21. Traveling Salesman Problem The traveling salesman problem (TSP) requires that we find the shortest path visitingeach of a given set of cities and returning to the starting point. http://www.delphiforfun.org/Programs/traveling_salesman.htm

Home Introduction About Delphi Feedback ... Site Search (bottom of page) Available Now (Click a level below to expand program list if it is collapsed - MS Internet Explorer only) Beginners 10 Easy Pieces Bouncing Ball A Card Trick? Cards #1 ... Multiple of 12? Intermediate 15 Puzzle #1 The 9321 Problem Aliquot Sums (Friendly If not Perfect) Alphametics ... Word Ladder Advanced 15 Puzzle #2 Akerue Brute Force Countdown Plus! ... WordStuff 2 Contact Feedback Send an e-mail with your comments about this program (or anything else). Search WWW Search delphiforfun.org Problem Description The Traveling Salesman Problem (TSP) requires that we find the shortest path visiting each of a given set of cities and returning to the starting point. Here's a program that lets you match your skill against the computer to define a path connecting a random set of U.S. cities. First, I want to thank fellow seeker Robert Harrold for suggesting this project. He runs a wide ranging website including this education page where he was kind enough to place a link to DFF. He says that a computer display similar to this program existed on the second floor of the National Aerospace Museum in Washington, DC during the 80's. It disappeared one day in 1988, and he's been looking for it ever since. Maybe this will help, Bob. Thanks for asking.

22. Traveling Salesman Problem traveling salesman problem. Session MA29 Date/Time Monday 08000930Type Contribute Sponsor Track Cluster Room Room 326 http://www.informs.org/Conf/WA96/TALKS/MA29.html

Go to INFORMS Page ... INFORMS Home What's New Info for Members Info for Nonmembers Conferences Continuing Education Education/Students Employment Prizes Publications Subdivisions Searchable Databases Links About this Web Site INFORMS Online Bookstore Discussion Search Traveling Salesman Problem Session: Date/Time: Monday 08:00-09:30 Type: Contribute Sponsor: Track: Cluster: Room: Room 326 Chair: John Mittenthal Chair Address: A Comparison of Global Search Heuristics for the TSSP+1 John Mittenthal Jose M. Pires, Luis Gouveia Prize-Collecting Traveling Salesman Problem Santosh N. Kabadi, A. Punnen Polyhedral Analysis for the Prize Collecting TSP Ann E. Bixby, David Simchi-Levi, Collette Coullard INFORMS Online Questions on membership, subscriptions and the like should go to INFORMS Customer Service Questions/comments of a general nature about this Web site should go to Editor, IOL

23. The Traveling Salesman Problem & A Reformulation Of The Miller-Tucker-Zemlin Con The traveling salesman problem a Reformulation of the MillerTucker-ZemlinConstraints. MA29.2 The traveling salesman problem http://www.informs.org/Conf/WA96/TALKS/MA29.2.html

Go to INFORMS Page ... INFORMS Home What's New Info for Members Info for Nonmembers Conferences Continuing Education Education/Students Employment Prizes Publications Subdivisions Searchable Databases Links About this Web Site INFORMS Online Bookstore Discussion Search Jose M. Pires, Luis Gouveia - ISCAL, Av. Miguel Bombarda 20, 1000 Lisboa, , Portugal INFORMS Online Questions on membership, subscriptions and the like should go to INFORMS Customer Service Questions/comments of a general nature about this Web site should go to Editor, IOL

24. Traveling Salesman Problem From FOLDOC traveling salesman problem. spelling US spelling of travelling salesmanproblem. (199612-13). Try this search on OneLook / Google. http://wombat.doc.ic.ac.uk/foldoc/foldoc.cgi?traveling salesman problem

25. The Traveling Salesman Problem The traveling salesman problem. This section introduces the travelingsalesman problem (TSP). It begins by developing two formulations http://rodin.wustl.edu/~kevin/dissert/node11.html

Next: Graphical Traveling Salesman Up: Mathematical Background Previous: Order Theory The Traveling Salesman Problem This section introduces the traveling salesman problem (TSP). It begins by developing two formulations of the symmetric case: the original formulation by Dantzig, Fulkerson and Johnson (DFJ) ( ) and the Miller, Tucker, and Zemlin (MTZ) ( ) formulation. The formulations are different in two ways. The polyhedral structure of the DFJ model is better understood resulting in better solution algorithms. the efficiency of facet generation algorithms. On the other hand , the MTZ model has additional variables allowing it express a greater variety of objective functions and side constraints. Basic Problem Statement Given a graph , the edge set is a traveling salesman tour if it is a simple cycle of length in G . In the context of the TSP, tours are Hamiltonian cycles. However in some variations on the problem, the definition of tour is altered. In this section, a tour is a traveling salesman tour. A cycle of length in G is called a subtour. A tour

