647 research outputs found
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
The Traveling Salesman Problem Under Squared Euclidean Distances
Let be a set of points in , and let be a
real number. We define the distance between two points as
, where denotes the standard Euclidean distance between
and . We denote the traveling salesman problem under this distance
function by TSP(). We design a 5-approximation algorithm for TSP(2,2)
and generalize this result to obtain an approximation factor of
for and all .
We also study the variant Rev-TSP of the problem where the traveling salesman
is allowed to revisit points. We present a polynomial-time approximation scheme
for Rev-TSP with , and we show that Rev-TSP is APX-hard if and . The APX-hardness proof carries
over to TSP for the same parameter ranges.Comment: 12 pages, 4 figures. (v2) Minor linguistic change
A Systematic Review of Approximability Results for Traveling Salesman Problems leveraging the TSP-T3CO Definition Scheme
The traveling salesman (or salesperson) problem, short TSP, is a problem of
strong interest to many researchers from mathematics, economics, and computer
science. Manifold TSP variants occur in nearly every scientific field and
application domain: engineering, physics, biology, life sciences, and
manufacturing just to name a few. Several thousand papers are published on
theoretical research or application-oriented results each year. This paper
provides the first systematic survey on the best currently known
approximability and inapproximability results for well-known TSP variants such
as the "standard" TSP, Path TSP, Bottleneck TSP, Maximum Scatter TSP,
Generalized TSP, Clustered TSP, Traveling Purchaser Problem, Profitable Tour
Problem, Quota TSP, Prize-Collecting TSP, Orienteering Problem, Time-dependent
TSP, TSP with Time Windows, and the Orienteering Problem with Time Windows. The
foundation of our survey is the definition scheme T3CO, which we propose as a
uniform, easy-to-use and extensible means for the formal and precise definition
of TSP variants. Applying T3CO to formally define the variant studied by a
paper reveals subtle differences within the same named variant and also brings
out the differences between the variants more clearly. We achieve the first
comprehensive, concise, and compact representation of approximability results
by using T3CO definitions. This makes it easier to understand the
approximability landscape and the assumptions under which certain results hold.
Open gaps become more evident and results can be compared more easily
Approximation Algorithms for Multi-Criteria Traveling Salesman Problems
In multi-criteria optimization problems, several objective functions have to
be optimized. Since the different objective functions are usually in conflict
with each other, one cannot consider only one particular solution as the
optimal solution. Instead, the aim is to compute a so-called Pareto curve of
solutions. Since Pareto curves cannot be computed efficiently in general, we
have to be content with approximations to them.
We design a deterministic polynomial-time algorithm for multi-criteria
g-metric STSP that computes (min{1 +g, 2g^2/(2g^2 -2g +1)} + eps)-approximate
Pareto curves for all 1/2<=g<=1. In particular, we obtain a
(2+eps)-approximation for multi-criteria metric STSP. We also present two
randomized approximation algorithms for multi-criteria g-metric STSP that
achieve approximation ratios of (2g^3 +2g^2)/(3g^2 -2g +1) + eps and (1 +g)/(1
+3g -4g^2) + eps, respectively.
Moreover, we present randomized approximation algorithms for multi-criteria
g-metric ATSP (ratio 1/2 + g^3/(1 -3g^2) + eps) for g < 1/sqrt(3)), STSP with
weights 1 and 2 (ratio 4/3) and ATSP with weights 1 and 2 (ratio 3/2). To do
this, we design randomized approximation schemes for multi-criteria cycle cover
and graph factor problems.Comment: To appear in Algorithmica. A preliminary version has been presented
at the 4th Workshop on Approximation and Online Algorithms (WAOA 2006
Asymmetric Traveling Salesman Path and Directed Latency Problems
We study integrality gaps and approximability of two closely related problems
on directed graphs. Given a set V of n nodes in an underlying asymmetric metric
and two specified nodes s and t, both problems ask to find an s-t path visiting
all other nodes. In the asymmetric traveling salesman path problem (ATSPP), the
objective is to minimize the total cost of this path. In the directed latency
problem, the objective is to minimize the sum of distances on this path from s
to each node. Both of these problems are NP-hard. The best known approximation
algorithms for ATSPP had ratio O(log n) until the very recent result that
improves it to O(log n/ log log n). However, only a bound of O(sqrt(n)) for the
integrality gap of its linear programming relaxation has been known. For
directed latency, the best previously known approximation algorithm has a
guarantee of O(n^(1/2+eps)), for any constant eps > 0. We present a new
algorithm for the ATSPP problem that has an approximation ratio of O(log n),
but whose analysis also bounds the integrality gap of the standard LP
relaxation of ATSPP by the same factor. This solves an open problem posed by
Chekuri and Pal [2007]. We then pursue a deeper study of this linear program
and its variations, which leads to an algorithm for the k-person ATSPP (where k
s-t paths of minimum total length are sought) and an O(log n)-approximation for
the directed latency problem
Approximability of the Multiple Stack TSP
STSP seeks a pair of pickup and delivery tours in two distinct networks,
where the two tours are related by LIFO contraints. We address here the problem
approximability. We notably establish that asymmetric MaxSTSP and MinSTSP12 are
APX, and propose a heuristic that yields to a 1/2, 3/4 and 3/2 standard
approximation for respectively Max2STSP, Max2STSP12 and Min2STSP12
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