traveling salesman

In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation

by William J. Cook  · 1 Jan 2011  · 245pp  · 12,162 words

we were mathematicians in need of a speedy computer, he looked us in the eye and warned, “You guys aren’t trying to solve that traveling salesman problem, are ya?" Quite a bit of foresight there. This was the first of many computers we ground to a halt, spending the better

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that are very easy to solve and yet have far more candidate solutions than the number of ways to route a salesman. What sets the traveling salesman problem apart is the fact that despite decades of research by top applied mathematicians around the world, in general it is not known how

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an efficient method or prove that this is impossible. The complexity question that is the subject of the Clay Prize is the Holy Grail of traveling-salesman-problem research and we may be far from seeing its resolution. But this is not to say mathematicians have thus far come away empty-

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used to compute optimal or near-optimal tours for a host of applied models on a daily basis. One of the enduring strengths of the traveling salesman problem has been its remarkable success as an engine of discovery in applied mathematics and its fields of operations research and mathematical programming. And many

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all, I thank my comrades David Applegate, Robert Bixby, and Vašek Chvátal, for over twenty years of fun and work toward unraveling part of the traveling salesman mystery. I also thank Michel Balinsky, Mark Baruch, Robert Bland, Sylvia Boyd, William Cunningham, Michel Goemans, Timothy Gowers, Nick Harvey, Keld Helsgaun, Alan Hoffman,

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School of Industrial and Systems Engineering at Georgia Tech and the Department of Operations Research and Financial Engineering at Princeton University. My work on the traveling salesman problem is funded by grants from the National Science Foundation (CMMI-0726370) and the Office of Naval Research (N00014-09-1-0048), and by

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Chicago, Illinois. Police officers Toody and Muldoon navigated Car 54 in a popular American television series. Their 33-city task is an instance of the traveling salesman problem, or TSP for short. In its general form, we are given a collection of cities and the distance to travel between each pair of

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challenge had been circulating through the mathematics community since the mid-1930s. Its solution was reported in Newsweek.5 Finding the shortest route for a traveling salesman—starting from a given city, visiting each of a series of other cities, and then returning to his original point of departure—is more than

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for treating the problem and impossibility results would also be valuable.’ —Merrill Flood, 1956.8 ‘I conjecture that there is no good algorithm for the traveling salesman problem.’ —Jack Edmonds, 1967.9 ‘In this paper we give theorems which strongly suggest, but do not imply, that these problems, as well as

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many others, will remain intractable perpetually.’ —Richard Karp, 1972.10 Challenges The authors of these remarks are three giants of traveling-salesman research. Merrill Flood rallied support for the problem in the 1940s; more than anyone else, Flood is responsible for the emergence of the TSP as

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It appears to have been discussed informally by mathematicians at mathematics meetings for many years. —George Dantzig, Ray Fulkerson, and Selmer Johnson, 1954.1 he traveling salesman problem is known far and wide, but the path it has taken to such mathematical prominence is somewhat obscure. For example, we cannot say for

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both in the south of the state. 21 22 Chapter 2 person making the rounds. Timothy Spears’s book 100 Years on the Road: The Traveling Salesman in American Culture cites an 1883 estimate by Commercial Travelers Magazine of 200,000 traveling salesmen working in the United States, and a further estimate

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the 1832 German handbook Der Handlungsreisende—Von einem alten Commis-Voyageur.3 The Commis-Voyageur describes the need for good tours.4 Business leads the traveling salesman here and there, and there is not a good tour for all occurring cases; but through an expedient choice and division of the tour

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always to visit as many localities as possible without having to touch them twice. This is an explicit description of the TSP, made by a traveling salesman himself! The Commis-Voyageur book presents five routes through regions of Germany and Switzerland. Four of these routes include return visits to an earlier city

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time. Numerous volumes written later in the century describe well-chosen routes in the United States, Britain, and other countries. The romantic image of the traveling salesman is captured, too, in stage, film, literature, Halle Sondershausen Leipzig Muehlhausen Dresden Eisenach Freiberg Chemnitz Salzungen Fulda Plauen Gelnhausen Frankfurt Hof Hanau Aschaffenburg Baireuth Bamberg

