152 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
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
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
Approximation Algorithms for Generalized MST and TSP in Grid Clusters
We consider a special case of the generalized minimum spanning tree problem
(GMST) and the generalized travelling salesman problem (GTSP) where we are
given a set of points inside the integer grid (in Euclidean plane) where each
grid cell is . In the MST version of the problem, the goal is to
find a minimum tree that contains exactly one point from each non-empty grid
cell (cluster). Similarly, in the TSP version of the problem, the goal is to
find a minimum weight cycle containing one point from each non-empty grid cell.
We give a and -approximation
algorithm for these two problems in the described setting, respectively.
Our motivation is based on the problem posed in [7] for a constant
approximation algorithm. The authors designed a PTAS for the more special case
of the GMST where non-empty cells are connected end dense enough. However,
their algorithm heavily relies on this connectivity restriction and is
unpractical. Our results develop the topic further
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
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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