26. Graphical Traveling Salesman Problem A Relaxation Of TSP() Graphical traveling salesman problem A Relaxation of TSP(). The TSP is definedas finding a minimum cost Hamiltonian cycle on the complete graph . http://rodin.wustl.edu/~kevin/dissert/node12.html

Next: Models of the Up: Mathematical Background Previous: The Traveling Salesman The TSP is defined as finding a minimum cost Hamiltonian cycle on the complete graph . However, in its usual physical interpretation, where the nodes of a graph are cities and the edges represent roads interconnecting them, the graph is most likely not complete. To remedy this situation, G is usually completed with the cost of the added edges equal to the cost of the shortest path in the original graph. There are two problems with embedding G in . First, because each edge of G represents a different binary variable in linear programming formulations, solving the problem on requires variables even if G has few edges. Second, the original problem on G might easily be solved directly by exploiting the underlying structure of the graph. Specialized algorithms have been developed to solve the TSP in linear time for serialparallel graphs ( ) and Halin graphs ( Gilmore et al. 1985 ). In fact, have completely categorized all graphs for which TSP( G ) is the feasible set of the DFJ formulation without the integrality requirement.

27. Interactive Genetic Algorithms For The Traveling Salesman Problem next Next INTRODUCTION Interactive Genetic Algorithms for the TravelingSalesman Problem. Sushil J. Louis Genetic Adaptive Systems Lab Dept. http://www.cs.unr.edu/~sushil/papers/conference/newpapers/99/gecco99/iga/GECCO/g

Next: INTRODUCTION Interactive Genetic Algorithms for the Traveling Salesman Problem Sushil J. Louis Genetic Adaptive Systems Lab Dept. of Computer Science/171 University of Nevada, Reno Reno, NV 89557 sushil@cs.unr.edu Rilun Tang Genetic Adaptive Systems Lab Dept. of Computer Science/171 University of Nevada, Reno Reno, NV 89557 tang@cs.unr.edu Abstract: We use an interactive genetic algorithm to divide and conquer large traveling salesperson problems. Current genetic algorithm approaches are computationally intensive and may not produce acceptable tours within the time available. Instead of applying a genetic algorithm to the entire problem, we let the user interactively decompose a problem into subproblems, let the genetic algorithm separately solve these subproblems and then interactively connect subproblem solutions to get a global tour for the original problem. Our approach significantly reduces the computing time to find high quality solutions for large traveling salesperson problems. We believe that an interactive approach can be extended to other visually decomposable problems. INTRODUCTION METHODOLOGY Genetic Algorithm for TSP Analysis ... About this document ...

28. The Traveling Salesman Problem next up previous Next Methodology Up Augmenting Genetic Algorithmswith Previous Introduction. The traveling salesman problem. http://www.cs.unr.edu/~sushil/papers/conference/papers/fea/97/fea/node2.html

Next: Methodology Up: Augmenting Genetic Algorithms with Previous: Introduction The Traveling Salesman Problem The traveling salesman problem(TSP) is: given N cities, if a salesman starting from his home city is to visit each city exactly once and then return home, find the order of a tour such that the total distance traveled is minimum. The TSP is a classical NP-complete problem which has extremely large search spaces and is very difficult to solve. People have tried to use both exact and heuristic or probabilistic methods to solve the TSP. The objective function for the N cities two dimensional Euclidean TSP is the sum of Euclidean distances between every pair of cities in the tour. That is: Where, are the coordinates of city i and equals . We also make some changes to the encoding, selection, and recombination. Recombination Our sequential path representation (a ordered list of cities to visit) means that traditional crossover and mutation operators are not suitable for TSPs. Instead we use Greedy Crossover [ ]. Greedy crossover selects the first city of one parent, compares the cities leaving that city in both parents, and chooses the closer one to extend the tour. If one city has already appeared in the tour, we choose the other city. If both cities have already appeared, we randomly select a non-selected city. Mutation is implemented by swapping two randomly chosen sites.