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sciences. Two of the leading figures of the era did, however, explore aspects of the TSP, and they are rightly viewed as the grandfathers of traveling-salesman research. Graph Theory and the Bridges of Königsberg The great Leonhard Euler wrote the most important of all early mathematical papers describing touring problems. The

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in the following passage from Dantzig, Fulkerson, and Johnson’s classic paper.19 Merrill Flood (Columbia University) should be credited with stimulating interest in the traveling-salesman problem in many quarters. As early as 1937, he tried to obtain near optimal solutions in reference to routing of school buses. Both Flood and

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to RAND, coinciding with Flood’s relocation to California. Princeton University’s Harold Kuhn writes the following in a December 2008 e-mail letter. The traveling salesman problem was known by name around Fine Hall by 1949. For instance, it was one of a number of problems for which the RAND corporation

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her contributions to the salesman provided the background for the RAND breakthrough several years later. Whence the TSP? In a 1949 research paper, Robinson uses “traveling salesman problem” in an offhand way, suggesting it was a familiar concept at the time. In fact, until a copy of the RAND prize list is

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person to have this information, but unfortunately he does not, as he explained to Albert Tucker. “I don’t know who coined the peppier name Traveling Salesman Problem for Whitney’s problem, but that name certainly caught on, and the problem has turned out to be of very fundamental importance.” Whatever the

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origin, except for small variations in spelling and punctuation, “traveling” versus “travelling,” “salesman” versus “salesman’s,” etc., by the mid-1950s the TSP name was in wide use, and the problem was beginning to pick up its notorious

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t make it less interesting to me—just the other way around. —George Dantzig, 1986.1 he name itself announces the applied nature of the traveling salesman problem. This has surely contributed to a focus on computational issues, keeping the research topic well away from perils famously described in John von Neumann

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13 cities in three weeks. When my publisher gave me the list of cities, I realized something amazing. I was actually going to live the Traveling Salesman Problem! I tried conveying my excitement to the publicity department, tried explaining to them the mathematical significance of all this, and how we could perhaps

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a tour depends on the direction of travel. And More The areas of application we have described by no means exhaust the reach of the traveling salesman. Indeed, intriguing new uses for the model appear regularly in the applied mathematics literature. Successful projects that have been reported include the following:12 •

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editor who went ahead and shifted the locations of the cities over to the 48 state capitals. The Discover caption is as follows. “The traveling salesman problem is one of math’s most enduring unsolved puzzles. Here’s the shortest route for a salesman—or 75 76 Chapter 4 Figure 4

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). We were both keenly aware of the fact that, although the complete set of faces (or constraints) in the linear programming formulation of the Traveling Salesman Problem was enormous, if you could find an optimal solution to a relaxed problem with a subset of the faces that is a tour, then

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you had solved the underlying Traveling Salesman Problem. What we need is a good understanding of how to work with the inequalities, that is, how to produce an inequality when it

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to face the possibility that billions of cuts might be needed. —Alan Hoffman and Philip Wolfe, 1985.1 he linear-programming relaxations associated with the traveling salesman problem are wildly complex: the simplex method is no match for problems with constraints numbering in the billions. Fortunately, Dantzig, Fulkerson, and Johnson put

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number, and thus the facets might too have a simple characterization. He writes: “At least we should hope they have, because finding a really good traveling salesman algorithm is undoubtedly equivalent to finding such a characterization.”22 This is a bold statement, but his insight was right on the money. Indeed, the

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-cut. Their paper concludes with the optimistic note, “the problem with 2392 cities should not be the end of the ongoing saga of the symmetric traveling salesman.” Figure 8.6 Top: Optimal tour of a 2,392-hole printed circuit board. Bottom: Manfred Padberg and Giovanni Rinaldi, 1985. Courtesy of Manfred

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do. We take a group of talented young people, and we expose them to the history and theory of some famous N P problem. The traveling salesman problem will do nicely. —Charles Sheffield, 1996.1 alesmen, lawyers, preachers, authors, and tourists have been plotting tours for years, not to mention all

