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The Traveling Salesman Problem Under Squared Euclidean Distances
[article]

2010
*
arXiv
*
pre-print

We denote

arXiv:1001.0236v3
fatcat:dk2ic7bm2rhalgzsfv5bwfnzwe
*the**traveling**salesman**problem**under*this*distance*function by TSP(d,α). ... We also study*the*variant Rev-TSP of*the**problem*where*the**traveling**salesman*is allowed to revisit points. ... Interestingly, these results do not carry over to Tsp (2, 2) with*squared**Euclidean**distances*. ...##
###
The Traveling Salesman Problem under Squared Euclidean Distances

2010
*
Symposium on Theoretical Aspects of Computer Science
*

We denote

doi:10.4230/lipics.stacs.2010.2458
dblp:conf/stacs/NijnattenSWWB10
fatcat:kbwl6mbmtjf2ziqr2xts575mju
*the**traveling**salesman**problem**under*this*distance*function by Tsp(d, α). ... We also study*the*variant Rev-Tsp of*the**problem*where*the**traveling**salesman*is allowed to revisit points. ... Interestingly, these results do not carry over to Tsp (2, 2) with*squared**Euclidean**distances*. ...##
###
THE TRAVELING SALESMAN PROBLEM UNDER SQUARED EUCLIDEAN DISTANCES

2010
*
Symposium on Theoretical Aspects of Computer Science
*
unpublished

We denote

fatcat:wrsuoz3zbndabp6wnhwrrdhksa
*the**traveling**salesman**problem**under*this*distance*function by Tsp(d, α). ... We also study*the*variant Rev-Tsp of*the**problem*where*the**traveling**salesman*is allowed to revisit points. ... Interestingly, these results do not carry over to Tsp (2, 2) with*squared**Euclidean**distances*. ...##
###
A PTAS for the Min-Max Euclidean Multiple TSP
[article]

2021
*
arXiv
*
pre-print

We present a polynomial-time approximation scheme (PTAS) for

arXiv:2112.04325v2
fatcat:u67tvjt6lvhdzoeu6mdfr4aoyi
*the*min-max multiple TSP*problem*in*Euclidean*space, where multiple*traveling*salesmen are tasked with visiting a set of n points and*the*objective ... Our algorithm introduces a rounding process to balance*the*allocation of path lengths among*the*multiple*salesman*. ...*The*length of a tour is just*the*sum of inter-city*distances**under**the**Euclidean*metric, and*the*objective is to minimize*the*maximum tour length over all k tours ...##
###
A Priori Bounds on the Euclidean Traveling Salesman

1995
*
SIAM journal on computing (Print)
*

It is proved that there are constants cl, c2, and c3 such that for any set S of n points in

doi:10.1137/s0097539792226771
fatcat:nuezkvnnnfcifivtpirdhqy37i
*the*unit*square*and for any minimum-length tour T of S (1)*the*sum of*squares*of*the*edge lengths of T is bounded ...*The*presence of*the*logarithmic term in (1) is engaging because such a term is not needed in*the*case of*the*minimum spanning tree and several analogous*problems*, and. furthermore, we know that there always ... It is well known that edges of an optimal*Euclidean**traveling**salesman*tour cannot intersect. ...##
###
Evolutionary Computational Approaches to Solving the Multiple Traveling Salesman Problem Using a Neighborhood Attractor Schema
[chapter]

2002
*
Lecture Notes in Computer Science
*

This paper presents a variation of

doi:10.1007/3-540-46004-7_16
fatcat:6pek7ynxd5fd3kjjsk4qi4phx4
*the**Euclidean**Traveling**Salesman**Problem*(TSP),*the*Multiple*Traveling**Salesman**Problem*(MTSP), and compares a variety of evolutionary computation algorithms and paradigms ... Techniques implemented, analyzed, and discussed herein with regard to MTSP include use of a neighborhood attractor schema (a variation on k-means clustering),*the*"shrink-wrap" algorithm for local neighborhood ... Conclusions and Future Research In this paper a variation of*the**Euclidean**Traveling**Salesman**Problem*(TSP),*the*Multiple*Traveling**Salesman**Problem*(MTSP), was presented and discussed. ...##
###
The curvature-constrained traveling salesman problem for high point densities

2007
*
2007 46th IEEE Conference on Decision and Control
*

In

doi:10.1109/cdc.2007.4434503
dblp:conf/cdc/NyFF07
fatcat:w63cofqnqvfz5b37nu63deo44u
*the*case of low point densities, i.e., when*the**Euclidean**distances*between*the*points are larger than*the*turning radius of*the*vehicle, various heuristics based on*the**Euclidean**Traveling**salesman*... We consider algorithms for*the*curvatureconstrained*traveling**salesman**problem*, when*the*nonholonomic constraint is described by Dubins' model. We indicate a proof of*the*NP-hardness of this*problem*. ... SURVEY OF RECENT RESULTS Let us state precisely*the*Dubins'*traveling**salesman**problem*(DTSP). ...##
###
The Travelling Salesman Problem and Minimum Matching in the Unit Square

