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Topic: Traveling salesman problem

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Bottleneck traveling salesman problem

Hamiltonian path problem

Vertex cover problem

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Shortest path problem

Approximation algorithm

Boolean satisfiability problem

Glossary of graph theory

Minimum spanning tree

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	Traveling Salesman Problem
		The Concorde TSP solver is used in a genome sequencing package from the National Institutes of Health.
		A collection of 25 TSP challenge problems consisting of cities in Argentina through Zimbabwe.
		A challenge problem consisting of the locations of 1,904,711 cities throughout the world.
	www.tsp.gatech.edu (284 words)

	TSPBIB Home Page
		A study of complexity transitions on the asymmetric traveling salesman problem, by W. Zhang and R.E. Korf.
		The traveling salesman problem, by V. Chvatal, who is one of the authors of a computer code that solved more than twenty previously unsolved instances.
		Polynomially Solvable Cases of the Traveling Salesman Problem and a New Exponential Neighbourhood, by R.E. Burkard and V.G. Deineko, Technische Universität Graz, SFB-Report 1, July 1994.
	www.densis.fee.unicamp.br /~moscato/TSPBIB_home.html (5518 words)

	[No title]
		The importance of the TSP is that it is representative of a larger class of problems known as combinatorial optimization problems.
		In the case of the traveling salesman problem, the mathematical structure is a graph where each city is denoted by a point (or node) and lines are drawn connecting every two nodes (called arcs or edges).
		As our understanding of the underlying mathematical structure of the TSP problem improves, and with the continuing advancement in computer technology, it is likely that many difficult and important combinatorial optimization problems will be solved using a combination of cutting plane generation procedures, heuristics, variable fixing through logical implications and reduced costs and tree search.
	iris.gmu.edu /~khoffman/papers/trav_salesman.html (2481 words)

	NationMaster - Encyclopedia: Traveling salesman problem
		The problem remains NP-hard even for the case when the cities are in the plane with Euclidean distances, as well as in a number of other restrictive cases.
		In the TSP with triangle inequality case it is possible to prove upper bounds in terms of the minimum spanning tree and design an algorithm that has a provable upper bound on the length of the route.
		Euclidean TSP, or planar TSP, is the TSP with the distance being the ordinary Euclidean distance.
	www.nationmaster.com /encyclopedia/Traveling-salesman-problem (3816 words)

	NationMaster - Encyclopedia: Hamiltonian cycle problem
		In the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem is the problem of determinining whether a Hamiltonian path or a Hamiltonian cycle exists in a given graph.
		The Hamiltonian cycle problem is a special case of the traveling salesman problem, obtained by setting the distance between two cities to unity if they are adjacent and infinity otherwise.
		In the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path or a Hamiltonian cycle exists in a given graph (whether directed or undirected).
	www.nationmaster.com /encyclopedia/Hamiltonian-cycle-problem (275 words)

	Bottleneck traveling salesman problem - Definition, explanation
		The decision problem version of this, "for a given length x, is there a Hamiltonian cycle in a graph g with no edge longer than x?", is NP-complete.
		Euclidean bottleneck TSP, or planar bottleneck TSP, is the bottleneck TSP with the distance being the ordinary Euclidean distance.
		An example would be a salesperson traveling by train with a special ticket that is valid for any number of trips between two cities up to a certain distance.
	www.calsky.com /lexikon/en/txt/b/bo/bottleneck_traveling_salesman_problem.php (383 words)

	Science Fair Projects - 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.
		Removing the condition of visiting each city "only once" does not remove the NP-hardness, since it is easily seen that in the planar case an optimal tour visits cities only once (otherwise, by the triangle inequality, a shortcut that skips a repeated visit will decrease the tour length).
		Modern methods can find solutions for extremely large problems (millions of cities) within a reasonable time which are provably 2-3% away from the optimal solution.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Traveling_salesperson_problem (1793 words)

	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.
		TSP belongs to a class of problems which for some non-obvious reason are called NP complete.
		For example: robotic travel problems like soldering or drilling operations on printed circuit boards, sequencing local genome maps to produce a global map, planning the order in which a satellite interferometer studies a sequence of stars, etc. A Google search will turn up lots more.
	www.delphiforfun.org /Programs/traveling_salesman.htm (1268 words)

