G. B. Dantzig, D. R. Fulkerson, and S. M. Johnson, Solution of a Large-scale Traveling Salesman Problem, Operations Research 2, 393--410, 1954 10  Home/Search   Document Not in Database   Summary   Related Articles   Check

....value 0. In order to prove that the solution of the restricted LP is optimal for the entire set of variables one has to check the reduced costs of the ignored variables. This step is called pricing. 2 Comparison to Extant Work The cutting plane procedure was invented in 1954 by Dantzig et al. DFJ54] for the traveling salesman problem. The branch and cut paradigm was rst formulated for and successfully applied to the linear ordering problem by Gr otschel et al. in 1984 [GJR84] The rst state of the art BCP algorithm was developed in 1991 by Padberg and Rinaldi [PR91] Since then, the ....

G.B. Dantzig, D.R. Fulkerson, and S.M. Johnson. Solution of a large scale traveling salesman problem. Operations Research, 2:393-410, 1954.

....by acroread (Acrobat Reader) For more information on L A T E Xsee http: www.latex project.org and http: www.tug.org. Pdflatex is a version of L A T E Xthat produces PDF files. The problem is here to solve the symmetric Traveling Salesman Problem (TSP) using a famous data set from Dantzig e.a. [3]. The GAMS model for this problem is: eolcom This model solves a Symmetric TSP using a simple algorithm that adds cuts to exclude subtours found in the previous solution. The data set is dantzig42 from TSPLIB. Reference: Dantzig, Fulkerson, Johnson, Solution of a ....

....using a Java Stored Procedure. The relevant code may look like: call gams String[ cmdArray = new String[5] Figure 5. GAMS ORAQUEUE Model Upload Facility cmdArray[0] C: Program Files GAMS 20.5 gams.exe ; cmdArray[1] D: TMP trnsport.gms ; cmdArray[2] WDIR=D: TMP ; cmdArray[3] = SCRDIR=D: TMP ; cmdArray[4] LO=2 ; Process p = Runtime.getRuntime( exec(cmdArray) p.waitFor( 10. SPAWNING GAMS FROM A WEB SERVER 105 Figure 6. GAMS ORAQUEUE Job Table Page 10. Spawning GAMS from a Web Server Running GAMS remotely using a Web based thin client architecture requires ....

G. Dantzig, R. Fulkerson, and S. Johnson, Solution of a large-scale traveling-salesman problem, Operations Research 2 (1954), 393--410.

....Travelling Salesman Problem (TSP) In this problem, a salesman has a list of cities that he must visit in some order and then return to the city from which he started. The distance between each pair of cities is known, and the objective is to #nd the shortest tour. Dantzig, Fulkerson and Johnson [28] solved a TSP instance with 49 cities using the simplex method and an idea based on cutting planes. In spite of this partial success, the TSP seemed harder than the other combinatorial optimization problems considered. New methods, especially cutting planes and a technique called branch and bound, ....

George B. Dantzig, D. Ray Fulkerson, and Selmer M. Johnson. Solution of a large-scale traveling-salesman problem. Operations Research, 2:393#410, 1954.

....value 0. In order to prove that the solution of the restricted LP is optimal for the entire set of variables one has to check the reduced costs of the ignored variables. This step is called pricing. 2 Comparison to Extant Work The cutting plane procedure was invented in 1954 by Dantzig et al. DFJ54] for the traveling salesman problem. The branch and cut paradigm was first formulated for and successfully applied to the linear ordering problem by Grotschel et al. in 1984 [GJR84] The first state of the art BCP algorithm was developed in 1991 by Padberg and Rinaldi [PR91] Since then, the ....

G.B. Dantzig, D.R. Fulkerson, and S.M. Johnson. Solution of a large scale traveling salesman problem. Operations Research, 2:393--410, 1954.

