# The Traveling Salesman Problem: A Computational Study

David L. Applegate
Robert E. Bixby
Vašek Chvatál
William J. Cook
Pages: 606
https://www.jstor.org/stable/j.ctt7s8xg

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. Preface
(pp. xi-xii)
4. Chapter One The Problem
(pp. 1-58)

Given a set of cities along with the cost of travel between each pair of them, thetraveling salesman problem, orTSPfor short, is to find the cheapest way of visiting all the cities and returning to the starting point. The “way of visiting all the cities” is simply the order in which the cities are visited; the ordering is called atourorcircuitthrough the cities.

This modest-sounding exercise is in fact one of the most intensely investigated problems in computational mathematics. It has inspired studies by mathematicians, computer scientists, chemists, physicists, psychologists, and a host of...

5. Chapter Two Applications
(pp. 59-80)

Much of the earliest work on the TSP was motivated by direct applications. For example, Flood [182] considers the planning of school bus routes, Mahalanobis [368] and Jessen [286] work with crop surveys, and Morton and Land [403] mention the routing of a laundry van.

Although the best argument for continued interest in the TSP is its wild success as a general engine of discovery, a steady stream of new applications breathes life into the research area and keeps the mathematical community focused on the computational aspects of the problem. In this chapter we cover some of the general applications...

6. Chapter Three Dantzig, Fulkerson, and Johnson
(pp. 81-92)

The origins of the study of the TSP as a mathematical problem are somewhat obscure. Karl Menger’s [389]Botenproblemwas presented in Vienna in 1930, and later in the decade the TSP was widely discussed at Princeton University. There is little mention of the problem, however, in the mathematical literature from this period.

In one of the earliest papers on the TSP, Merrill Flood [182] made the following statement.

This problem was posed, in 1934, by Hassler Whitney in a seminar talk at Princeton University.

This points to Whitney as the founding father of the TSP at Princeton. Hoffman and...

7. Chapter Four History of TSP Computation
(pp. 93-128)

The triumph of Dantzig, Fulkerson, and Johnson in 1954 set off a flurry of activity in mathematics and operations research circles, building toward Gomory’s fundamental work in mixed-integer linear programming at the end of the decade. Interestingly, however, the TSP literature from the 1950s does not include any attempts to extend or automate the Dantzig et al. solution procedure; from the narrow perspective of the TSP, this appears to be the dark ages of the cutting-plane method. This inactivity slowed the overall progress on the problem, and it would be another 16 years before a larger TSP instance was solved,...

8. Chapter Five LP Bounds and Cutting Planes
(pp. 129-158)

In the next six chapters we describe methods for finding cutting planes for the TSP. To set the stage for these discussions, we gather together in this chapter some common notation and themes concerning linear programming, separation routines, and the cutting-plane method.

A general instance of ann-city TSP is specified by then(n− 1)/2 travel costs between the pairs of cities. When dealing with these data it is often convenient to adopt the language of graph theory. In this section we give only basic definitions that are needed later in the book; for further information we refer the...

9. Chapter Six Subtour Cuts and PQ-Trees
(pp. 159-184)

In the application of the cutting-plane method to the TSP, first and foremost among possible cuts are the subtour inequalities. Cuts that match the subtour template are the means to force the LP solutionxto be a single connected piece, an obvious requirement forxto approximate a tour vector.

Fortunately, the subtour separation problem is very well understood, including a range of efficient exact separation methods. In this chapter we discuss fast subtour heuristics, as well as the exact method of Padberg and Rinaldi [445] that is used in Concorde. We also discuss an effective mechanism for building...

10. Chapter Seven Cuts from Blossoms and Blocks
(pp. 185-198)

Comb inequalities have a long history in the TSP, going back to the by-hand computations of Dantzig, Fulkerson, and Johnson [151]. The general comb template is now a workhorse in TSP codes. An important open question, however, is to determine the complexity of the exact separation of comb inequalities, given a vectorxsatisfying all subtour constraints. No polynomial-time algorithm is known for the problem, but neither is it known to be${{\mathcal N}{\mathcal P}}$-hard.

A practical method for the exact separation of combs would likely have a dramatic impact on our ability to solve large-scale instances of the TSP. The...

11. Chapter Eight Combs from Consecutive Ones
(pp. 199-220)

In this chapter we present a heuristic separation algorithm for 3-toothed comb inequalities, exploiting the relationship between these inequalities and the following notion:

A family${{\mathcal F}}$of subsets of a set W is said to have the consecutive ones property if W can be endowed with a linear orderso that each S in${{\mathcal F}}$has the form{yW:ayb}with a,bW.

To describe this relationship, let us select once and for all a city denoted ext for “exterior” and let us writeVintforV− {ext}.

If${{\mathcal F}}$is...

12. Chapter Nine Combs from Dominoes
(pp. 221-240)

The subject of this chapter is a heuristic separation algorithm for comb inequalities,$\eta (H,x)+\sum\limits_{j=1}^{t}{\eta ({{T}_{j}}},x)\ge t-1. \caption {(9.1)}$

Earlier we gave a combinatorial proof that (9.1) is satisfied by all toursxthroughV. The algorithm of the present chapter was motivated by an algebraic proof of the same fact.