	Traveling Salesman Problem
		Imagine a traveling salesman who has to visit each of a given set of cities by car.
		Implementations: The world-record-setting traveling salesman program is by Applegate, Bixby, Chvatal, and Cook [ABCC95], which has solved instances as large as 7,397 vertices to optimality.
		Algorithm 608 [Wes83] of the Collected Algorithms of the ACM is a Fortran implementation of a heuristic for the quadratic assignment problem, a more general problem that includes the traveling salesman as a special case.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK4/NODE175.HTM (1630 words)

	Traveling Salesman Problem
		TSPLIB is a library of sample instances for the TSP (and related problems) from various sources and of various types.
		This problem is an asymmetric traveling salesman problem with additional constraints.
		The problem is to find tours for the trucks of minimal total length that satisfy the node demands without violating truck capacity constraint.
	www.iwr.uni-heidelberg.de /groups/comopt/software/TSPLIB95 (437 words)

	Traveling Salesman Problem Using Genetic Algorithms
		In the Traveling Salesman Problem, the goal is to find the shortest distance between N different cities.
		The two complex issues with using a Genetic Algorithm to solve the Traveling Salesman Problem are the encoding of the tour and the crossover algorithm that is used to combine the two parent tours to make the child tours.
		Again, when debugging problems it is useful to be able to run the algorithm with the same exact parameters.
	www.lalena.com /ai/tsp (1355 words)

	The Travelling Salesman Problem
		The Travelling Salesman Problem (TSP) consists in the problem of determining the shortest circuit which can be made visiting a list of cities, in such a way that each city is visited (once and only once).
		This is an "NP-complete" optimization problem, which means that, for a problem with a reasonable dimension, there are so many hypothesis to consider that it is generally unpractical to look for an optimum solution, because the computation cost is too high.
		The representation scheme used to display the circuits of the TSP is based in a permutation matrix: there is a line for each city and each column represents the position of the city in the circuit.
	to-campos.planetaclix.pt /neural/hope.html (389 words)

	[No title]
		The Traveling Salesman Problem (TSProblem) is a very interesting minimization problem in which a salesman wishes to visit N cities.
		For the symmetric problem where distance (cost) from city A to city B is the same as from B to A, the number of possible paths to consider is given by (N-1)!/2.
		The problem is restricted to the Euclidian case where the TSP can be formulated as follows: Given n cities in the plane and their Euclidian distances, the problem is to find the shortest TSP-tour, i.e.
	www.lycos.com /info/traveling-salesman-problem.html (488 words)

	Traveling Salesman Problem - Permutation City (Site not responding. Last check: )
		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.
		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.
		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.
	www.permutationcity.co.uk /projects/mutants/tsp.html (737 words)

	Travelling salesman problem Summary (Site not responding. Last check: )
		Assume that the salesman can start from any city on his itinerary and that the distance from city X to city Y is the same as the distance from Y to X (i.e., there exist no one-way shortcuts).
		The TSP is equivalent to a class of mathematical problems called N-P ("nondeterministic polynomial") complete problems, which are often used to create security features on computers and networks.
		Most TSP loop families grow polynomially Private web page shows that a method exists for obtaining a set of optimal "traveling salesman" routes that is a member of a family that grows no faster than about 2
	www.bookrags.com /Travelling_salesman_problem (2651 words)

	COP2500 - Traveling Salesman Problem (Site not responding. Last check: )
		In this problem, the nodes of the graph represent locations and the edges represent the cost (in money or time or distance) to travel from one location to another.
		Your job is to design a route of travel so that he visits every location exactly once and returns to node 1 while making his total trip a minimum cost.
		Assume one of your classmates gave you their optimized solution to the traveling salesman problem but not the values of the edges for the graph they used.
	www.cs.ucf.edu /courses/cop2500/assignments/cop2500lab11.html (320 words)

	CSERD Resources: Models: Parallel Traveling Salesman Problem
		The traveling salesman problem, like many nonlinear optimization problems with a large number of parameters, is typically solved using an approach that utilizes randomness, rewards improved optimization, and allows for diversity in possible solutions to reduce the chance of becoming stuck in a local minimum.
		The parallelization of this problem is not entirely trivial, as having 10 processes attempt 10 trials in parallel is not equivalent to having 1 process calculate 100 trials.
		This traveling salesman model is designed to run on a UNIX machine from a command line, with X-Windows used for visualization.
	www.shodor.org /refdesk/Resources/Models/TravelingSalesman/theory.php (360 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: )
		The basic premise behind a TSP (as you probably already know) is to minimize the distance that a salesman has to travel to get to each of a set of cities, visiting each only once.
		For an explanation of why that is, see: The Travelling Salesman Problem - Robert Dakin http://www.pcug.org.au/~dakin/tsp.htm The number of paths becomes excessively large as n increases.
		The TSP is one of a special class of problems known as "NP-complete." Any NP-complete problem could theoretically be solved by converting it to another, equivalent NP-complete problem.
	mathforum.org /dr.math/problems/anderson.5.24.01.html (739 words)