....of the TSP and adding linear inequalities that cut off undesirable parts of the polytope until the optimum solution to the relaxed problem is integral. One set of inequalities that has been very useful is subtour elimination constraints, first introduced by Dantzig, Fulkerson, and Johnson [15]. The problem of identifying a subtour elimination constraint can be rephrased as the problem of finding a minimum cut in a graph with real valued edge weights. Thus, cutting plane algorithms for the traveling salesman problem must solve a large number of minimum cut problems (see [43] for a ....

G. B. Dantzig, D. R. Fulkerson, and S. M. Johnson. Solution of a Large-Scale Traveling Salesman Problem. Oper. Res., 2:393--410, 1954.

....It will be convenient to let ck be the cost of 1 tree Tk, di be the degree of node i in T, and vi = di 2. So then nn[ck Thus we can express max w(c) as Interestingly enough, 1.1) was the basis of one of the earbest attempts to grapple with the TSP. Dantzig, Fulkerson, and Johnson [5] started with the node degree constraints xij = 2 and added subtour elimination constraints as necessary. CHAPTER 1. INTRODUCTION subject to: w k q L1 7riVik, Vk 1, t. 1.2) The dual of this linear program will also be an equivalent formulation of the heuristic. Taking the dual of (1.2) ....

....between points i and j, since the Euclidean metric is symmetric and obeys the triangle inequality. We will call this subcase the Euclidean TSP. One of the earliest papers on solving the TSP dealt with the Euclidean TSP, finding a tour through cities of the 48 continental states of the U.S. [5]. The Euclidean TSP is A P complete [29] 2.1 Planarity of Solutions We begin this chapter by showing that for any instance of the TSP 1 which has an embedding of its nodes in the plane that obeys certain properties, the Held Karp heuristic has an optimal solution that is planar. We will then ....

G. Dantzig, R. Fulkerson, S. Johnson (1954). Solution of a large-scale traveling- salesman problem. Oper. Res. 2, 393-410.

....of the TSP and adding linear inequalities that cut off undesirable parts of the polytope until the optimum solution to the relaxed problem is integral. One set of inequalities that has been very useful is subtour elimination constraints, first introduced by Dantzig, Fulkerson, and Johnson [15]. The problem of identifying a subtour elimination constraint can be rephrased as the problem of finding a minimum cut in a graph with real valued edge weights. Thus, cutting plane algorithms for the traveling salesman problem must solve a large number of minimum cut problems (see [43] for a ....

G. B. Dantzig, D. R. Fulkerson, and S. M. Johnson. Solution of a Large-Scale Traveling Salesman Problem. Oper. Res., 2:393--410, 1954.

.... subtour note that the loop over tt is just syntax: tt contains one element loop(tt, tour(tt,reach) yes; s(reach) no; reach(i,j) no; if no remaining solutions, we are done if (card(s) 0, done=1) new subtour tt(t) tt(t 1) display tour; Consider the data set from [1] which is electronically available from TSPLIB (see [4] It consists of 42 cities in the U.S. The model solves in about 10 cycles (it depends a bit on which solver is used and what options as there are multiple solutions the solver can choose from during the process) In each cycle a handful of ....

Dantzig, G., Fulkerson, R., Johnson, S., Solution of a large-scale TravelingSalesman Problem, Operations Research, 2, 1954, pp.393--410.

....for Integer Linear Programming (ILP) In recent decades, a considerable research e ort has been 1 devoted to the de nition of e ective separation procedures for families of well structured cuts. This line of research was originated by the pioneering work of Dantzig, Fulkerson and Johnson [12] on the Traveling Salesman Problem (TSP) and led to the very successful branch and cut approach introduced by Padberg and Rinaldi [28] Most of the known methods have been originally proposed for the TSP, a prototype in combinatorial optimization and integer programming. In spite of the large ....

G. Dantzig, D. Fulkerson, S. Johnson (1954). Solution of a large scale travelingsalesman problem. Oper. Res. 2, 393-410.

....practical insight achieved in the study of TSP can often be useful in the solution of other problems in this area. In fact, much progress in combinatorial optimization can be traced back to research on TSP. The now well known computing method, branch and bound, was first used in the context of TSP [3, 4]. It is also worth mentioning that research on TSP was an important driving force in the development of the computational complexity theory in the beginning of the 1970s [5] However, the interest in TSP not only stems from its practical and theoretical importance. The intellectual challenge of ....