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some cases successfully capturing the mathematical essence that has brought so much attention to the salesman. W Julian Lethbridge I was delighted to discover the Traveling Salesman paintings by Julian Lethbridge. Lethbridge is a celebrated contemporary artist, whose work is included in collections of the National Gallery of Art in Washington,

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York Times, 1995.2 Lethbridge’s abstraction is cerebral, often based on mathematical or natural principles. —ULAE, 1997.3 200 Chapter 11 Figure 11.1 Traveling Salesman, Julian Lethbridge, 1995, lithograph, 43.75 × 42 inches. Image courtesy of Julian Lethbridge and United Limited Art Editions. His vocabulary often originates from random

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of his artwork, but moved quickly to general mathematics, where Lethbridge has interest in, and intuition for, the aesthetics of our discipline. When discussing his Traveling Salesman series, Lethbridge explained that he came upon a description of the TSP in a journal and was struck immediately by the economy of thought and

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compare these to the map paintings of Jasper Johns and to Jean Dubuffet’s Compagnie Fallacieuse.6 Such references would be Aesthetics Figure 11.2 Traveling Salesman 4, Julian Lethbridge, 1995, oil on linen, 72 × 72 inches. The Robert and Jane Meyerhoff Collection, photograph by Adam Reich. disappointing to fans of

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and music. Concerning his TSP pieces, Galanter made the following comments in an introduction to his work at an exhibition in Lima, Peru.10 The Traveling Salesman mural series explores emergence in a different way. The design for each mural is uniquely generated for each specific wall. A large number of random

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even philosophy. It is no wonder that the interplay of these activities displays some affinity to artistic works. We can express similar feelings for the traveling salesman problem, and hope to see further connections between the TSP and art, as progress is made in understanding the problem’s fundamental complexity. 12:

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he beauty of the TSP will no doubt continue to attract mathematicians and computer scientists for years to come. Christos Papadimitriou told me that the traveling salesman problem is not a problem, it’s an addiction. —Jon Bentley, 1991.2 It’s addictive. No matter how much progress you make, you

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the study of the TSP, both with an eye on the million-dollar complexity prize and on the practical, step-by-step, computational attack. The traveling salesman problem is as tough as it gets, but, as Rashers Ronald would say, bash on regardless. Notes Chapter 1. Challenges 1. IBM Corporation press

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release from January 2, 1964, describing a new computer program for solving small instances of the traveling salesman problem. The computer program was developed by Michael Held, Richard Karp, and Richard Shareshian. 2. Menger (1931). 3. T0P500 Supercomputer List, June 2009. 4.

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software/TSPLIB95/. http://www2.research.att.com/∼dsj/chtsp/about.html. Johnson, D. S., L. A. McGeogh. 2002. In: G. Gutin, A. Punnen, eds. The Traveling Salesman Problem and Its Variations. Kluwer, Boston, MA. 369–443. http://www.tsp.gatech.edu/vlsi/index.html. http://www.tsp.gatech.edu/world/countries.html

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is, a onedimensional plane. Grötschel, M., M. W. Padberg. 1979. Math. Program. 16, 265–80. Naddef, D. 2002. In: G. Gutin, A. Punnen, eds. The Traveling Salesman Problem and Its Variations. Kluwer, Boston, Massachusetts. 29–116. Schrijver, A. 2002. Math. Program. 91, 437–45. The subtour inequality for S is identical to

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The father of linear programming. The College Mathematics Journal 17, 293–314. [2] Applegate, D. L., R. E. Bixby, V. Chvátal, W. Cook. 2006. The Traveling Salesman Problem: A Computational Study. Princeton University Press, Princeton, New Jersey. [3] Arora, S., B. Barak. 2009. Computational Complexity: A Modern Approach. Cambridge University Press, New

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D. S. Johnson. 1979. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco, California. [13] Gomory, R. E. 1966. The traveling salesman problem. Proceedings of the IBM Scientific Computing Symposium on Combinatorial Problems. IBM, White Plains, New York. 93–121. [14] Grötschel, M., O. Holland. 1991. Solution

