21. 1.5.4 Traveling Salesman Problem Excerpt from The Algorithm Design Manual The traveling salesman problem is the most The best book available for this problem is The Traveling Salesman http://www.cs.sunysb.edu/~algorith/files/traveling-salesman.shtml

1.5.4 Traveling Salesman Problem INPUT OUTPUT Input Description: A weighted graph G Problem: Find the cycle of minimum cost visiting all of the vertices of G exactly once. Excerpt from The Algorithm Design Manual : The traveling salesman problem is the most notorious NP-complete problem. This is a function of its general usefulness, and because it is easy to explain to the public at large. Imagine a traveling salesman who has to visit each of a given set of cities by car. Although the problem arises in transportation applications, its most important applications arise in optimizing the tool paths for manufacturing equipment. For example, consider a robot arm assigned to solder all the connections on a printed circuit board. The shortest tour that visits each solder point exactly once defines the most efficient path for the robot. A similar application arises in minimizing the amount of time taken by a graphics plotter to draw a given figure. The best book available for this problem is The Traveling Salesman Problem : A Guided Tour of Combinatorial Optimization by E.L. Lawler (Editor) and A. H. Rinnooy-Kan.

22. The Traveling Salesman Problem This page has moved to http//www.cs.concordia.ca/~chvatal/tsp/tsp.html Back to Vašek Chvátal s home page. http://www.cs.rutgers.edu/~chvatal/tsp.html

This page has moved to http://www.cs.concordia.ca/~chvatal/tsp/tsp.html

23. 711:611 Cutting-plane Method And Traveling Salesman Problem a computer code for solving traveling salesman problems (developed by D. V. Chvátal, and W. Cook, On the Solution of traveling salesman problems http://www.cs.rutgers.edu/~chvatal/611/intro.html

Spring 2004 Instructor: Email: c h v a t a l (at) c s (dot) r u t g e r s (dot) e d u Phone: (732) 445-3908 Office: Hill 486 Office hours: Mondays 3:30 5:30. Time and place: Tuesdays 2:50 5:50 in RUTCOR Building , Room 139. Outline: Concorde 13,509-city instance in May 1998 and a 15,112-city instance in April 2001. Topics: Spotting violated subtour inequalities (parametric connectivity, shrinking heuristic, subtour cuts from tour intervals, Padberg-Rinaldi exact-separation procedure). Hypergraph templates of facet-inducing inequalities. Spotting violated blossom inequalities. Tightening and teething. Local cuts. The core LP relaxation. Cut storage. Edge pricing. Reading: On the Solution of Traveling Salesman Problems TSP cuts which do not conform to the template paradigm Implementing the Dantzig-Fulkerson-Johnson Algorithm for Large Traveling Salesman Problems M. Bender and M. Farach-Colton. " The LCA Problem Revisited LATIN '00 Related web sites: TSP home page Ivars Peterson's The Traveling Monkey and The Shoelace Problem Julian Lethbridge's prints Prerequisites: Linear programming, computer programming.

24. Traveling Salesman Problem This problem is an asymmetric traveling salesman problem with additional constraints. Given a set of n nodes and distances for each pair of nodes, http://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/

University Informatik Home People ... Internal TSPLIB TSPLIB is a library of sample instances for the TSP (and related problems) from various sources and of various types. Frequently asked questions We have a small collection of answers to frequently asked questions (FAQ) Please read this and the description of the library before reporting problems with TSPLIB Symmetric traveling salesman problem (TSP) Given a set of n nodes and distances for each pair of nodes, find a roundtrip of minimal total length visiting each node exactly once. The distance from node i to node j is the same as from node j to node i. -> TSP data Best known solutions for symmetric TSPs Hamiltonian cycle problem (HCP) Given a graph, test if the graph contains a Hamiltonian cycle or not. -> HCP data Asymmetric traveling salesman problem (ATSP) Given a set of n nodes and distances for each pair of nodes, find a roundtrip of minimal total length visiting each node exactly once. In this case, the distance from node i to node j and the distance from node j to node i may be different. -> ATSP data Best known solutions for asymmetric TSPs Sequential ordering problem (SOP) This problem is an asymmetric traveling salesman problem with additional constraints. Given a set of n nodes and distances for each pair of nodes, find a Hamiltonian path from node 1 to node n of minimal length which takes given precedence constraints into account. Each precedence constraint requires that some node i has to be visited before some other node j.