G. B. Dantzig, D. R. Fulkerson & S. M. Johnson, Solution of a large-scale traveling-salesman problem, Oper. Res., 2 , 393-410 (1954).

....algorithm in pseudo code: Algorithm 4.1.2 k : 1 t : 1 visited : # tour(i) # #i for i : 1. n do choose j such that x kj = 1 tour(t) tour(t) x kj visited : visited k k : j if j # visited then t : t 1 choose k ## visited fi od Consider the data set from [4] which is electronically available from TSPLIB (see [12] It consists of 42 cities in the U.S. When translate the above algorithm in GAMS: find and display tours set t tours t1 t17 ; abort (card(t) card(i) Set t is possibly too small parameter tour(i,j,t) subtours; sets from(i) ....

Dantzig, G., Fulkerson, R., Johnson, S., Solution of a large-scale TravelingSalesman Problem, Operations Research, 2, 1954, pp.393--410.

....of all prescribed templates. Combining this technique with the traditional template approach was a crucial step in our solutions of a 13,509 city TSP instance and a 15,112 city TSP instance. 1 The Cutting Plane Method and Its Descendants The groundbreaking work of Dantzig, Fulkerson, and Johnson [19] on the traveling salesman problem introduced the cutting plane method , which can be used to attack any problem minimize c T x subject to x 2 S; 1) where S is a finite subset of some Euclidean space IR m , provided that an efficient algorithm to recognize points of S is available. This ....

Dantzig, G., Fulkerson, R., Johnson, S.: Solution of a large-scale traveling salesman problem. Operations Research 2, 393--410, 1954.

....cutting planes for separation purposes. Such cuts are generated algebraically, i.e. they ignore the combinatorial structure of the problem. Valid inequalities implied by integrality constraints were first introduced in a study of the traveling salesman problem by Dantzig, Fulkerson Johnson [2, 3]. # Supported by a Gerhard Hess Forschungsforderpreis (WE 1462 2 2) of the German Science Foundation (DFG) awarded to R. Weismantel. # Supported by grants FKZ 0037KD0099 and FKZ 2495A 0028G of the Kultusministerium of Sachsen Anhalt. 1 The theory of general cutting planes is due to the ....

G. B. Dantzig, D. R. Fulkerson, and S. M. Johnson, Solution of a Large-scale Traveling Salesman Problem, Operations Research 2, 393--410, 1954 10

....to the TSP than does one pass of the F M heuristic for the VLSI NPP. However, the Croes algorithm is more computationally expensive than one pass of F M. Experimental data used for evaluating the TSSA TSP system consists of the following: the 42 city instance of Dantzig, Fulkerson, and Johnson [21]; a randomly generated 50 city instance; c i ( d j j 1 , j 1 = n 1 d n 1 , 66 the 57 city instance of Karg and Thompson [55] a randomly generated 100 city instance; the 318city instance of Lin and Kernighan [39] and the 532 city instance of Padberg and Rinaldi [78] The results ....

G.B. Dantzig, D.R. Fulkerson, and S.M. Johnson, "Solution of a Large Scale TravelingSalesman Problem," Operations Research, vol. 2, 393-410, 1954.

....i.e. y(V ) 1. Constraints (2) are called Generalized subtour elimination constraints (GSECs) PRIZE COLLECTING STEINER PROBLEM IN GRAPHS 3 They guarantee that the solution is cycle free. GSECs generalize subtour elimination constraints (SECs) introduced by Dantzig, Fulkerson, and Johnson [6] for the traveling salesman problem) Notice that if y s = 1, for all s # S, then (2) reduces to a SEC. The set of feasible solutions for (5) corresponds to the set of all trees of G. The above formulation can be seen as a generalization of the spanning tree polytope [8] by noting that if the ....

G.B. Dantzig, D.R. Fulkerson, and S.M. Johnson. Solution of a large scale traveling salesman problem. Operations Research, 2:393--410, 1954.