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D.C. [25] Schrijver, A. 2003. Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin, Germany. [26] Spears, T. B. 1994. 100 Years on the Road: The Traveling Salesman in American Culture. Yale University Press, New Haven, Connecticut. Index Aaronson, Scott, 189 Adams, Douglas, 178 Adleman, Leonard, 184–186 Agarwala, Richa, 50 algorithm, 6

The Everything Blueprint: The Microchip Design That Changed the World

by James Ashton  · 11 May 2023  · 401pp  · 113,586 words

to lead the band of engineers that formed Arm. Robin Saxby was impatient, enthusiastic, confident, dishevelled, brusque and something of a namedropper. He was a travelling salesman with grey-blue eyes and mousy brown hair plastered down across his forehead. And he had a knack for forming relationships built to last. As

The Golden Ticket: P, NP, and the Search for the Impossible

by Lance Fortnow  · 30 Mar 2013  · 236pp  · 50,763 words

need computers not just to search the data we already have but to search for possible solutions to problems. Consider the plight of Mary, a traveling salesman working for the US Gavel Corporation in Washington, D.C. Starting in her home city, she needs to travel to the capitals of all the

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. Mary sketched a simple drawing on a map, played with it for a while, and came up with a pretty good route. Figure 1-1. Traveling Salesman Problem Map. The travel department wanted Mary to see if she could come up with a different route, one that used less than 11,000

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the basic question of this book. The P versus NP problem asks, among other things, whether we can quickly find the shortest route for a traveling salesman. P and NP are named after their technical definitions, but it’s best not to think of them as mathematical objects but as concepts. “NP

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to which we can find a solution quickly. “P = NP” means we can always quickly compute these solutions, like finding the shortest route for a traveling salesman. “P ≠ NP” means we can’t. The Partition Puzzle Consider the following thirty-eight numbers: 14,175, 15,055, 16,616, 17,495, 18,072

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, we could use this procedure to do much more than solve these math puzzles. We could use it to solve everything, even to find a traveling salesman’s shortest route. This simple puzzle captures the P versus NP problem: any program that can solve large versions of this puzzle can compute just

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opposite sides of the Cold War. Richard Karp followed up with a list of twenty-one important problems that capture P versus NP, including the traveling salesman problem and the partition puzzle mentioned earlier. After Karp, computer scientists started to realize the incredible importance of the P versus NP problem, and it

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algorithms, then satisfiability doesn’t either. Not only does clique reduce to Satisfiability, but so do the other NP problems we discussed earlier, including the traveling salesman problem, the Hamiltonian path, max-cut, and map coloring. What Steve Cook did was show that every problem in NP has a reduction to satisfiability

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was one of the hardest problems in NP, he also showed that nineteen other important problems were also among these hardest, including the partition puzzle, traveling salesman, Hamiltonian path, map coloring, and max-cut. Solve any of these problems efficiently and you have an efficient algorithm for all of them and have

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to visit many places with minimal effort is such a common problem that it has its own nickname: the Traveling Salesman problem. Many scientists have tried to find the best solution for the traveling salesman problem. Others have tried to find great algorithms for the job scheduling problem. Others have tried for clique, max

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machines. Large Sudoku games are NP-complete. Think you are good at Sudoku? If you can reliably solve large puzzles, then you can solve satisfiability, traveling salesman problems, and thousands of other NP-complete problems. Sudoku is hardly the only NP-complete solitaire game. Consider the Minesweeper game built into Microsoft Windows

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use 2009 technology than to solve your problem using 1971 technology. For some other NP problems we can do ever larger examples. The NP-complete traveling salesman problem tries to find the best way to visit a large number of cities using the shortest total distance. Using something called the cutting-plane

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method we can solve traveling salesman problems on 10,000 cities in a short amount of time. Here is a solution for 13,509 cities of population at least 500. Figure

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the problem also increases greatly with the problem size. Don’t expect algorithms to solve all satisfiability problems on 150 variables or to find optimal traveling salesman tours on 20,000 cities any time in the near future. Heuristics Woodworkers in the seventeenth century would often use the width of their thumbs