29. Traveling Salesman Problem - Wikipedia traveling salesman problem. The traveling salesman problem (TSP) is a prominentillustration of a class of problems in computational complexity theory. http://www.wikipedia.org/wiki/Traveling_salesman_problem

Main Page Recent changes Edit this page Older versions Special pages Set my user preferences My watchlist Recently updated pages Upload image files Image list Registered users Site statistics Random article Orphaned articles Orphaned images Popular articles Most wanted articles Short articles Long articles Newly created articles All pages by title Blocked IP addresses Maintenance page External book sources Printable version Talk Log in Help Traveling salesman problem From Wikipedia, the free encyclopedia. The traveling salesman problem (TSP) is a prominent illustration of a class of problems in computational complexity theory . The problem can be stated as: 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 exactly once and then returns to the starting city? The most direct answer would be to try all the combinations and see which one is cheapest, but given that the number of combinations is N! (the factorial of the number of cities), this solution rapidly becomes impractical. How fast are the best known deterministic algorithms?

30. KLUWER Academic Publishers | The Traveling Salesman Problem And Its Variations Books » The traveling salesman problem and Its Variations. The Traveling SalesmanProblem and Its Variations. Add to cart. edited by Gregory Gutin Dept. http://www.wkap.nl/prod/b/1-4020-0664-0

Title Authors Affiliation ISBN ISSN advanced search search tips Books The Traveling Salesman Problem and Its Variations The Traveling Salesman Problem and Its Variations Add to cart edited by Gregory Gutin Dept. of Computer Science, University of London, UK Abraham P. Punnen Dept. of Mathematics, Statistics and Computer Science, University of New Brunswick, St. John, Canada Book Series: COMBINATORIAL OPTIMIZATION Volume 12 This volume, which contains chapters written by reputable researchers, provides the state of the art in theory and algorithms for the traveling salesman problem (TSP). The book covers all important areas of study on TSP, including polyhedral theory for symmetric and asymmetric TSP, branch and bound, and branch and cut algorithms, probabilistic aspects of TSP, thorough computational analysis of heuristic and metaheuristic algorithms, theoretical analysis of approximation algorithms, including the emerging area of domination analysis of algorithms, discussion of TSP software and variations of TSP such as bottleneck TSP, generalized TSP, prize collecting TSP, maximizing TSP, orienteering problem, etc. Audience: Researchers, practitioners, and academicians in mathematics, computer science, and operations research. Appropriate as a reference work or as a main or supplemental textbook in graduate and senior undergraduate courses and projects.

31. KLUWER Academic Publishers | The Traveling Salesman Problem And Its Variations Books » The traveling salesman problem and Its Variations. The TravelingSalesman Problem and Its Variations edited by Gregory Gutin Dept. http://www.wkap.nl/prod/b/1-4020-0664-0?a=1

32. Citations: The Traveling Salesman Problem And Minimum Spanning Trees: Part II - M. Held and RM Karp, The traveling salesman problem and minimum spanning treesPart II, Mathematical Programming 6 (1971), 6288. 132 citations found. http://citeseer.nj.nec.com/context/88915/0

133 citations found. Retrieving documents... M. Held and R.M.Karp (1971), " The Traveling Salesman Problem and Minimum Spanning Trees: Part II ", Mathematical Programming 1, 625. Home/Search Document Not in Database Summary Related Articles Check This paper is cited in the following contexts: First 50 documents Next 50 Exact Solution of the - Quadratic Knapsack Problem (Correct) (2) 6) Anyway, exact solution of this LP relaxation would be computationally very expensive due to the large number of variables and constraints involved (see also [6] Our approach determines a near optimal multiplier matrix by a standard subgradient optimization procedure; see Held and Karp and Held, Wolfe and Crowder [16] The procedure generates a series ; of matrices, where : and, for k 0, is defined from as follows. Let (x; y) denote an optimal solution of the Lagrangian relaxation associated with . The corresponding subgradient vector is given .... M. Held and R.M.Karp (1971), " The Traveling Salesman Problem and Minimum Spanning Trees: Part II ", Mathematical Programming 1, 625.