....Croes algorithm are on average 10 closer to E in terms of standard deviation units s than final SA solutions. Experimental data used for evaluating the TSSA TSP system consists of the following instances: the 20 city problem of Croes [5] the 42 city problem of Dantzig, Fulkerson, and Johnson [6]; a randomly generated 50 city problem; the 57 city problem of Karg and Thompson [18] a randomly generated 100 city problem; and the 318 city problem of Lin and Kernighan [22] The results are given in Tables 8 and 9. Again, significant speedup is noted over standard SA with no loss in solution ....

G.B. Dantzig, D.R. Fulkerson, and S.M. Johnson, "Solution of a Large Scale Traveling -Salesman Problem," Operations Research, vol. 2, 393-410, 1954.

.... Following World War II, OR methods were used to analyze a a broad spectrum of civilian problems, such as determining an ideal schedule for maintenance of a fleet of airplanes, purchasing an optimal number of ambulances for a county, or minimizing the number of oil tankers to meet a fixed schedule (Dantzig and Fulkerson, 1954). Investigators in the OR community developed a set of numerically intensive methods, often referred to as OR techniques, for modeling and solving optimization and scheduling problems (Dantzig 1963) These OR techniques included linear programming, nonlinear programming, integer programming, ....

....on each subproblem recursively, until we reach goal states or cannot apply any more operators. The subgoals in such a search tree define the search space for a theorem proving problem. 9 Consider, as another application, the classic search problem, called the traveling salesperson problem (TSP) (Dantzig et al. 1954; Christofides 1979; Lawler et al. 1985) The TSP requires us to identify the shortest path connecting a set of cities each of which the salesperson must visit once before returning to a home base. In generating the search tree for the TSP, we apply a MOVE operator to generate all possible trips ....

:393-410.

....linear program is known as the subtour polytope S n , the program itself as Held Karp relaxation: Any of the constraints corresponding to a moat variable is a so called subtour elimination constraint. These subtour elimination constraints were first introduced by Dantzig, Fulkerson and Johnson [6]. Grotschel and Padberg showed that they are facet inducing for n 4. Grotschel, Lov asz and Schrijver [14] and Karp and Papadimitriou [17] showed that a polynomial method for solving the separation problem for a polytope yields a polynomial method for optimization by means of the ellipsoid ....

G.B.Dantzig, D.R.Fulkerson, S.M.Johnson, Solutions of a large-scale traveling salesman problem, Operations Research, 2 (1954), 393--410.

.... 2 for all W V with ; 6= W 6= V; 30) 0 x e 1 for all e 2 E: The integral vectors in P are the incidence vectors of Hamiltonian circuits in G; the problem of maximizing a linear function over this set of integral vectors is the traveling salesman problem (TSP) Dantzig, Fulkerson, and Johnson [5] introduced P as a relaxation of the TSP and developed the cutting plane method for optimizing over P I . The most successful algorithms for solving large TSP instances all adopt the Dantzig, Fulkerson, and Johnson approach (see Junger, Reinelt, and Rinaldi [12] for a survey of this work) ....

G. B. Dantzig, R. Fulkerson, and S. M. Johnson. Solution of a large-scale traveling salesman problem, Operations Research 2 (1954) 393--410.

....by at least k vehicles, where k is an integer that is no larger than k(S) the least number of vehicles needed to serve each customer in S. These inequalities are a natural generalization of the subtour cuts introduced for the traveling salesman problem (TSP) by Dantzig, Fulkerson, and Johnson [12]; they were studied in the context of vehicle routing by Laporte, Nobert, and Desrochers [29] and dubbed k path cuts by Kohl, Desrosiers, Madsen, Solomon, and Soumis [26] in their work on the VRPTW. Given a nonempty set S C, let ffi(S) denote the set of outgoing arcs from S, that is ffi(S) ....

G. B. Dantzig, R. Fulkerson, and S. M. Johnson, Solution of a large-scale traveling salesman problem. Operations Research 2 (1954) 393--410.

