with lines in is APX-hard for any . More generally, this implies that TSP with -dimensional flats does not admit a PTAS for any unless , which gives a complete classification of the approximability of these problems, as there are known PTASes for (i.e., points) and (hyperplanes). We are able to give a stronger inapproximability factor for by showing that TSP with lines does not admit a -approximation in dimensions under the unique games conjecture. On the positive side, we leverage recent results on restricted variants of the group Steiner tree problem in order to give an -approximation algorithm for the problem, albeit with a running time of
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