33. Citations: The Traveling Salesman Problem - Lawler, Lenstra, Rinnooy-Kan, Shmoys Similar pages More results from citeseer.nj.nec.com traveling salesman problem traveling salesman problem. Given a set of cities, find the shortest routethat visits each city exactly once and returns to the home city. http://citeseer.nj.nec.com/context/2688/0

163 citations found. Retrieving documents... Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., and Shmoys, D.B., The Traveling Salesman Problem , Wiley, Chichester, 1985. Home/Search Document Not in Database Summary Related Articles Check This paper is cited in the following contexts: First 50 documents Next 50 Using The Meta-Raps Approach To Solve Combinatorial Problems - Gail Depuy Ph (Correct) ....9.04 N A N A N A Simulated Annealing by Malek et al. 21] 0.09 N A N A N A N A Tabu Search by Malek et al. 21] 0.33 N A N A N A N A Tabu Search 3 opt by Tsubakitani and Evans [30] 1. 37 N A N A N A N A Shaded cells indicate Meta RaPS CI was outperformed as reported by Lawler et al. as implemented by Modares et al. 23] Table 2: Comparison of the Meta RaPS to Other Heuristic Results for all 110 Patterson Problems Average Std. Dev. Maximum Percentage Solution Method Difference Difference Difference Optimal from Optimal from Optimal from Optimal Solutions .... Lawler, E.L.; Lenstra, J.K.; Rinnooy Kan, A.H.G.; Shmoys, D.B., 1985, The Traveling Salesman Problem Experimental Study of Minimum Cut Algorithms - Levine (1997) (1 citation) (Correct) ....and Johnson [15] The problem of identifying a subtour elimination constraint can be rephrased as the problem of finding a minimum cut in a graph with real valued edge weights. Thus, cutting plane algorithms for the traveling salesman problem must solve a large number of minimum cut problems (see

34. Generation5.org - AISolutions: Genetic Algorithm And Traveling Salesman Problem Genetic Algorithm and traveling salesman problem. Konstantin Boukreev GeneticAlgorithm and traveling salesman problem. About traveling salesman problem. http://www.generation5.org/aisolutions/tspapp.shtml

Home Essays Interviews Programs ... Search Search: Genetic Algorithm and Traveling Salesman Problem Konstantin Boukreev Title: Genetic Algorithm and Traveling Salesman Problem Author: Konstantin Boukreev Email: konstantin@mail.primorye.ru Environment: VC++ 6.0. SP5, Win2k SP2, MS Platform SDK April 2001 Description: The example of using Genetic Algorithm for solving Traveling Salesman Problem Download TSPApp executable - 155 Kb Download TSPApp source - 119 Kb contents Genetic Algorithm Theory GA and TSP Base implementation, Template class GA and GA Selection classes Genome of Travel TSP Application GA thread UI interface Environment Reference I am not a GA guru and I do not have any degree in GA so this article can't be used as GA book or GA tutorial. There aren't any mathematics nor logic nor algebra about GA. It's only a programmer's view on Genetic Algorithms and only example of GA coding. Use it carefully! Any comments and criticism are highly appreciated. Genetic Algorithm, Theory There are so many books and so many resources on the WEB about Genetic Algorithms. The best that I can do is quote some nice descriptions from my preferred sites. Definition from Marek Obitko's Site "Genetic algorithms are a part of evolutionary computing, which is a rapidly growing area of artificial intelligence. As you can guess, genetic algorithms are inspired by Darwin's theory about evolution. Simply said, solution to a problem solved by genetic algorithms is evolved."

35. Sensitivity Analysis For The Euclidean Traveling Salesman Problem Origin Sensitivity Analysis for the Euclidean traveling salesman problem. Introduction. SensitivityAnalysis for The Euclidean traveling salesman problem. http://physics.hallym.ac.kr/education/TIPTOP/VLAB/Sales/tsp.html

Origin Sensitivity Analysis for the Euclidean Traveling Salesman Problem Introduction The Euclidean traveling salesman problem can be stated as follows. Given n cities located in the plane, find the shortest route that visits all the cities exactly once and returns to the starting city. The problem forms one of the canonical problems in the field of combinatorial optimization. Many new algorithmic techniques first saw application to the traveling salesman problem. Hundreds of heuristics have been developed that can produce good, but not necessarily optimal routes. Sensitivity Analysis Sensitivity Analysis asks the question: what happens when the input values change? For many types of optimization, for example, linear programming, sensitivity analysis is well understood. For combinatorial optimization problems, including the traveling salesman problem, sensitivity analysis has hardly been explored. Sensitivity Analysis for The Euclidean Traveling Salesman Problem For the Euclidean Traveling Salesman Problem, we consider the question of how the optimal solution changes when a city is moved. Of course, cities do not move, so this particular problem may be moot. Given the extensive research on the traveling salesman problem, results developed for it may be applicable to other combinatorial optimization problems. An Example The following image shows a six city travelling salesman problem solved using common heuristic called Farthest Insertion.