....speed over that same period these codes represent an approximate four orders of magnitude improvement in our ability to solve linear programming problems. The second major development is in so called cutting plane technology. Motivated by work of Dantzig, Johnson and Fulkerson in the 50 s [12], Padberg, Groetschel and others have shown how cutting plane techniques could be used to strengthen the linear programming relaxations of many pure 0 1 integer programming problems [25] The strengthening is effected by studying the facets of the underlying polytope generated by the convex hull ....

G. B. Dantzig, D. R. Fulkerson, and S. M. Johnson. Solution of a large scale traveling salesman problem. Operations Research, 7:58--66, 1954.

....for Integer Linear Programming (ILP) In recent decades, a considerable research effort has been devoted to the definition of effective separation procedures for families of well structured cuts. This line of research was originated by the pioneering work of Dantzig, Fulkerson and Johnson [11] on the Traveling Salesman Problem (TSP) and led to the very successful branch and cut approach introduced by Padberg and Rinaldi [22] Most of the known methods have been originally proposed for the TSP, which acted as a prototype problem in combinatorial optimization and integer programming. In ....

G. Dantzig, D. Fulkerson, S. Johnson (1954). Solution of a large scale traveling-salesman problem. Oper. Res. 2, 393--410.

....in cutting plane based techniques for ILPs. In recent decades, a considerable research effort has been devoted to the definition of effective separation procedures for families of well structured cuts. This line of research was originated by the pioneering work of Dantzig, Fulkerson and Johnson [26] on the traveling salesman problem and led to the very successful branch and cut approach introduced by Padberg and Rinaldi [48] In spite of the large research effort, however, polynomial time exact separation procedures are known for only a few classes of cuts. In this section, we outline the ....

G. Dantzig, D. Fulkerson and S. Johnson, "Solution of a large scale traveling-salesman problem" Operations Research 2 (1954) 393--410.

....inequalities. 0 x 1 (1) x(ffi(v) 2 (2) These latter equalities are called degree constraints. Not all integer solutions to this system are tours: two or more disconnected cycles covering the vertex set al..so obey these constraints. To eliminate these solutions, Dantzig, Fulkerson, and Johnson [9] introduced subtour elimination constraints. Subtour elimination constraints or subtour constraints are among the simplest facet defining inequalities, and are of the form x(ffi(A) 2; 8A ae V; 2 jAj n Gamma 2: 3) With these added constraints, all integer solutions are tours. There are ....

G. B. Dantzig, D. R. Fulkerson, and S. M. Johnson. Solution of a large-scale traveling-salesman problem. Operations Research, 2:393--410, 1954.

....speed over that same period these codes represent an approximate four orders of magnitude improvement in our ability to solve linear programming problems. The second major development is in so called cutting plane technology. Motivated by work of Dantzig, Johnson and Fulkerson in the 50 s [DFJ54] Padberg, Groetschel and others have shown how cutting plane techniques could be used to strengthen the linear programming relaxations of many 0 1 integer programming problems [PR91] The strengthening is effected by studying the facets of the underlying polytope generated by the convex hull of ....

G. B. Dantzig, D. R. Fulkerson, and S. M. Johnson. Solution of a large scale traveling salesman problem. Operations Research, 7:58--66, 1954.

....speed over that same period these codes represent an approximate four orders of magnitude improvement in our ability to solve linear programming problems. The second major development is in so called cutting plane technology. Motivated by work of Dantzig, Johnson and Fulkerson in the 50 s [DFJ54] Padberg, Groetschel and others have shown how cutting plane techniques could be used to strengthen the linear programming relaxations of many pure 0 1 integer programming problems [GH91, PR91, HP92] The strengthening is effected by studying the facets of the underlying polytope generated by ....

G. B. Dantzig, D. R. Fulkerson, and S. M. Johnson. Solution of a large scale traveling salesman problem. Operations Research, 7:58--66, 1954.

