502 research outputs found
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
On the complexity of the multiple stack TSP, kSTSP
The multiple Stack Travelling Salesman Problem, STSP, deals with the collect
and the deliverance of n commodities in two distinct cities. The two cities are
represented by means of two edge-valued graphs (G1,d2) and (G2,d2). During the
pick-up tour, the commodities are stored into a container whose rows are
subject to LIFO constraints. As a generalisation of standard TSP, the problem
obviously is NP-hard; nevertheless, one could wonder about what combinatorial
structure of STSP does the most impact its complexity: the arrangement of the
commodities into the container, or the tours themselves? The answer is not
clear. First, given a pair (T1,T2) of pick-up and delivery tours, it is
polynomial to decide whether these tours are or not compatible. Second, for a
given arrangement of the commodities into the k rows of the container, the
optimum pick-up and delivery tours w.r.t. this arrangement can be computed
within a time that is polynomial in n, but exponential in k. Finally, we
provide instances on which a tour that is optimum for one of three distances
d1, d2 or d1+d2 lead to solutions of STSP that are arbitrarily far to the
optimum STSP
Recommended from our members
Fully dynamic maintenance of euclidean minimum spanning trees
We maintain the minimum spanning tree of a point set in the plane, subject to point insertions and deletions, in time O(n^5/6 log1^2/2 n) per update operation. No nontrivial dynamic geometric minimum spanning tree algorithm was previously known. We reduce the problem to maintaining bichromatic closest pairs, which we also solve in the same time bounds. Our algorithm uses a novel construction, the ordered nearest neighbors of a sequence of points. Any point set or bichromatic point set can be ordered so that this graph is a simple path
On trip planning queries in spatial databases
In this paper we discuss a new type of query in Spatial Databases, called Trip Planning Query (TPQ). Given a set of points P in space, where each point belongs to a category, and given two points s and e, TPQ asks for the best trip that starts at s, passes through exactly one point from each category, and ends at e. An example of a TPQ is when a user wants to visit a set of different places and at the same time minimize the total travelling cost, e.g. what is the shortest travelling plan for me to visit an automobile shop, a CVS pharmacy outlet, and a Best Buy shop along my trip from A to B? The trip planning query is an extension of the well-known TSP problem and therefore is NP-hard. The difficulty of this query lies in the existence of multiple choices for each category. In this paper, we first study fast approximation algorithms for the trip planning query in a metric space, assuming that the data set fits in main memory, and give the theory analysis of their approximation bounds. Then, the trip planning query is examined for data sets that do not fit in main memory and must be stored on disk. For the disk-resident data, we consider two cases. In one case, we assume that the points are located in Euclidean space and indexed with an Rtree. In the other case, we consider the problem of points that lie on the edges of a spatial network (e.g. road network) and the distance between two points is defined using the shortest distance over the network. Finally, we give an experimental evaluation of the proposed algorithms using synthetic data sets generated on real road networks
- …