61. Network Models Network Models. Edited by MO Ball, TL Magnanti, CL Monma and GL Nemhauser. CHAPTER4. The traveling salesman problem. M. Ringer, G. Reinelt and G. Rinaldi. http://www1.elsevier.com/homepage/sae/orms/horms/vol7/chap4.html

Network Models Edited by M.O. Ball, T.L. Magnanti, C.L. Monma and G.L. Nemhauser CHAPTER 4 The Traveling Salesman Problem M. Ringer, G. Reinelt and G. Rinaldi 1. Introduction 2. Related problems 3. Practical applications 4. Approximation algorithms 5. Relaxations 6. Finding optimal and provably good solutions 7. Computation References The first two pages of the chapters are available as PDF file. [Back]

62. 8.4 Traveling Salesman Problem 8.4 traveling salesman problem. REF. The traveling salesman problem.John Wiley Sons, 1985. Tour (Hamilton) (Hamiltonian cycle) http://lcm.csa.iisc.ernet.in/dsa/node185.html

63. 8.7.2 Traveling Salesman Problem Graph Algorithms. 8.7.2 traveling salesman problem. This assignmentinvolves implementing the following five problems. 1. Generate http://lcm.csa.iisc.ernet.in/dsa/node192.html

64. Problem 54: Traveling Salesman Problem In Solid Grid Graphs Problem 54 traveling salesman problem in Solid Grid Graphs. Statement Whatis the complexity of finding a shortest tour in a solid planar grid graph? http://cs.smith.edu/~orourke/TOPP/P54.html

Next: Problem 55: Pallet Loading Up: The Open Problems Project Previous: Problem 53: Minimum-Turn Cycle Problem 54: Traveling Salesman Problem in Solid Grid Graphs Statement What is the complexity of finding a shortest tour in a solid planar grid graph? A planar grid graph is a graph whose vertices are any set of points on the planar integer lattice and whose edges connect every pair of vertices at unit distance. Distances between nodes correspond to induced shortest-path distances in the graph, which corresponds to ``Manhattan'' distances. A grid graph is solid if it does not have any holes, i.e., its complement in the planar integer lattice is connected. Origin ] show that the problem is NP-complete in general planar grid graphs. Status/Conjectures Open. Motivation Partial and Related Results ] show that Hamiltonicity of a solid grid graph can be decided in polynomial time. Thus we can decide whether there is a tour of length equal to the number of vertices. In contrast, deciding Hamiltonicity is NP-hard in general planar grid graphs [ ABD+01 ] observe that finding the shortest tour is polynomially solvable when restricted to thin grid graphs, i.e., grid graphs that do not contain an induced

65. Traveling Salesman Problem - InformationBlast traveling salesman problem Information Blast. traveling salesman problem. Thebottleneck traveling salesman problem is also NP-hard. Algorithms. http://www.informationblast.com/Traveling_salesman_problem.html

Traveling salesman problem The traveling salesman problem TSP ), also known as the traveling salesperson problem , is a problem in discrete or combinatorial optimization . It is a prominent illustration of a class of problems in computational complexity theory which are hard to solve. Given a number of cities and the costs of travelling from one to the other, what is the cheapest roundtrip route that visits each city and then returns to the starting city? An equivalent formulation in terms of graph theory is: Find the shortest Hamiltonian cycle in a weighted graph It can be shown that the requirement of returning to the starting city does not change the computational complexity of the problem. A related problem is the Bottleneck traveling salesman problem (bottleneck TSP): Find the Hamiltonian cycle in a weighted graph with the minimal length of the longest edge The problem is of considerable practical importance, apart from evident transportation and logistics areas. A classical example is in printed circuit manufacturing scheduling of a route of the drill machine to drill holes in a PCB. In robotic machining or drilling applications, the "cities" are parts to machine or holes (of different sizes) to drill, and the "cost of travel" includes time for retooling the robot (single machine job sequencing problem). The most direct solution would be to try all the combinations and see which one is cheapest, but given that the number of combinations is N! (the