1983
*
SIAM journal on computing (Print)
*

We show that

doi:10.1137/0212009
fatcat:sikverobmzdovd5xw6aewshbee
*the*cost (length) Of*the*shortest*traveling**salesman*tour through n points in*the*unit*square*is, in*the*worst case, aopt v/n + o (x/-n), where 1.075 atsPopt <= 1.414. ...*The*cost of*the*minimum 4+ O(4), where matching of n points in*the*unit*square*is shown to be, in*the*worst case, a opt 0.537 mat <0.707 Furthermore, for each of these two*problems*there is an almost linear ...*The*(*Euclidean*)*traveling**salesman*(respectively, matching)*problem*is to find a minimum cost tour (respectively, matching). ...##
###
Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems

1998
*
Journal of the ACM
*

For every fixed c Ͼ 1 and given any n nodes in 2 , a randomized version of

doi:10.1145/290179.290180
fatcat:r3vsoxqhq5hato3oytpnlcgw64
*the*scheme finds a (1 ϩ 1/c)-approximation to*the*optimum*traveling**salesman*tour in O(n(log n) O(c) ) time. ... We also give similar approximation schemes for some other NP-hard*Euclidean**problems*: Minimum Steiner Tree, k-TSP, and k-MST. ... Finally, I am indebted to Warren Smith and Michel Goemans for suggesting*the*use of Steiner points or "portals" in*the*description of*the*PTAS for TSP; this greatly simplified*the*exposition and*the*proof ...##
###
Page 3688 of Mathematical Reviews Vol. , Issue 90F
[page]

1990
*
Mathematical Reviews
*

*The*

*travel*cost between x; and x; has to be

*the*

*Euclidean*

*distance*. Let m =[n/2], 2< ! <[(m+1)/2], / an integer. ... An asymptotically exact algorithm for

*the*

*traveling*

*salesman*

*problem*for a maximum in

*Euclidean*space. (Russian) Upravlyaemye Sistemy No. 27 (1987), 79-87, 90. Consider n points x;,---,x, from R*. ...

##
###
Generalized travelling salesman problem through n sets of nodes: the asymmetrical case

1987
*
Discrete Applied Mathematics
*

This paper presents an exact algorithm for a generalized version of

doi:10.1016/0166-218x(87)90020-5
fatcat:a3p3nxg4fbgovovx32takkpkvi
*the**Travelling**Salesman**Problem*which consists of finding*the*shortest Hamiltonian circuit through n clusters of nodes, in*the*case where ...*the**distance*matrix is asymmetrical. ... In*the*generalized*travelling**salesman**problem*(GTSP) , it is assumed that N is*the*union of n clusters or sets of nodes S,. ...##
###
On properties of geometric random problems in the plane

1995
*
Annals of Operations Research
*

*The*paper specifically concentrates on

*the*

*traveling*

*salesman*and minimum spanning tree

*problems*, even though most of

*the*results apply to other

*problems*such as

*the*Steiner tree

*problem*and

*the*minimum ... In this paper, we present results dealing with properties of well-known geometric random

*problems*in

*the*plane, together with their motivations. ... Another analysis of partitioning algorithms for

*the*

*Euclidean*

*traveling*

*salesman*

*problem*is contained in Halton and Terada [11 ] . ...

##
###
Computational Geometry Column 37
[article]

1999
*
arXiv
*
pre-print

Open

arXiv:cs/9908007v1
fatcat:xbqfp3fsorbinbonpiisbunsfm
*problems*from*the*15th Annual ACM Symposium on Computational Geometry. ... Simplicity and hardness of*the*Maximum*Traveling**Salesman**Problem**under*geometric*distances*. Proc. 10th Annu. ACM-SIAM Sympos. Discrete Algorithms, 1999, 337-345. ...*The*maximum*traveling**salesman**problem**under*polyhedral norms. Proc. 6th Internat. Integer Program. Combin. Optim. Conf., Springer-Verlag, Lecture Notes in Comput.Sci. 1412Sci. , 1998.[F99] S. P. ...##
###
Nearly linear time approximation schemes for Euclidean TSP and other geometric problems
[chapter]

1997
*
Lecture Notes in Computer Science
*

For any xed c > 1 and any s e t o f n nodes in

doi:10.1007/3-540-63248-4_5
fatcat:lk3ymxklznfsbep22nqzr7fxqq
*the*plane,*the*new scheme nds a (1 + 1 c )approximation to*the*optimum*traveling**salesman*tour in O(n(log n) O(c) ) time. ...*The*algorithm generalizes to*the*same set of*Euclidean**problems*handled by t h e previous algorithm, including Steiner Tree, k-TSP, Minimum degree-restricted spanning tree, k-MST, etc, although for k-TSP ... Acknowledgements I thank Tom Leighton for useful discussions at*the*start of this project, and Mihalis Yannakakis for helpful comments. ...##
###
A Priori Optimization

1990
*
Operations Research
*

We consider four

doi:10.1287/opre.38.6.1019
fatcat:jxljmtkm7bcutjkdziqg77wmue
*problems*:*the**traveling**salesman**problem*(TSP),*the*minimum spanning tree, vehicle routing, and*traveling**salesman*facility location. ... We introduce*the*idea of a priori optimization as a strategy competitive to*the*strategy of reoptimization,*under*which*the*combinatorial optimization*problem*is solved optimally for every instance. ... ACKNOWLEDGMENT*The*research of*the*first and third authors is partially supported by*the*National Science Foundation*under*grant ECS-8717970. ...
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