25. Traveling Salesman Problem - Wikipedia, The Free Encyclopedia traveling salesman problem Wikipedia, the free encyclopediaThe traveling salesman problem or travelling salesman problem (TSP), also known as the A related problem is the Bottleneck traveling salesman problem http://en.wikipedia.org/wiki/Traveling_salesman_problem

Traveling salesman problem From Wikipedia, the free encyclopedia. The traveling salesman problem or travelling 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. Contents Problem statement Computational complexity NP-hardness Algorithms ... edit Problem statement Given a number of cities and the costs of traveling from any city to any other city, what is the cheapest round-trip route that visits each city once and then returns to the starting city? An equivalent formulation in terms of graph theory is: Given a complete weighted graph (where the vertices would represent the cities, the edges would represent the roads, and the weights would be the cost or distance of that road), find the Hamiltonian cycle with the least weight. 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 classic example is in

26. Oefen traveling salesman problem (TSP) using Simulated Annealing This applet attempts to solve the traveling salesman problem by simulated annealing. http://www.math.uu.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.

27. Traveling Salesman Problem (TSP) JAVA implementation for the symmetric traveling salesman problem (TSP). Exact traveling salesman problem Fast exhaustive version, runs up to 150 cities http://home.planet.nl/~onno.waalewijn/tspfast.html

JAVA Traveling Salesman Problem (TSP) JAVA implementation for the symmetric Traveling Salesman Problem (TSP). In this problem a salesperson has to find the shortest possible route to visit all the cities. more information See also: Exact Traveling Salesman Problem Fast exhaustive version, runs up to 150 cities links to related sites combinatorics home

28. Exact Traveling Salesman Problem - TSP Exact implementation for the traveling salesman problem See also traveling salesman problem heuristic version, up to 20000 cities with many TSPLIB http://home.planet.nl/~onno.waalewijn/tspx.html

Exact Traveling Salesman Problem (TSP) Exact implementation for the Traveling Salesman Problem Also solves Asymmetric TSP's - see demo's The following actions are performed to compute the shortest path: - make an initial path by greedy or nearest neighbour assignments - improve this by Opt4, Opt2 and Opt3 moves - improve this again by Simulated Annealing - then start an exhaustive search, with the upperbound found in the previous step - here we backtrack if the lowerbound found by computing the Minimum Spanning Tree for the unassigned cities exceeds the upperbound - improve the lowerbound by Lagrangean Relaxation - for 2d problems I also try to exclude crossings in the path using opt2 (still thinking about opt3/4) - I also try to find points close to but not in the path yet, that can be proven to belong in the path usage: While computing the optimal solution intermediate results are shown at 5 second intervals: The black line shows current path, red line shows MST, green is best found yet Interrupt by pressing the Break button The knights 5*6 and 8*8 are not true TSP problems, but the Knights problem for a 5*6 and a 8*8 chessboard. The Knight has to make a closed tour, touching each square exact once

29. Traveling Salesman Problem Karla Hoffman George Mason University Introduction The traveling salesman problem (TSP) is one which has commanded In one example of an 16 city traveling salesman problem the problem of http://iris.gmu.edu/~khoffman/papers/trav_salesman.html

Traveling Salesman Problem Karla Hoffman George Mason University Manfred Padberg New York University Introduction: The traveling salesman problem (TSP) is one which has commanded much attention of mathematicians and computer scientists specifically because it is so easy to describe and so difficult to solve. The problem can simply be stated as: if a traveling salesman wishes to visit exactly once each of a list of m cities (where the cost of traveling from city i to city j is c ij ) and then return to the home city, what is the least costly route the traveling salesman can take? A complete historical development of this and related problems can be found in Hoffman and Wolfe (1985). The importance of the TSP is that it is representative of a larger class of problems known as combinatorial optimization problems . The TSP problem belongs in the class of combinatorial optimization problems known as NP-complete. Specifically, if one can find an efficient algorithm (i.e., an algorithm that will be guaranteed to find the optimal solution in a polynomial number of steps) for the traveling salesman problem, then efficient algorithms could be found for all other problems in the NP-complete class. To date, however, no one has found a polynomial-time algorithm for the TSP. Does that mean that it is impossible to solve any large instances of such problems? Many practical optimization problems of truly large scale are solved to optimality routinely. In 1994, Applegate, et. al. solved a traveling salesman problem which models the production of printed circuit boards having 7,397 holes (cities), and, in 1998, the same authors solved a problem over the 13,509 largest cities in the U.S. So, although the question of what it is that makes a problem "difficult" may remain open, the computational record of specific instances of TSP problems coming from practical applications is optimistic.

