21. Exact Traveling Salesman Problem - TSP Exact traveling salesman problem (TSP). Exact implementation for the TravelingSalesman Problem Also solves Asymmetric TSP s see demo s. 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

22. 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

23. 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.

24. 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 duck below to expand program list if it is collapsed - MS Internet Explorer only) There is also a computer generated program index page here 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 Airport Simulation Akerue Astronomy Demo ... 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.

25. Oefen traveling salesman problem (TSP) using Simulated Annealing. This applet attemptsto 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.

26. TSPLIB TSPLIB. Symmetric traveling salesman problem (TSP). Given a not. HCP data Asymmetric traveling salesman problem (ATSP). Given a http://www.iwr.uni-heidelberg.de/iwr/comopt/software/TSPLIB95/

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.

27. DIMACS TSP Challenge 8th DIMACS Implementation Challenge The traveling salesman problem. ChallengeNews Still Open for Business! (Including New DoIt-Yourself Feature). 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.

28. Neil Simonetti's Traveling Salesman Problem Page Neil Simonetti s traveling salesman problem Page. Results for the JobShop Scheduling Problems. More traveling salesman problem Links http://www.andrew.cmu.edu/~neils/tsp/

Neil Simonetti's Traveling Salesman Problem Page Results for the Job Shop Scheduling Problems Test data is found here K-comparisons: K-comparisons with local search: Summary of results. 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 twdist.c twtime.c uscity.c (Instructions are contained in an appendix to my thesis) Building the Auxiliary Structure: auxgraph.c Selected Test Data Sites for the TSP: U. S. Cities : Symmetric TSP's based on the layout of America's Largest Cities. (Neil Simonetti) TSPLIB : Largest catalog of TSP related problems on the web. (Gerhard Reinelt) [

29. Traveling Salesman Problem traveling salesman problem. To formulate the symmetric traveling salesman problem,one notes that the direction traversed is immaterial, so that c ij = c ji . 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. Euclidean Traveling Salesman Problem Definition of Euclidean traveling salesman problem, possibly with links to more informationand implementations. NIST. Euclidean traveling salesman problem. http://www.nist.gov/dads/HTML/euclidntrvls.html

Euclidean traveling salesman problem (classic problem) Definition: Find a path of minimum Euclidean distance between points in a plane which includes each point exactly once and returns to its starting point. See also traveling salesman spanning tree Note: This can be generalized to higher dimensions, for instance, points in a 3-dimensional space. This problem is a special case of traveling salesman since the cost between points is the planar distance instead of arbitrary weights. Author: PEB Go to the Dictionary of Algorithms and Data Structures home page. If you have suggestions, corrections, or comments, please get in touch with Paul E. Black (paul.black@nist.gov). Entry modified Fri Oct 15 10:04:13 1999. HTML page formatted Thu Jan 15 16:37:56 2004. This page's URL is http://www.nist.gov/dads/HTML/euclidntrvls.html

31. The Travelling Salesman Problem The Travelling Salesman Problem. Program by Peter Meyer. The WorldRecordtraveling salesman problem for 3038 Cities Solved. In http://www.hermetic.ch/misc/ts3/ts3demo.htm

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. Most important, it has applications in science and engineering. For example, in the manufacture of a circuit board, it is important to determine the best order in which a laser will drill thousands of holes. An efficient solution to this problem reduces production costs for the manufacturer. World-Record Traveling Salesman Problem for 3038 Cities Solved In 1992 I came upon an article in a physics journal (I don't remember where or by whom it was written) which described the use of the so-called Simulated Annealing Algorithm to solve this problem. The algorithm is so named because it can be seen as a simulation of the physical process of annealing (in which a hot material cools slowly, allowing its constituent atoms to assume arrangements that would not be possible with rapid cooling). I then wrote a program to implement this algorithm. A good explanation of the simulated annealing algorithm is given by Robert C. Williams:

32. Generation5 - Genetic Algorithm And Traveling Salesman Problem Genetic Algorithm and traveling salesman problem. About Traveling SalesmanProblem. Again a quotation from http//www.math.princeton.edu/tsp/. http://www.generation5.org/aisolutions/tspapp.shtml

Home Articles Reviews Interviews ... Contribute Login Login: Password: Keep me logged in. ( help Register Why register? Lost your password? ... Binary Decision Tree implementation in C++ Friends and Affiliates Home Articles Genetic Algorithms > Applications/Code Genetic Algorithm and Traveling Salesman Problem By Konstantin Boukreev Printable Version 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