66. Tsp.html Solving the traveling salesman problem. Bruno Guerrieri. Florida A MUniversity. bruno.guerrieri@famu.edu. Initialization. The Traveling http://www.mapleapps.com/categories/mathematics/Operations Research/html/tsp.htm

tsp.mws Initialization Needed Procedures used by all Algorithms Procedure Used by The Nearest Neighbor Algorithm ... References Solving the Traveling Salesman Problem Bruno Guerrieri bruno.guerrieri@famu.edu Initialization restart: with(plots): with(plottools): Warning, the name changecoords has been redefined Warning, the name arrow has been redefined Needed Procedures used by all Algorithms We will need certain standard procedures for an experiment of that sort. We decided to keep the program simple, consider the symmetric case, and let the computer randomly generate N city positions within a 1x1 square. So, the procedure "city_position" does just that. global N; local k, toss, city_pos; toss := rand(0..10^7); for k from 1 to N do od; RETURN(city_pos); end: "mixup" is a procedure which randomly selects a particular tour. In general, this procedure is invoked at the beginning of the process so as to start with a given tour on which to improve i.e. incrementally evolving to a better tour by removing certain edges and adding others while maintaining the integrity of the tour . If the incoming parameter "val" is set to 1, then there is no rearranging in the initial order of the cities at the time of the call. Otherwise, there is. mixup := proc(val) global N;

67. Wiley Canada::The Traveling Salesman Problem: A Guided Tour Of Combinatorial Opt Wiley Canada Mathematics Statistics Discrete Mathematics The TravelingSalesman Problem A Guided Tour of Combinatorial Optimization. http://www.wiley.ca/WileyCDA/WileyTitle/productCd-0471904139.html

Shopping Cart My Account Help Contact Us By Keyword By Title By Author By ISBN By ISSN Wiley Canada Discrete Mathematics The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization Related Subjects General Interest Computer Science General Computer Engineering Finite Mathematics General Statistics ... Special Topics in Mathematics Related Titles More By These Authors Scheduling Theory and Its Applications (Hardcover) Discrete Mathematics Simulated Annealing and Boltzmann Machines: A Stochastic Approach to Combinatorial Optimization and Neural Computing (Hardcover) by Emile Aarts, Jan Korst Optimal Control: Basics and Beyond (Hardcover) by Peter Whittle Stochastic Programming Problems with Probability and Quantile Functions (Hardcover) by Andrey I. Kibzun, Yuri S. Kan Computational Learning and Probabilistic Reasoning (Hardcover) by A. Gammerman (Editor) Modern Heuristic Search Methods (Hardcover) by V. J. Rayward-Smith (Editor), I. H. Osman (Editor), C. R. Reeves (Editor), G. D. Smith (Editor) Discrete Mathematics The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization E. L. Lawler, Jan Karel Lenstra, A. H. G. Rinnooy Kan, D. B. Shmoys

68. Traveling Salesman Problem - Permutation City genetic algorithm banner Travelling Salesman Problem. Salesman sJob. For time. Figure 12 Ulysses16 Travelling Salesman Problem. http://www.permutationcity.co.uk/projects/mutants/tsp.html

Travelling Salesman Problem Salesman's Job For the first demonstration of the toolkit's power the purpose was to explore the workings of the toolkit rather than develop a new genetic algorithm. Several problems are well known in the genetic algorithm field, one of these is the Travelling Salesman Problem which is also a common test for other search space techniques. Because of this large sets of data readily available for test purposes with optimal solution already found by exhaustive search. As a bonus one of standard implementations is both simple and will exercise different toolkit components than The Blues. The travelling salesman problem involves planning a route from city to city for a salesman to travel so that each city is visited once and only once apart from one city which should be both start and finish point. Additionally the total cost of travelling on the journey should be minimised. Another way to describing this problem is as finding the minimum cost tour among all permutations of the n cities in the salesman's itinerary. Although expressed in terms of a salesman travel plans this problem is equivalent to many other distance minimisation problems. "Circuit board drilling applications with up to 17,000 cities are mentioned... , X-ray crystallography instances with up to 14,000 cities are mentioned... , and instances arising in VLSI fabrication have been reported with as many as 1.2 million cities.... Moreover, 5 hours on a multi-million dollar computer for an optimal solution may not be cost-effective if one can get within a few percent in seconds on a PC. Thus there remains a need for heuristics"