36. The Traveling Salesman Problem The traveling salesman problem. This classic optimization problem canbe very simply stated a salesman has to visit N cities, and http://www.npac.syr.edu/users/paulc/lectures/montecarlo/node144.html

Next: Implementing Annealing Algorithms Up: Examples of Optimization Previous: A Graph Problem The Traveling Salesman Problem This classic optimization problem can be very simply stated - a salesman has to visit N cities, and wants to take the shortest possible route. Here, the cost function is the length of the tour. The change a configuration (a particular tour) is not as simple as a single spin flip in the spin glass problem, or moving an element from one side to the other in the graph partitioning problem. Each tour can be presented as a permutation of the numbers 1 to N , which represent the cities. The simplest change to the tour is to swap pairs of cities, and measure the change in the tour path. Paul Coddington, Northeast Parallel Architectures Center at Syracuse University, paulc@npac.syr.edu

37. Oefen traveling salesman problem (TSP) using Simulated Annealing. This applet attemptsto solve the traveling salesman problem by simulated annealing. http://www.math.ruu.nl/people/beukers/anneal/anneal.html

Traveling salesman problem (TSP) using Simulated Annealing Author: Frits Beukers Before starting choose at least three cities. This can be done by clicking in the black panel. Or by entering a value for #cities and then press the zap or grid button. This applet attempts to solve the traveling salesman problem by simulated annealing. In the black window one can select a set of cities in the following manner. Click in it with the mouse. A small dot depicting a city will appear. Repeat until you think you have enough cities. Note that the city counter has increased while doing so. Another method is to set the city counter with a number and then press the button `zap' or `grid'. If desired one can add a few more cities with the mouse. The number of cities should be below 100, otherwise the program becomes very slooooow... When you are happy with the arrangement push `start'. A path will appear which is usually far from shortest. However, it will gradually improve. If so desired one can stop the process to change the temperature. Unfortunately the program's reaction to the stop button is often slow, especially with more than 50 cities. So be patient. A good starting temperature is T about 10 or 20. The value T=0 forces a greedy behaviour of the system, in which it is easy to get stuck in local minima. When T>50 you really cook the system to get only randomish paths. To leave the process altogether push `stop' and then `reset'. The city counter is then set to zero and you can start all over again.

38. A Traveling Salesman Problem A traveling salesman problem. The Museum. IM Lzee has just been hiredas the night guard in the Museum for Discrete Mathematics. Her http://bhs.broo.k12.wv.us/discrete/Traveling.htm

A Traveling Salesman Problem The Museum I. M. Lzee has just been hired as the night guard in the Museum for Discrete Mathematics. Her job is to check on all exhibits once per hour, by passing through all the doorways. Knowing that you are interested in discrete mathematics, she has asked if you can determine a route through all the doors so she will not have to retrace her steps. She would also like to end her trip where she started. If that is not possible, through what door(s) must she pass through more often (how often?) to accomplish this task? Where must she start? Can you draw a graph, where the doors are the vertices, and the room s are the edges, to justify your answer? Links Euler Paths and Circuits West Virginia goals and objectives for discrete math Counting Vertices Answer ... See my Style Sheet

39. 4. Levinthal's Paradox In The Traveling Salesman Problem 4. LEVINTHAL'S PARADOX IN THE traveling salesman problem. This is thetraveling salesman problem. The problem is stated as follows. http://konf2.ims.ac.jp/review/sec4.html

Predicting Protein Tertiary Structures from the First Principles, Yuko Okamoto 4. "LEVINTHAL'S PARADOX" IN THE TRAVELING SALESMAN PROBLEM The difficulty in the prediction of protein tertiary structures belongs to a common class of difficulty encountered in systems with frustrations (e.g., traveling salesman problem, spinglass, optimum electrical circuit wiring problem, etc.) The problem is mathematically classified as NP complete (nonpolynomical complete). Here, NP complete means that for a system with size n , the time, T n ), it takes to solve the problem grows faster than any power of n . (That is, T n ) grows not like n a but e.g., a n .) This kind of problem is impossible to solve when n becomes large. In fact, for a protein with N amino acids the rough estimate of the necessary computation time was T N N We now discuss a completely different problem that belongs to the same class, NP complete, as in the protein folding problem. This is the traveling salesman problem. The problem is stated as follows. "A salesman travels N cities by starting from a certain city and visiting each city once, and comes back to the starting city. Find the shortest path." Here, the number of possible paths is (

40. Traveling Salesman Problem Application next up previous Next Scoring of MSAs without Up Methods Previous Circulartours traveling salesman problem Application. The probability http://www.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

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