....about the TSP, we suggest a recent text, JM97] It discusses several optimization methods applied to the TSP, including genetic algorithms, and can be used as backlink to earlier references. For remarks on the origins of the TSP see chapter 10 in [Mic96] and the footnote at the beginning of [DFJ54] The latter text is also the earliest reference we found; it is shown that a tour across 49 US cities has the shortest road distance . This work is divided into two main parts: section 2 informally describes the TSP and GAs, section 3 successively presents the conversion to their RelFun ....

G. Dantzig, R. Fulkerson, and S. Johnson. Solution of a Large Scale Traveling Salesman Problem. Operations Research, 2:393--410, 1954.

.... previous papers [Kar97b] exploits an extremely tight connection between minimum cuts and network reliability) Minimum cut computations are used to find the subtour elimination constraints that are needed in the implementation of cutting plane algorithms for solving the traveling salesman problem [DFJ54, LLKS85]. Padberg and Rinaldi [PR90] and Applegate [App92] have reported that solving min cut problems was the computational bottleneck in their state of the art cuttingplane based TSP algorithm, as well as other cutting plane based algorithms for combinatorial problems whose solutions induce connected ....

G. B. Dantzig, D. R. Fulkerson, and S. M. Johnson. Solution of a large-scale traveling salesman problem. Operations Research, 2:393--410, 1954.

....this perspective, arguing the difficulty of the TSP by harping on the large number of tours is not entirely convincing: the number of spanning trees in a complete graph is much larger than the number of tours. A breakthrough came in 1954, when George Dantzig, Ray Fulkerson, and Selmer Johnson [7] published a description of a method for solving the TSP and illustrated the power of this method by solving an instance with 49 cities, an impressive size at that time. Riding the wave of excitement over the numerous applications of the simplex method (designed by George Dantzig in 1947) they ....

....linear inequalities satisfied by all x in S and say in a footnote, We are indebted to I. Glicksberg of Rand for pointing out relations of this kind to us . These two inequalities read x(f15; 16; 18; 19g) 2x f14;15g x f16;17g x f19;20g 6 (1. 6) actually, this constraint is presented in [7] as our (1.6) minus the sum of the three equations x(fvg; V 0 fvg) 2 with v = 15; 16; 19) and X a e x e 42 (1.7) with a f22;23g = 2 and a e = 1 for all other e except that a e = 0 when (i) e = f25; 26g, or (ii) x 3 e = 0 and je f10; 11; 28gj = 1, or (iii) x 3 e = 0 and je ....

G. B. Dantzig, R. Fulkerson, and S. M. Johnson, "Solution of a large-scale traveling salesman problem", Oper. Res. 2 (1954), 393--410.

....formulation of TSP and adding linear inequalities that cut off undesirable parts of the polytope until the optimum solution to the relaxed problem is integral. One set of inequalities that has been very useful is subtour elimination constraints, first introduced by Dantzig, Fulkerson and Johnson [12]. The problem of identifying a subtour elimination constraint can be rephrased as the problem of finding a minimum cut in a graph with real valued edge weights. Thus, cutting plane algorithms for the traveling salesman problem must solve a large number of minimum cut problems (see [31] for a ....

G. B. Dantzig, D. R. Fulkerson, and S. M. Johnson. Solution of a Large-Scale Traveling Salesman Problem. Oper. Res., 2:393--410, 1954.

....the mathematician Merrill Flood popularized it among his colleagues at the RAND Corporation. Supported by ONR Grant N00014 98 1 0014. Eventually, the TSP gained notoriety as the prototype of a hard problem in combinatorial optimization. A breakthrough came when Dantzig, Fulkerson, and Johnson [1954] published a description of a method for solving the TSP and illustrated the power of this method by solving an instance with 49 cities, an impressive size at that time. Riding the wave of excitement over the numerous applications of the simplex method (designed by Dantzig in 1947) and following ....

G.B. Dantzig, R. Fulkerson, and S.M. Johnson, "Solution of a large-scale traveling salesman problem", Operations Research 2 (1954) 393--410.