69. Newsgroups Sci.math,sci.op-research From David@research.att.com att.com (David Applegate) Subject Re TSP CONTEST Winners Writeups Date Tue,28 Mar 1995 052205 GMT Keywords traveling salesman problem, TSP, asymmetric http://www.math.niu.edu/~rusin/known-math/95/tsp

99 cities: Length 78448, found and proved in 471 seconds: 110 cities: Length 87045, found and proved in 7202 seconds: 111 cities: Length 85879, found and proved in 1406 seconds: In article , Bruno W. Repetto wrote: >Chad: You wrote: > >> In article > >> -skito >> >> >> here i am >> here i am >> waiting to hold >> churritz@cts.com This Mortal Coil you > >Bruno W. Repetto >br0w+andrew.cmu.edu >

70. TRAVELING SALESMAN PROBLEM - Meaning And Definition Of The Word traveling salesman problem Dictionary Entry and Meaning. Computing Dictionary.Definition US spelling of travelling salesman problem. http://www.hyperdictionary.com/computing/traveling salesman problem

English Dictionary Computer Dictionary Thesaurus Dream Dictionary ... Medical Dictionary Search Dictionary: TRAVELING SALESMAN PROBLEM: Dictionary Entry and Meaning Computing Dictionary Definition: US spelling of travelling salesman problem See Also: spelling HOME ABOUT HYPERDICTIONARY

71. Traveling Salesman Problem traveling salesman problem. Approximation Algorithms. This is the infamoustraveling salesman problem (aka TSP) problem (formal defintion). http://valis.cs.uiuc.edu/~sariel/research/CG/applets/tsp/TspAlg.html

Home Bookmarks Search Computational Geometry ... Papers Traveling Salesman Problem Approximation Algorithms Applet By Kreimer Natasha. Problem: A traveling salesman has to travel through a bunch of cities, in such a way that the expenses on traveling are minimized. This is the infamous Traveling Salesman Problem (aka TSP ) problem ( formal defintion ). It belongs to a family of problems, called NP-complete problem. It is conjectured that all those problems requires exponential time to solve them. In our case, this means that to find the optimal solution you have to go through all possible routes, and the numbers of routes increases exponential with the numbers of cities. If you want to get a notion of what numbers we are talking about look at this: the number of routes with 50 cities is (50-2)!, which is An alternative approach would be to compute a solution which is not optimal, but is guarenteed to be close the optimal solution. We present here an applet that implements such an approximation algorithm for the Euclidean TSP problem. In our case we have points in the plane (i.e. cities) and the cost of the traveling between two points is the distance between them. In other words, we have a map with cities, any two of which are connected by a direct straight road (yeh, sure!) and we want to find a shortest tour for our poor traveling salesman, who "wants" to visit every city.

72. The Traveling Salesman Problem The traveling salesman problem. The traveling salesman problem is theproblem of finding the shortest cyclical itinerary for a traveling http://home.ku.edu.tr/~dyuret/pub/aitr1569/node23.html

Next: Summary and future Up: Results and related Previous: Refraction tomography The traveling salesman problem The traveling salesman problem is the problem of finding the shortest cyclical itinerary for a traveling salesman who must visit each of N cities in turn. This problem has a very different nature from the above problems, it is an example of combinatorial optimization . There is an objective function to be minimized, but the search space is the discrete space of all permutations of N cities. The combinatorial optimization problems are challenging because the number of elements in the search space is factorially large, thus they cannot be exhaustively searched. Also it is hard to define a concept of ``direction'' in a permutation space. The representation one chooses to use might make it possible to define concepts of ``distance'' and ``direction'' in non-cartesian spaces. One can carefully define various mutations such that ``moving further in the same direction'' is meaningful. It is also possible to define a notion of ``size'' in carefully selected operations. I will not present a detailed study of the application of my program on combinatorial problems. However, I thought it desirable to include one example, to show that the same key ideas apply even when one does not have a cartesian space. I compared the results I obtained from a very simple implementation of my algorithm with results from simulated annealing. The simulated annealing program [