The algebraic proof proceeds in two stages in the spirit of the framework for describing Gomory cuts that has been propounded by Chvátal [125]. In the first stage, we show that every nonnegativexwhich satisfies all subtour inequalities must satisfy the inequality$\eta (H,x)+\sum\limits_{j=1}^{t}{\eta ({T_j,x})\ge t-2;} \caption {(9.2)}$the second stage amounts to observing that every tourx...

13. Chapter Ten Cut Metamorphoses
(pp. 241-270)

Watching a run of the cutting-plane method is like viewing a tug-of-war between the LP solver and the cut finder. When a cutting plane is added to the LP relaxation, the solver often reacts by shifting the defect inxprohibited by the cut to an area just beyond the cut’s control. This may be one explanation for the footnote in the Dantzig et al. [150] technical report remarking that E. W. Paxson called their procedure the “finger in the dike” method.

One way to deal with the shifting LP solution is to respond to each slight adjustment ofx...

14. Chapter Eleven Local Cuts
(pp. 271-344)

The template paradigm of identifying inequalities having a prescribed structure, as outlined in Section 5.6, has been a dominant theme in TSP computation since the work of Grötschel and Padberg [237] on combs. This work has been guided by research into classes of facet-inducing inequalities for the convex hull of tours through the set of citiesV. In this chapter we present a separation method that disdains all understanding of the TSP polytope and bashes on regardless of all prescribed templates.

The cut-finding technique described in this chapter is an examination of a number of mappingsσfromVonto...

15. Chapter Twelve Managing the Linear Programming Problems
(pp. 345-372)

The solution of linear programming problems is a crucial part of the Dantzig-Fulkerson-Johnson approach to the TSP. Although the individual LPs are often not particularly difficult to solve, the great number of them that arise make the time spent in the LP solver a dominant part of a TSP computation.

In this chapter we describe machinery to handle the flow of cuts and edges into and out of the LP relaxations, as well methods used for interacting with an LP solver. The actual solution of the LPs will be described in Chapter 13.

The LPs that need to be solved...

16. Chapter Thirteen The Linear Programming Solver
(pp. 373-410)

The computational histories of the TSP and linear programming (LP) are closely related. Central parts of the computational procedure of Dantzig et al. [151] depend in a fundamental way on our ability to solve LP problems, usually sequences of closely related LP problems. Conversely, developments of codes for LP have also been strongly influenced by the TSP. For example, the work described in this monograph has been the direct source of many of the important ideas behind the development of the callable library routines for the CPLEX commercial LP solver.

It is interesting at this point to note a statement...

17. Chapter Fourteen Branching
(pp. 411-424)

The branch-and-cut algorithm embeds the cutting-plane method within a branch-and-bound search. An outline of the general framework is given in Section 5.7. In this chapter we describe details of its specialization to the TSP.

The first computer implementation of the branch-and-cut algorithm is due to Hong [270]. Branching is carried out in his work by selecting a fractional variable$x_{e}^{*}$from the LP solutionx, and creating child subproblems with the additional constraintsxe= 0 andxe= 1. The choice of the variablexeis guided by Hong’s implementation of the dual simplex algorithm. Concerning this rule, Hong...

18. Chapter Fifteen Tour Finding
(pp. 425-488)

The study of heuristic algorithms for the TSP is a popular topic, having a large and growing literature devoted to its various aspects. Our treatment is restricted to tour finding that is applicable to solution methods for the TSP, namely, finding near-optimal tours within a reasonable amount of computing time. Other topics, such as finding tours very quickly, can be found in Bentley [56], Johnson and McGeoch [291], and Reinelt [475]; an important theoretical result concerning the approximate solution of Euclidean instances of the TSP can be found in Arora [24].

At the heart of the most successful tour-finding approaches...

19. Chapter Sixteen Computation
(pp. 489-530)

The algorithmic components described thus far in the book are brought together in the Concorde computer code for the TSP. The design of Concorde is focused on the solution of larger, more difficult, instances of the problem, rather than, say, concentrating on the fastest solution of small examples. This decision is based on our belief that by attempting to extend the reach of TSP solvers we can best continue the success of the problem as an engine of discovery, as well as making a direct contribution to the long-term research of the TSP itself.

In this chapter we describe Concorde...

20. Chapter Seventeen The Road Goes On
(pp. 531-540)

The solution of the TSPLIB test problems is by no means the end of the journey for the TSP. The general model will no doubt continue to inspire researchers and productively serve as an engine of discovery in applied mathematics. In this chapter we give a short survey of recent work by various research groups, aimed at pushing the limits of TSP computation.

The quality of the root LP bounds obtained by Concorde is very high, but the solution process for the largest instances involves the repeated application of branch-and-cut runs to gather cuts at great computational expense. This points...

21. Bibliography
(pp. 541-582)
22. Index
(pp. 583-593)