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. Approximation Maybe we can’t get the best possible solution to a problem, but often a not so bad solution is good enough. Consider the traveling salesman problem, an NP-complete problem where one wants to visit a number of cities using the least total distance. If I have to take a

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to 1,803,000 miles, I can be back in the black even though I never did find the absolute best route. Even though the traveling salesman problem on a map is NP-complete and presumably difficult to solve exactly, we can find tours that get very close to the best solution

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. Sanjeev Arora and Joe Mitchell give an algorithm that subdivides the map into small pieces, finding good solutions for the traveling salesman problem in those small pieces and then connecting them all up again in a clever way. Consider the map of 71,009 Chinese cities. Figure

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6-11. Chinese Cities. We put a tight grid on top and solve the traveling salesman problem in each grid and piece them together. If there are too many cities in a small region, we will build smaller grids in those

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areas. Using this scheme, we can find traveling salesman tours within a few percent of optimum in a reasonable amount of time. Figure 6-12. Chinese Cities Grid. If all NP problems had such

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and allow us to solve cliques exactly. Many NP problems are not as hard to approximate as clique or as easy to approximate as the traveling salesman problem on a map. Let’s look again at very cozy groups. In a small example like this we can just check all the possibilities

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for every NP problem. Alice can convince Bob that there is a large clique in Frenemy, a three-coloring of a map, or a short traveling salesman tour, all without revealing any information about the clique, coloring, or tour other than that they exist. In a common attack in cryptography, a person

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research on an anatomically correct testbed hand incorporates information presented at a talk given at the 2010 CRA Snowbird Conference on July 18, 2010. The traveling salesman problems were generated using software from Mark Daskin, http://sitemaker.umich.edu/msdaskin/software. Chapter 2 Nearly everything in this chapter, except the section on

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Normal Forms for Boolean Functions by Algorithms of a Certain Class, Doklady Akademii Nauk SSSR 132 (1960): 504–6. Chapter 6 The traveling salesman example is from “CRPC Researchers Solve Traveling Salesman Problem for Record-Breaking 13,509 Cities,” a 2003 press release from the Center for Research on Parallel Computation at Rice University

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movies, 13 Tetris, 63, 63 theoretical cybernetics, 79–85 tracking, over Internet, 159–60 Trakhtenbrot, Boris, 83–84 transistors, in circuits, 113 translation, 18, 23 traveling salesman problem: approximation of, 99–100, 100, 101; description of, 2–4, 3; size of problem, 91, 91 Tsinghua University, 12 Turing, Alan, 73–74; in

Buffett

by Roger Lowenstein  · 24 Jul 2013  · 612pp  · 179,328 words

tightly buttoned and noticeably creased, with an attaché and an oversized valise and something of the appearance of a down-at-the-heels but earnest traveling salesman. Buffett made for the office on Cove Street, and Seabury emerged from the ivory tower for a final time. Calling the meeting to order, Seabury

Uncommon Grounds: The History of Coffee and How It Transformed Our World

by Mark Pendergrast  · 2 Jan 2000  · 564pp  · 153,720 words

—at his tender age he has more sense than most of us.” Soon, however, Folger sold out and rejoined Bovee, now as a clerk and traveling salesman. The same miner’s 1858 diary entry noted that Folger was “in business for himself down in Frisco and selling coffee to every damned diggings

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. While still in his teens he started a hardware store in Independence, Kansas, selling it a year later for a profit. He worked as a traveling salesman of farm implements, then invented and manufactured farm equipment on his own, obtaining patents for a seed planter, sulky plow, harrow, hay stacker, and various

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triple. Joel Cheek Creates Maxwell House After attending college, Joel Owsley Cheek went to Nashville, Tennessee, in 1873 to seek his fortune. Hired as a traveling salesman, or drummer, for a wholesale grocery firm there, he moved back to his home state of Kentucky to open new territory, generally riding on horseback

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the moral in it on my part to profit him, is an immoral transaction.” He conveyed what appeared to be a real concern for the traveling salesman, explaining that he had been on the road himself for twenty-eight years. “Bear with him in his weaknesses and shortcomings. Encourage him as much