73. Beats Biblionetz - Begriffe: Traveling Salesman Problem Traveling SalesmanProblem . Bemerkungen zum Begriff traveling salesman problem . http://beat.doebe.li/bibliothek/w01592.html

Beats Biblionetz: Begriffe Home Themen Personen Texte ... Changes lokal A B C D ... Meistgesuchte Traveling Salesman Problem Definitionen des Begriffs "Traveling Salesman Problem" Zu einer gegebenen Menge von Städten und Entfernungen zwischen allen Paaren von Städten ist eine Tour durch alle Städte zu finden, deren Länge kleiner als M ist. von Robert Sedgewick im Buch Algorithmen auf Seite 721 Ein weiteres NP-vollständiges Problem ist das "Problem des Handlungsreisenden"; es ähnelt dem Problem des Hamiltonschen Kreises, nur weist man den verschiedenen Kanten Zahlenwerte zu und sucht denjenigen Hamiltonschen Kreis, für den die Summe der Zahlen (die vom Handlungsreisenden zurückgelegte "Entfernung") ein Minimum ist. (Genaugenommen benötigen wir hier eine Ja/Nein-Version wie: Gibt es für das Problem des Handlungsreisenden einen Weg, dessen Länge kleiner ist als ein bestimmter Wert?) von Roger Penrose im Buch Computerdenken im Text Wahrheit, Beweis und Erkenntnis Bemerkungen zum Begriff "Traveling Salesman Problem" Aufgrund der NP-Vollständigkeit ist das Problem des Handlungsreisenden also in der Praxis nicht lösbar - zumindest bei unserem derzeitigen Wissensstand nicht.

74. Application: Traveling Salesman Problem next up previous contents Next Improving blind search Up Simulating discreteevents Previous Blind search. Application traveling salesman problem. http://math.uc.edu/~brycw/probab/books/smplbook/node16.html

Next: Improving blind search Up: Simulating discrete events Previous: Blind search Application: Traveling salesman problem The following program searches for the shortest way to pass through the first n cities in the USA in alphabetic ordering. Planning such a tour is easy by hand for 3-4 cities. For longer tours some help is needed. To check the performance of the blind search, you may want to know what the usual algorithms involve. A greedy method starts with the shortest distance, and then keeps going to the closest city not yet visited. Another heuristic method is to to select a pair of closest cities and accept the best (shortest) connection among those that do not complete a shorter loop, and do not introduce a spurious third connection to a city. Eventually, the disjoint pieces will form a path that often can be further improved upon inspection. The program is longer but not at all sophisticated - it just selects paths at random. Notice that a solution to Exercise how to generate a random permutation - is given in one of the subprograms (which one?). The method for the latter is simple-minded - the algorithm attempts to place consecutive numbers at random spots until an empty spot is selected.

75. The Geometric Maximum Traveling Salesman Problem (ResearchIndex) We consider the traveling salesman problem when the cities are points in Rd forsome fixed d and distances are computed according to geometric distances http://citeseer.ist.psu.edu/543374.html

76. Citations The Traveling Salesman Problem - Lawler, Lenstra EL Lawler, JK Lenstra, AHG RinnooyKan, and DB Shmoys (Eds.), The travelingsalesman problem. New York Wiley and Sons, 1985. 163 citations found. http://citeseer.ist.psu.edu/context/2688/0

77. Using Traveling Salesman Problem Algorithms For Evolutionary Tree Construction next Next Introduction Using traveling salesman problem Algorithmsfor Evolutionary Tree Construction. Chantal Korostensky and Gaston http://www2.inf.ethz.ch/personal/gonnet/papers/Construction/

Next: Introduction Using Traveling Salesman Problem Algorithms for Evolutionary Tree Construction Chantal Korostensky and Gaston H. Gonnet Institute for Scientific Computing ETH Zurich, Switzerland e-mail: Date: Abstract: We present a new tree construction method that constructs a tree with minimum score for a given set of sequences. To do this, the problem of tree construction is reduced to the Traveling Salesman Problem (TSP). The input for the TSP algorithm are the pairwise distances of the sequences and the output is a circular tour through the optimal, unknown tree plus the minimum score of the tree. The circular order and the score of the optimal tree can be used to construct the topology of the tree in time where n is the number of input sequences. We can guarantee that we reconstruct a correct evolutionary tree if the error for each distance measurement is smaller than , where x is the shortest edge in the tree. For data sets with large errors, a dynamic programming approach is used to reconstruct the tree. Keywords : tree construction, Traveling Salesman, circular order, evolution