30. The Code Project - Genetic Algorithms And The Traveling Salesman Problem - C++ / Genetic Algorithm and traveling salesman problem The traveling salesman problem, or TSP for short, is this given a finite number of cities along http://www.codeproject.com/cpp/tspapp.asp

View our advertisers Advertise with us document.write(""); All Topics MFC / C++ C++ / MFC Algorithms Genetic Algorithms and the Traveling Salesman Problem By Konstantin Boukreev An example of using Genetic Algorithms for solving the Traveling Salesman Problem Advanced C++ (VC6) Windows (Win95, Win98, NT4, Win2K) Win32, STL, VS Dev Posted 27 Sep 2001 Articles by this author views Search: Articles Authors Help! Articles Message Boards StoreFront ... Send to a friend Sign in / Sign up Email Password Remember me Lost your Password? document.write(""); 48 members have rated this article. Result: Popularity: 7.19 . Rating: out of 5. Download source files - 119 Kb Download demo executable - 159 Kb contents Genetic Algorithm Theory GA and TSP 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

31. Traveling Salesman Problem 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. http://www.delphiforfun.org/Programs/traveling_salesman.htm

Home Delphi Techniques Utilities DFF Library ... Math Topics Available Now All Programs Delphi Techniques Math Topics Utilities ... Library Units 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. TSP belongs to a class of problems which for some non-obvious reason are called NP complete . These problems have no known efficient algorithms for solving them. "Not efficient" here means that the time to solve the problem increases exponentially with problem size, i.e. time to solve is an expression that contains N as an exponent. This is a much faster growth rate than any polynomial time problem. (Compare values of

32. TOI Search Engine Redirector traveling salesman problem Enpsychlopediatraveling salesman problem. (Redirected from Travelling salesman). The traveling salesman problem or travelling salesman problem (TSP), also known as the http://jochen.pleines.bei.t-online.de/

33. DIMACS TSP Challenge 8th DIMACS Implementation Challenge. The traveling salesman problem has now been published in The traveling salesman problem and Its Variations, http://www.research.att.com/~dsj/chtsp/

8th DIMACS Implementation Challenge: The Traveling Salesman Problem Challenge News: Still Open for Business! (Including New Do-It-Yourself Feature) The original deadline for submitting results to TSP Challenge has passed, and the Johnson-McGeoch chapter that summarizes the Challenge results (as of 1 July 2001), ``Experimental analysis of heuristics for the STSP,'' has now been published in The Traveling Salesman Problem and Its Variations , G. Gutin and A. P. Punnen (Editors), Kluwer Academic Publishers, 2002, Boston, 369-443. A near-final draft, differing from the published version only in pagination and the correction of a few typographical errors, can be downloaded: ( 80 pages postscript PDF ). The deadline has also passed for the DIMACS technical report that will cover all the Challenge submissions and describe this website in detail. However, this report has not yet been completed, so there may still be time for late results to be included (new deadline: 1 September 2002). Even after the report is written we will continue to welcome submissions, and will periodically add them to the Results Page below. We hope to maintain this site indefinitely so that future TSP researchers will have a ready set of benchmarks to which they can compare their results. Moreover, software is now available ( tarfile zipfile ) so that users of UNIX/LINUX systems can generate figures and comparison charts for their data in our standard formats (gif and postscript) before they submit it to the site. Also included is code that will generate normalized results like those on our Results page, and this code should work on most platforms.

34. TSP Problem. EXAMPLE Heuristic algorithm for the traveling salesman problem (TSP) . This is one of the most known problems ,and is often called as a difficult problem. http://students.ceid.upatras.gr/~papagel/project/tspprobl.htm

The T.S.P. Example. EXAMPLE: Heuristic algorithm for the Traveling Salesman Problem (T.S.P) This is one of the most known problems ,and is often called as a difficult problem.A salesman must visit n cities, passing through each city only once,beginning from one of them which is considered as his base,and returning to it.The cost of the transportation among the cities (whichever combination possible) is given.The program of the journey is requested,that is the order of visiting the cities in such a way that the cost is the minimum. Let's number the cities from 1 to n ,and let city 1 be the city-base of the salesman.Also let's assume that c (i,j) is the visiting cost from i to j .There can be c (i,j) (j,i) Apparently all the possible solutions are (n-1)! .Someone could probably determine them systematically,find the cost for each and everyone of these solutions and finally keep the one with the minimum cost.These requires at least (n-1)! steps. If for example there were 21 cities the steps required are (n-1)!=(21-1)!=20! steps.If every step required a

35. Neil Simonetti's Traveling Salesman Problem Page Neil Simonetti s traveling salesman problem Page My Thesis Applications of a Dynamic Programming Approach to the traveling salesman problem http://www.andrew.cmu.edu/~neils/tsp/

Neil Simonetti's Traveling Salesman Problem Page Results for the Job Shop Scheduling Problems Results here were found using a Sun Ultra 60 (UltraSPARC-II processors, 360MHz) Test data for standard problems are found here Test data for semiconductor problems are found here Results for standard problems: Results for semiconductor problems: Summary of results Test data for t2-ps and t2-pss instances are here Results for t2-ps and t2-pss instances Code for the Multi-Run SB-RGLS10 procedure Dynamic Programming Code for the TSP: Linear Time Dynamic Programming Algorithms for New Classes of Restricted TSPs: A Computational Study gzipped postscript Adobe pdf INFORMS Journal on Computing My Thesis: Applications of a Dynamic Programming Approach to the Traveling Salesman Problem gzipped postscript Adobe pdf Conference Procedings, IPCO V: Implementation of a Linear Time Algorithm for Certain Generalized TSP's gzipped postscript Adobe pdf The Code: ag.h agtw.h dynopt.h solver.h (Instructions are contained in an appendix to my thesis) Sample Shells: regtsp.c

