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The Traveling Salesman Problem Under Squared Euclidean Distances [article]

Mark de Berg and Fred van Nijnatten and René Sitters and Gerhard J. Woeginger and Alexander Wolff
2010 arXiv   pre-print
We denote 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.  ... 
arXiv:1001.0236v3 fatcat:dk2ic7bm2rhalgzsfv5bwfnzwe

The Traveling Salesman Problem under Squared Euclidean Distances

Fred Van Nijnatten, René Sitters, Gerhard J. Woeginger, Alexander Wolff, Mark De Berg, Marc Herbstritt
2010 Symposium on Theoretical Aspects of Computer Science  
We denote 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.  ... 
doi:10.4230/lipics.stacs.2010.2458 dblp:conf/stacs/NijnattenSWWB10 fatcat:kbwl6mbmtjf2ziqr2xts575mju

THE TRAVELING SALESMAN PROBLEM UNDER SQUARED EUCLIDEAN DISTANCES

Mark De Berg, Fred Van Nijnatten, René Ren´, René Sitters, Gerhard Woeginger, Alexander Wolff
2010 Symposium on Theoretical Aspects of Computer Science   unpublished
We denote 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.  ... 
fatcat:wrsuoz3zbndabp6wnhwrrdhksa

A PTAS for the Min-Max Euclidean Multiple TSP [article]

Mary Monroe
2021 arXiv   pre-print
We present a polynomial-time approximation scheme (PTAS) for 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  ... 
arXiv:2112.04325v2 fatcat:u67tvjt6lvhdzoeu6mdfr4aoyi

A Priori Bounds on the Euclidean Traveling Salesman

Timothy Law Snyder, J. Michael Steele
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 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.  ... 
doi:10.1137/s0097539792226771 fatcat:nuezkvnnnfcifivtpirdhqy37i

Evolutionary Computational Approaches to Solving the Multiple Traveling Salesman Problem Using a Neighborhood Attractor Schema [chapter]

Donald Sofge, Alan Schultz, Kenneth De Jong
2002 Lecture Notes in Computer Science  
This paper presents a variation of 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.  ... 
doi:10.1007/3-540-46004-7_16 fatcat:6pek7ynxd5fd3kjjsk4qi4phx4

The curvature-constrained traveling salesman problem for high point densities

Jerome Le Ny, Emilio Frazzoli, Eric Feron
2007 2007 46th IEEE Conference on Decision and Control  
In 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).  ... 
doi:10.1109/cdc.2007.4434503 dblp:conf/cdc/NyFF07 fatcat:w63cofqnqvfz5b37nu63deo44u

The Travelling Salesman Problem and Minimum Matching in the Unit Square

Kenneth J. Supowit, Edward M. Reingold, David A. Plaisted
1983 SIAM journal on computing (Print)  
We show that 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).  ... 
doi:10.1137/0212009 fatcat:sikverobmzdovd5xw6aewshbee

Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems

Sanjeev Arora
1998 Journal of the ACM  
For every fixed c Ͼ 1 and given any n nodes in 2 , a randomized version of 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  ... 
doi:10.1145/290179.290180 fatcat:r3vsoxqhq5hato3oytpnlcgw64

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

Gilbert Laporte, Hélène Mercure, Yves Nobert
1987 Discrete Applied Mathematics  
This paper presents an exact algorithm for a generalized version of 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,.  ... 
doi:10.1016/0166-218x(87)90020-5 fatcat:a3p3nxg4fbgovovx32takkpkvi

On properties of geometric random problems in the plane

Patrick Jaillet
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 ] .  ... 
doi:10.1007/bf02098279 fatcat:de7ovxblyvddfec5eiir42djte

Computational Geometry Column 37 [article]

Erik D. Demaine, Joseph O'Rourke
1999 arXiv   pre-print
Open 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.  ... 
arXiv:cs/9908007v1 fatcat:xbqfp3fsorbinbonpiisbunsfm

Nearly linear time approximation schemes for Euclidean TSP and other geometric problems [chapter]

Sanjeev Arora
1997 Lecture Notes in Computer Science  
For any xed c > 1 and any s e t o f n nodes in 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.  ... 
doi:10.1007/3-540-63248-4_5 fatcat:lk3ymxklznfsbep22nqzr7fxqq

A Priori Optimization

Dimitris J. Bertsimas, Patrick Jaillet, Amedeo R. Odoni
1990 Operations Research  
We consider four 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.  ... 
doi:10.1287/opre.38.6.1019 fatcat:jxljmtkm7bcutjkdziqg77wmue
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