	The Travelling Salesman Problem
		The travelling salesman problem consists in finding the shortest (or a nearly shortest) path connecting a number of locations (perhaps hundreds), such as cities visited by a travelling salesman on his sales route.
		The Traveling Salesman Problem is typical of a large class of "hard" optimization problems that have intrigued mathematicians and computer scientists for years.
		The Travelling Salesman, Version 3.10 This is the demonstration version (and at present the only version) of "The Travelling Salesman" program, written by Peter Meyer.
	www.hermetic.ch /misc/ts3/ts3demo.htm (767 words)

	The Traveling Salesman Problem
		Problem: Given a complete undirected graph G=(V, E) that has nonnegative integer cost c(u, v) associated with each edge (u, v) in E, the problem is to find a hamiltonian cycle (tour) of G with minimum cost.
		Note that the TSP problem is NP-complete even if we require that the cost function satisfies the triangle inequality.
		Observe that a TSP with one edge removed is a spanning tree (not necessarily MST).
	www.personal.kent.edu /~rmuhamma/Algorithms/MyAlgorithms/AproxAlgor/TSP/tsp.htm (639 words)

	Methods Applied to the Traveling Salesman Problem
		When solving optimization problems with computers, often the only possible approach is to calculate every possible solution and then choose the best of those as the answer.
		An example of a heuristic approach to the TSP might be to remove the most weighted edge from each node to reduce the size of the problem.
		The main idea of a heuristic approach to a problem is that, although there is exponential growth in the number of possible solutions to the problem, evaluating how good a solution is can be done in polynomial time.
	www.cs.nmsu.edu /~dcook/thesis/paper2.html (4774 words)

	The Traveling Salesman Problem (ProgrammingLand MOO) (Site not responding. Last check: )
		The basic problem is to find the minimal cost that the saleman has to pay in order to visit all ten cities.
		The problem with this problem is that there is no known algorithm that finds the best path all the time without testing all of the possible routes.
		Worse yet there are very many problems that are equivalent to the traveling salesman problem in complexity and many of them are quite common.
	euler.vcsu.edu:7000 /963 (304 words)

	The Math Forum - Math Library - Optimization
		The problem: given a finite number of cities and the cost of travel between each pair of them, find the cheapest way of visiting them all and returning to your starting point.
		One of the classic problems of planning ahead concerns a traveling salesman who must visit customers in a number of cities scattered across the country and then return home.
		Algorithms to solve the travelling salesman's problem (a travelling salesman is to visit a number of cities; how to plan the trip so every city is visited once and just once and the whole trip is as short as possible?) Links to Java applets; the shell...more>>
	www.mathforum.org /library/topics/optimization?keyid=16690899&start_at=101&num_to_see=50 (704 words)

	Premium Solver Platform for Excel - Alldifferent - Traveling Salesman Problem
		Problems involving ordering or permutations of choices are very difficult to model using conventional constraints, even with integer variables.
		An example is the famous Traveling Salesman Problem (TSP), where a salesman must choose the order of cities to visit so as to minimize travel time, and each city must be visited exactly once.
		This allows you to model the problem in a high-level way, and try a variety of Solver engines to see which one yields the best performance on your problem.
	www.solver.com /xlsplatform4.htm (249 words)

	Math Forum Discussions - Re: genetic algorithm Traveling Salesman Problem data representation
		problems that the problem is (considered to be) in
		problems are as hard to solve as all that is still only conjectural.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.com /kb/thread.jspa?forumID=13&threadID=1391395&messageID=4762146 (1178 words)

	Traveling Salesman Problem: Zur Geschichte der ZIP-Methode
		Wenn selbst Kenner des TSP wie Grötschel und Padberg 1999 in der "Optimierten Odyssee" schreiben, dass kombinatorisch Rundreisen mit 25 Knoten nicht optimal lösbar seien, dann scheint die ZIP-Methode wirklich neu zu sein.
		Jeder, der sich mit dem TSP beschäftigt, ist unmittelbar gefangen von der geographischen Vorstellung einer Rundreise.
		Man hat sofort die Vorstellung einer Aneinanderreihung von Strecken.
	www.jochen-pleines.de /anmerkungen/an_hist.htm (496 words)

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