Ashes to Ashes: America's Hundred-Year Cigarette War, the Public Health, and the Unabashed Triumph of Philip Morris

by Richard Kluger  · 1 Jan 1996  · 1,157pp  · 379,558 words

stint at the Upmann cigar factory in Cuba, learning the leaf and manufacturing end of the trade, followed by a more intensive spell as a traveling salesman for the Webster-Eisenlohr company, peddling their line of cigars on the East Coast. Although the Cullmans sold the Webster people the tobacco they used

Andrew Carnegie

by David Nasaw  · 15 Nov 2007  · 1,230pp  · 357,848 words

authorizing the construction of seven bridges across the Mississippi River and one across the Missouri River at Kansas City.12 Carnegie refashioned himself into a traveling salesman for Keystone Bridge, hawking his wares to local boosters, bankers, promoters, and politicians up and down the Mississippi, Missouri, and Ohio Rivers who hoped to

Reaganland: America's Right Turn 1976-1980

by Rick Perlstein  · 17 Aug 2020

a dear friend, so McAteer spoke frankly: “Howard, I think it’s absolutely essential.” McAteer was the nascent Christian right’s most indefatigable organizer. A traveling salesman in Colgate-Palmolive’s toilet articles division who had worked his way up to sales manager for the entire Southeast, in the early 1970s he

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Wycliffe Bible Translators, making contacts with evangelicals all over the world. In 1976, John Conlan and Bill Bright offered him a new job as their traveling salesman for their efforts turning preachers into precinct organizers. After it folded, McAteer became field director for Howard Phillips’s Conservative Caucus. But he missed working

Money and Power: How Goldman Sachs Came to Rule the World

by William D. Cohan  · 11 Apr 2011  · 1,073pp  · 302,361 words

of Harvard without a degree because he had trouble seeing, had a vision of Goldman Sachs as a leading securities underwriter. He had been a traveling salesman after leaving Harvard but had joined the family business at age twenty-eight and would help lead a transformation of the firm into the underwriting

Emergence

by Steven Johnson  · 329pp  · 88,954 words

of their movement. They are thinking like a swarm. 7 See What Happens For years mathematicians have puzzled over a classic brainteaser known as the “traveling salesman problem.” Imagine you’re a salesman who has to visit fifteen cities during a business trip—cities that are distributed semirandomly across the map. What

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maddeningly difficult to establish. Even with the number of cities set at a relatively modest fifteen, billions of potential routes exist for our traveling salesman. For complicated reasons, the traveling salesman problem is almost impossible to solve definitively, and so historically mathematicians—and traveling salesmen, presumably—have settled for the next best thing: routes

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that are tolerably short, but not necessarily the shortest possible. This might sound like an arcane issue, given the real-world decline of the traveling salesman, but the core elements of the problem lie at the epicenter of the communications revolution. Think of those traveling salesmen as bits of data, and

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Internet, where there may be thousands of “cities” on any given route, instead of just fifteen. The traveling salesman may finally have been killed off for good by online retailers like Amazon.com, but the traveling salesman problem has become even more critical to the digital world. In late 1999, Marco Dorigo of the

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Free University of Brussels announced that he and his colleagues had hit upon a way of reaching “near-optimal” solutions to the traveling salesman problem that was notably more time-efficient that any traditional approach. Dorigo’s secret: let the ants do the work. Not literal ants, of course

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heavily dosed with pheromones. After several repeated sessions, the salesmen swarm starts homing in on the shortest routes, reaching a near-optimal solution to the traveling salesman problem without using anything resembling traditional calculus or a central problem-solver. Since Dorigo’s announcement of his results, France Telecom, British Telecommunications, and MCI

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, 165–66, 178–80, 181 Tracker program, 59–63, 65 trade, 101–2, 104–7, 109, 110 traffic patterns, 97, 166, 204, 230–31, 232 traveling salesman problem, 227–29 tumors, brain, 119 Turing, Alan, 14, 18, 42–45, 49, 53, 54, 62–65, 67, 206, 236n, 242n, 254n–56n, 263n Turing

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