36. The Traveling Salesman Problem The traveling salesman problem, or TSP for short, is this given a finite number of ``cities along with the cost of travel between each pair of them, http://www.cs.concordia.ca/~chvatal/tsp/tsp.html

The traveling salesman problem The traveling salesman problem, or TSP for short, is this: given a finite number of ``cities'' along with the cost of travel between each pair of them, find the cheapest way of visiting all the cities and returning to your starting point. (Here, we consider just the symmetric TSP, where traveling from city X to city Y costs the same as traveling from Y to X; the ``way of visiting all the cities'' is simply the order in which the cities are visited.) To put it differently, the data consist of integer weights assigned to the edges of a finite complete graph; the objective is to find a hamiltonian cycle (that is, a cycle passing through all the vertices) of the minimum total weight. In this context, hamiltonian cycles are commonly called tours TSPLIB is Gerhard Reinelt's library of 110 instances of the traveling salesman problem. Some of these instances arise from the task of drilling holes in printed circuit boards and others have been constructed artificially. (A popular way of constructing a TSP instance is to choose a set of actual cities and to define the cost of travel from X to Y as the distance between X and Y.) None of them (with a single exception) is contrived to be hard and none of them is contrived to be easy; some of them have been solved (a few of these are shown here ) and others have not.

37. Branch And Cut For The Traveling Salesman Problem Branch Cut and Price Applications traveling salesman problem traveling salesman problem Links. Data Sets from TSPLIB Reference Materials http://branchandcut.org/TSP/

Branch Cut and Price Applications : Traveling Salesman Problem Home Software Applications FAQ ... Links We now offer a basic TSP solver, implemented with SYMPHONY , that uses separation subroutines from the CONCORDE TSP Solver of Applegate, Bixby, Chvatal, and Cook. Traveling Salesman Problem Links Data Sets from TSPLIB Reference Materials Source Code for the Solver This page maintained by Ted Ralphs ( ted@branchandcut.org Last updated October 9, 2003

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

39. Chapter 6 Paths And Networks Paths and Networks. Applet 9. traveling salesman problem Calculator methods for producing approximate solutions to the traveling salesman problem. http://www.wiley.com/college/mat/gilbert139343/java/java09_s.html

Choose a Chapter Chapter 1: Voting Methods Chapter 2: Apportionment Chapter 3: The Mathematics of Money Chapter 4: Probability Chapter 5: Statistics Chapter 6: Paths and Networks Chapter 7: Tilings and Polyhedra Chapter 8: Number Theory Chapter 9: Game Theory Chapter 6 Paths and Networks Applet 9. Traveling Salesman Problem Calculator The applet illustrates implements heuristic methods for producing approximate solutions to the Traveling Salesman Problem. By experimenting with various methods and variants of methods one can successively improve the route obtained. Start... Click on the panel to place cites on the map (use the "Background" menu to select the background map). Drag a city if you want to move a city once you've placed it. Click once on a city and press the delete key to remove it. To change the scale of the map (indicated in the top right corner of the window), select an edge by clicking once on each endpoint and then pressing the Set Scale button (the third from the left on the top row). Enter the length you want that edge to be, and all the other distances will rescale proportionately. Use the Clear button (the last button in the top row) to clear a circuit or erase an entire graph. When edges or a circuit are highlighted, the clear button erases them, but leaves the underlying graph in place. When no edges are selected, the Clear button erases the whole graph.

40. Traveling Salesman Problem: Information From Answers.com traveling salesman problem The traveling salesman problem or travelling salesman problem ( TSP ), also known as the traveling salesperson problem , http://www.answers.com/topic/traveling-salesman-problem

showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Wikipedia Best of Web Mentioned In Or search: - The Web - Images - News - Blogs - Shopping traveling salesman problem Wikipedia traveling salesman problem The traveling salesman problem or travelling 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. Problem statement Given a number of cities and the costs of traveling from any city to any other city, what is the cheapest round-trip route that visits each city once and then returns to the starting city? An equivalent formulation in terms of graph theory is: Given a complete weighted graph (where the vertices would represent the cities, the edges would represent the roads, and the weights would be the cost or distance of that road), find the Hamiltonian cycle with the least weight. It can be shown that the requirement of returning to the starting city does not change the computational complexity of the problem.

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