33. CodeGuru: Genetic Algorithm And Traveling Salesman Problem Genetic Algorithm and traveling salesman problem. About TravelingSalesman Problem. 7. Solving traveling salesman problem. Downloads. http://www.codeguru.com/Cpp/misc/misc/article.php/c3795/

CodeGuru.com -Articles -Comments -Forums Earthweb User ID: Password: Remember Me: Forgot Password? Not a member? Click here for more information and to register. internet.commerce Cheap Web Hosting News Feeds Digital Cameras Web Site Hosting ... Web Search RSS Feeds All VC++/C++ .NET/C# VB Home Visual C++ / C++ Miscellaneous Miscellaneous ... Attend a Microsoft ® Developer Network (MSDN) Webcast to get tips from industry experts. Pick from hundreds of webcasts. Live or on-demand. All at no cost. Register! http://www.microsoft.com/webcasts Genetic Algorithm and Traveling Salesman Problem Rating: none Konstantin Boukreev view profile September 27, 2001 (continued) Click Here Environment: VC++ 6.0. SP5, Win2k SP2, MS Platform SDK April 2001 Table of 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 genetic algorighm (GA) guru and I do not have any degree in GA so this article can't be used as a GA GA tutorial. There are not any mathematics, logic, or algebra about GA. It's only a programmer's view on Genetic Algorithms and only an example of GA coding. Use it carefully! Any comments and criticism are highly appreciated. Genetic Algorithm, Theory

34. GSRC Calibrating Achievable Design Bookshelf: Traveling Salesman Problem Slot MARCO GSRC Calibrating Achievable Design Bookshelf. traveling salesman problemSlot. Work in progress last updated Mon Dec 16 2002 (see other slots). http://vlsicad.ucsd.edu/GSRC/Bookshelf/Slots/TSP/

MARCO GSRC Calibrating Achievable Design: Bookshelf Traveling Salesman Problem Slot Work in progress : last updated Mon Dec 16 2002 see other slots Ken Boese, Andrew B. Kahng Mike Oliver and Tim Walters Contents I. Introduction II. Data Formats III. Publicly available instances, solutions and reference performance results IV. Executable Utilities (converters, generators, statistics browsers, evaluators, constraint verifiers) V. Optimizers and other non-trivial executables VI. Common in-memory representations, parsers and other source codes I. Introduction This slot gives pointers to various TSP codes, benchmarks, and instance generators. II. Data Formats To come III. Publicly available instances, solutions and reference performance results TSPLIB DIMACS Benchmark Instances Bill Cook's TSP page IV. Executable Utilities To come V. Optimizers and non-trivial executables To come VI. Source codes DIMACS Benchmark Code and Instance Generation Codes Ken Boese's implementation of the Lin-Kernighan heuristic Keld Helsgaun's implementation of the Lin-Kernighan heuristic The Concorde TSP code (Applegate, Bixby, Chvátal, and Cook) ... abk@ucsd.edu,oliver@cs.ucla.edu

35. Branch And Cut For The Traveling Salesman Problem Branch Cut and Price Applications traveling salesman problem. Applegate,Bixby, Chvatal, and Cook. traveling salesman problem Links. 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

36. 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 ...

37. 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.

38. 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://www.rpi.edu/~mitchj/math1900/lp/node6.html

Next: Car 54 Up: Applications of LP Previous: Telecommunications Traveling Salesman Problem Given a set of cities, find the shortest route that visits each city exactly once and returns to the home city. Applications include: Vehicle routing eg, how do we route school buses to pick up children, how do Federal Express and UPS assign packages to trucks? VLSI chip board manufacturing: holes need to be drilled on the board. How do we route the drill to visit all these holes so it takes the shortest possible route? Website: http://www.keck.caam.rice.edu/tsp/ Largest problem solved to date has more than 13000 cities. John E Mitchell

39. Chapter 6 Paths And Networks Chapter 6 Paths and Networks. Applet 9. traveling salesman problemCalculator The applet illustrates implements heuristic methods 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. Wiley::The Traveling Salesman Problem: A Guided Tour Of Combinatorial Optimizati Wiley Mathematics Statistics Discrete Mathematics The TravelingSalesman Problem A Guided Tour of Combinatorial Optimization. http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471904139.html

Shopping Cart My Account Help Contact Us By Keyword By Title By Author By ISBN By ISSN Wiley 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) Join a 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

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