78. Traveling Salesman Problem Application next up previous Next Scoring of MSAs without Up Methods Previous Circulartours traveling salesman problem Application. The probability http://www2.inf.ethz.ch/personal/gonnet/papers/MAScoring/node6.html

Next: Scoring of MSAs without Up: Methods Previous: Circular tours Traveling Salesman Problem Application The probability (exponential of the score) derived from pairwise alignments are now the key to identifying a circular tour. For a set of protein sequences it is computationally simple to obtain a set of pairwise scores by aligning each sequence with every other sequence using a DP algorithm to obtain the Optimal Pairwise Alignment. We shall refer to these as OPA scores [ Carillo and Lipman, 1988 ], to distinguish (see below) these from a pairwise alignment inferred from an MSA. Our goal is to be able to find a circular tour without the need of constructing an evolutionary tree. As we have shown above, a circular tour is the shortest possible tour for a tree (see Definition ). Note that a shorter distance corresponds to a higher score. A non-circular tour has at least some edge of the tree traversed more than twice, and no edge less than twice. In this paper we have so far always been using some tree, which we don't have in reality. The only information available to us is just the sequences and the OPA scores. So suppose now we do not have any information about the tree for that given set of sequences. But we know that the tree has some circular tour. We also know that a circular tour is the shortest possible tour through that tree, and that the best tree has the shortest total path length (sum of all edges) of all possible trees, since we want the tree with the maximum probability (see also section

79. Traveling Salesman Problem traveling salesman problem. The prototypical complex optimisationproblem is the traveling salesman problem (TSP). The task is to http://www.tcm.phy.cam.ac.uk/~tmf20/PHYSICS/thesis/node57.html

Next: Probabilistic Traveling Salesman Problem Up: HIERARCHICAL OPTIMISATION PROBLEMS Previous: HIERARCHICAL OPTIMISATION PROBLEMS Traveling Salesman Problem The prototypical complex optimisation problem is the traveling salesman problem ( TSP ). The task is to determine the minimal length tour through a set of cities such that each city is visited once and the path returns to its starting point (Figure a,b). The TSP has been the subject of an enormous amount of literature and is the standard testing ground for methods of combinatoric optimisation. We formulate the traveling salesman problem as follows. Consider a set A of N cities on a Euclidian plane, labelled . Pair-wise separations are given by the distance matrix D , where is the distance between city i and city j . Cities are visited according to a tour t where is a permutation of the cities . We represent the tour by the connection matrix T such that if cities and are neighbours along the tour and otherwise. The total distance traveled by tour t may be expressed The objective of the TSP is the ground state tour which minimises d Thomas Fink Tue Jun 16 17:14:36 BST 1998

80. Probabilistic Traveling Salesman Problem OPTIMISATION PROBLEMS Previous traveling salesman problem. Probabilistictraveling salesman problem. The introduction of probabilistic http://www.tcm.phy.cam.ac.uk/~tmf20/PHYSICS/thesis/node58.html

Next: Hierarchical Optimisation Problems Up: HIERARCHICAL OPTIMISATION PROBLEMS Previous: Traveling Salesman Problem Probabilistic Traveling Salesman Problem The introduction of probabilistic elements to combinatoric optimisations problems was made by Jaillet [ ] in 1985 and soon after extended by Bertsimas [ ]. Both considered, in particular, an extension of the TSP known as the probabilistic traveling salesman problem ( PTSP ), which has since become a paradigm problem of probabilistic combinatoric optimisation. Unlike the TSP , the PTSP applies to a set of cities for which, on any given instance, only a stochastically chosen subset need be visited. The objective of the PTSP is the construction of an a priori tour through all the cities such that, upon realising the active subset, the collapsed tour is of minimal length in the expected sense (by collapsed we mean that the reduced tour visit the subset of cities in the same order as the original). We label the set of N cities , with pair-wise separations . Each city is to be visited with probability

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