mathematics

In Pursuit of the Traveling Salesman

In Pursuit of the Traveling Salesman is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand In Pursuit of the Traveling Salesman rather than just read about it. In short: In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation is a book on the travelling salesman problem, by William J. Cook, published in 2011 by the Princeton University Press, with a paperback reprint in 2014.

Key takeaways

In Pursuit of the Traveling Salesman belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect In Pursuit of the Traveling Salesman to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of In Pursuit of the Traveling Salesman from memory before moving on to harder problems.

Reference excerpt

In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation is a book on the travelling salesman problem, by William J. Cook, published in 2011 by the Princeton University Press, with a paperback reprint in 2014. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries.

Topics The travelling salesman problem asks to find the shortest cyclic tour of a collection of points, in the plane or in more abstract mathematical spaces. Because the problem is NP-hard, algorithms that take polynomial time are unlikely to be guaranteed to find its optimal solution; on the other hand a brute-force search of all permutations would always solve the problem exactly but would take far too long to be usable for all but the smallest problems. Threading a middle ground between these too-fast and too-slow running times, and developing a practical system that can find the exact solution of larger instances, raises difficult questions of algorithm engineering, which have sparked the development of "many of the concepts and techniques of combinatorial optimization". The introductory chapter of the book explores the limits of calculation on the problem, from 49-point problems solved by hand in the mid-1950s by George Dantzig, D. R. Fulkerson, and Selmer M. Johnson to a problem with 85,900 points solved optimally in 2006 by the Concorde TSP Solver, which Cook helped develop. The next chapters covers the early history of the problem and of related problems, including Leonhard Euler's work on the Seven Bridges of Königsberg, William Rowan Hamilton's Icosian game, and Julia Robinson first naming the problem in 1949. Another chapter describes real-world applications of the problem, ranging "from genome sequencing and designing computer processors to arranging music and hunting for planets". Reviewer Brian Hayes cites "the most charming revelation" of the book as being the fact that one of those real-world applications has been route planning for actual traveling salesmen in the early 20th century. Chapters four through seven, "core of the book", discuss methods for solving the problem, leading from heuristics and metaheuristics, linear programming relaxation, and cutting-plane methods, up to the branch and bound method that combines these techniques and is used by Concorde. The next two chapters also cover technical material, on the performance of computer implementations and on the Computational complexity theory of the problem. The remaining chapters are more human-centered, covering human and animal problem-solving strategies, and the incorporation of TSP solutions into the artworks of Julian Lethbridge, Robert A. Bosch, and others. A short final summary chapter suggests possible future directions, including the possibility of progress on the P versus NP problem.

Audience The book is intended for a non-specialist audience, avoids technical detail and is written "in an easy to understand style". It includes many historical asides, examples, applications, and biographical information and photographs of key players in the story, making it accessible to readers without a mathematical background. Although In Pursuit of the Traveling Salesman is not a textbook, reviewer Christopher Thompson suggests that some of its material on the use of linear programming and on applications of the problem "would be well-suited for classroom use", citing in particular the way it links multiple fields including numerical analysis, graph theory, algorithm design, logic, and statistics. Reviewer Stan Wagon writes that "any reader with an interest in combinatorial algorithms will find much of value in this book". Jan Karel Lenstra and David Shmoys write that "The writing is relaxed and entertaining; the presentation is excellent. We greatly enjoyed reading it." And reviewer Haris Aziz concludes "The book is highly recommended to any one with a mathematical curiosity and interest in the development of ideas.".

Related works More details of Cook's work with Concorde, suitable for more serious researchers on the problem and on related topics, can be found in an earlier book by Cook with David Applegate, Robert E. Bixby and Václav Chvátal, The Traveling Salesman Problem: A Computational Study (2007). Other books on the travelling salesman problem, also more technical than In Pursuit of the Traveling Salesman, include The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization (by Lawler, Lenstra, Rinnooy Kan, and Shmoys, 1985) and The Traveling Salesman Problem and Its Variations (by Gutin and Punnen, 2006).

References

Further reading Ellenberg, Jordan (March 10, 2012), "The fuzzy path may be shortest (review of In Pursuit of the Traveling Salesman)", The Wall Street Journal McLemee, Scott (March 21, 2012), "Algorithm of a salesman (review of In Pursuit of the Traveling Salesman)", Inside Higher Education

Worked examples

Example 1 — a first encounter with In Pursuit of the Traveling Salesman

Start with the simplest possible case. Write down what In Pursuit of the Traveling Salesman claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to In Pursuit of the Traveling Salesman before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about In Pursuit of the Traveling Salesman ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of In Pursuit of the Traveling Salesman

In research: In Pursuit of the Traveling Salesman appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses In Pursuit of the Traveling Salesman in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: In Pursuit of the Traveling Salesman is common in secondary-school and first-year university syllabi. It links to neighbouring topics 2011 non-fiction books, Mathematics books, Princeton University Press books, so understanding it makes those chapters shorter.
In everyday life: Look for In Pursuit of the Traveling Salesman outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study In Pursuit of the Traveling Salesman in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what In Pursuit of the Traveling Salesman means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain In Pursuit of the Traveling Salesman out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is In Pursuit of the Traveling Salesman in simple terms?

Why does In Pursuit of the Traveling Salesman matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study In Pursuit of the Traveling Salesman?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on In Pursuit of the Traveling Salesman.