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

William J. Cook
Pages: 272
https://www.jstor.org/stable/j.ctt7t8kc

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. Preface
(pp. xi-xvi)
4. 1: Challenges
(pp. 1-18)

An advertising campaign by Procter & Gamble caused a stir among applied mathematicians in the spring of 1962. The campaign featured a contest with a $10,000 prize. Enough to purchase a house at the time. From the official rules: Imagine that Toody and Muldoon want to drive around the country and visit each of the 33 locations represented by dots on the contest map, and that in doing so, they want to travel the shortest possible route. You should plan a route for them from location to location which will result in the shortest total mileage from Chicago, Illinois back to... 5. 2: Origins of the Problem (pp. 19-43) The 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 certain when the problem’s lively name first came into use. Nevertheless, most of the story can be told, albeit with the help of an educated guess here and there. Its telling serves the useful side purpose of getting our TSP feet wet before jumping in with details of current attempts to crack the notorious problem. As a practical matter, the TSP was tackled by humans long before it became a fashionable topic... 6. 3: The Salesman in Action (pp. 44-61) The 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’s essay “The Mathematician”. “In other words, at a great distance from its empirical source, or after much ‘abstract’ inbreeding, a mathematical subject is in danger of degeneration”. Indeed, a strength of TSP research is the steady stream of practical applications that breathe new life into the area. In our roundup of TSP applications, let’s begin with a sample of tours taken by humans, including... 7. 4: Searching for a Tour (pp. 62-93) A salesman on the road will not be impressed by a claim of TSP unsolvability. She will nonetheless start up the car and get on with the task of visiting customers. This practical mind-set argues for an alternative approach to the problem: let’s give up for now the notion that only the absolute best solution will do, and focus on delivering, as quickly as possible, a near-optimal route. Such a view opens the door to all sorts of creative ideas for getting the salesman home in time for dinner. Indeed, some of the techniques developed and employed in this branch... 8. 5: Linear Programming (pp. 94-126) Selecting the best tour through a set of points and knowing it is the best is the full challenge of the TSP. Users of a brute-force algorithm that sorts through all permutations can be certain they have met the challenge, but such an approach lacks both subtlety and, as we know, practical efficiency. What is needed is a means to guarantee the quality of a tour, short of inspecting each permutation individually. In this context, the tool of choice islinear programming, an amazingly effective method for combining a large number of simple rules, satisfied by all tours, to obtain... 9. 6: Cutting Planes (pp. 127-145) The 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 forth an elegant idea for handling such complexity. Their cutting-plane method does not attempt to solve the full LP problem in a stroke, but rather computes LP bounds on a pay-as-you-go basis, generating specific TSP inequalities only as they are needed. This is a game changer, and not for the salesman alone. The road to Dantzig et al.’s tour through the United States begins with the degree LP... 10. 7: Branching (pp. 146-152) In the Dark Age of TSP cutting planes, between the work of Dantzig et al. in 1954 and Chvátal in 1973, researchers focused on a variety of alternative solution methods. Chief among these is the divide-and-conquer approach known as branch-and-bound, another general-purpose tool developed first in the context of the salesman. In state-of-the-art TSP software, branch-and-bound is combined with the cutting-plane method to produce a powerhouse capable of solving instances of the problem having thousands of cities. The search for an optimal tour hidden in an LP relaxation is a search for the best needle in a large haystack. With... 11. 8: Big Computing (pp. 153-167) The combination of ever-improving mathematical techniques, careful algorithm engineering, and powerful computing platforms has taken the TSP to dizzying heights, but the struggle with the salesman is far from over. Let’s see where we stand. When it comes to TSP records, nothing comes close to topping the work of Dantzig, Fulkerson, and Johnson. It is absolutely astonishing that the three authors were able to find an optimal solution of such a large TSP instance and to prove its optimality by manual computation. —George Nemhauser and Martin Grötschel, 2008.² Dantzig, Fulkerson, and Johnson showed a way to solve large instances of... 12. 9: Complexity (pp. 168-190) The pursuit of the salesman through ever larger numbers of cities has led to breakthroughs in mathematics, computing, and engineering, as well as advances in numerous practical applications. This is the pride and joy of TSP researchers. But the step-by-step approach has not answered the mother-of-all complexity questions: can we efficiently solveeveryinstance of the TSP? The fate of the salesman from this complexity point of view is tied to that of many other problems, including general integer programming, via the theory of Stephen Cook and Richard Karp. Indeed, the TSP is rolled into the$\cal{P}$vs.$\cal{NP}\$question,...

13. 10: The Human Touch
(pp. 191-198)

Salesmen, lawyers, preachers, authors, and tourists have been plotting tours for years, not to mention all of those tennis players collecting balls after long practice sessions. With all this experience, could the human mind be a viable non-computer platform for cracking the general TSP?

Like any good sporting event, the 1997 chess match between World Champion Gary Kasparov and IBM’s Deep Blue drew vocal supporters for both contestants. Those hoping to keep machines at bay for a few more years pulled for the human, while hardware and software fans aligned themselves with the computer. Science-fiction writer Charles Sheffield, covering the...

14. 11: Aesthetics
(pp. 199-210)

When a mathematician refers to a particular item of study as beautiful, it comes without any implication that the beauty can be realized in a physical form. This holds true for the TSP. A tour through a set of points may have a pleasing shape, but it is the combined beauty of the geometry and complexity of the problem, not the tour itself, that attracts mathematicians. Nevertheless, the TSP has been adopted in several engaging works of art, in some cases successfully capturing the mathematical essence that has brought so much attention to the salesman.

I was delighted to discover...

15. 12: Pushing the Limits
(pp. 211-212)

The 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 always have the nagging feeling that you still did not nail down a couple of hunches that could bring about another quantum leap.

—Vašek Chvátal, 1998.³

We offer no tips for breaking a TSP addiction. Far from it. I wouldn’t hesitate to include small TSP challenges on the backs of candy wrappers,...

16. Notes
(pp. 213-222)
17. Bibliography
(pp. 223-224)
18. Index
(pp. 225-228)