....largely due to connections with combinatorial problems such as the assignment problem (AP) the transportation problem (TP) and other LP problems, seemingly similar to these problems but harder to solve, presenting researchers with a challenge. The seminal paper on the TSP by Dantzig et al. [6] in 1954 was of great importance in the history of combinatorial optimisation. Occurring at the same time as the explosion of interest in LP, TSP resisted being solved by these new methods, even with the development of some of the first practical computers 12 . To understand their paper some ....

....are required. With these extra constraints vertices of the new polyhedron R are no longer guaranteed to give integer results, so we need to ensure variables only take the values 0 or 1. Thus the TSP becomes an IP problem, a class that had no solution method in the early 1950s. Dantzig et al. [6] suggested a new method to overcome these difficulties, essentially a branch and bound algorithm. They proposed that starting from an optimal or near optimal tour it might be possible to prove optimality by invoking only a few extra inequalities (cuts) Using this technique to find the optimal ....

[Article contains additional citation context not shown here]

Dantzig, Fulkerson, and Johnson. Solution of a Large--Scale Traveling-- Salesman Problem. Journal of the Operations Research Society of America, 1954.

....solve linear programming problems. Further algorithmic improvements are expected in linear programming, in preprocessing, and in branchand bound heuristics [5] The second major development is in so called cutting plane technology. Motivated by work of Dantzig, Johnson and Fulkerson in the 50 s [6], Padberg, Groetschel and others have shown how cutting plane techniques could be used to strengthen the linear programming relaxations of many 0 1 integer programming problems [7] The third major area of improvement has come in the application of parallel processing to handle the branching when ....

G. B. Dantzig, D. R. Fulkerson, and S. M. Johnson. Solution of a large scale traveling salesman problem. Operations Research, 7:58--66, 1954.

....formulation of TSP and adding linear inequalities that cut off undesirable parts of the polytope until the optimum solution to the relaxed problem is integral. The inequalities that have been very useful are subtour elimination constraints, first introduced by Dantzig, Fulkerson and Johnson [12]. The problem of identifying a subtour elimination constraint can be rephrased as the problem of finding a minimum cut in a graph with real valued edge weights. Thus, cutting plane algorithms for the traveling salesman problem must solve a large number of minimum cut problems (see [29] for a ....

G. B. Dantzig, D. R. Fulkerson, and S. M. Johnson. Solution of a Large-Scale Traveling Salesman Problem. Oper. Res., 2:393--410, 1954.

.... on a linear formulation of the problem, and perform linear optimization and cutting plane generation within a branch and bound scheme [16] Linear programming approaches have made great progress in the past decades, from the first 49 city example solved in 1954 by Dantzig, Fulkerson and Johnson [8] and the 64 city example of Held and Karp [12] in 1970 to the 120 city example of Gr tschel in 1980 [11] and to the large problems solved by Padberg and Rinaldi [16] 532 cities in 1987 and 2392 in 1991) 4.2. Improved bounding As mentioned in the historical section above, Held and Karp ....

G.B. Dantzig, D.R. Fulkerson, S.M. Johnson. Solution of a Large-Scale Traveling Salesman Problem, Operations Research 2, 1954

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G. B. Dantzig, D. R. Fulkerson, and S. M. Johnson, Solution of a Large-scale Traveling Salesman Problem, Operations Research 2, 393--410, 1954 10

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Dantzig, Fulkerson, and Johnson. Solution of a Large--Scale Traveling-- Salesman Problem. Journal of the Operations Research Society of America, 1954.

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G. B. Dantzig, D. R. Fulkerson, and S. M. Johnson, "Solution of a Large-Scale Traveling Salesman Problem," Operations Research, vol. 2, pp. 393--410, 1954.

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G.B. Dantzig, R. Fulkerson, and S.M. Johnson, "Solution of a large-scale traveling salesman problem", Operations Research 2 (1954) 393--410.

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G. B. Dantzig, D. R. Fulkerson, and S. M. Johnson. Solution of a Large-Scale Traveling Salesman Problem. Oper. Res., 2:393--410, 1954.

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Dantzig, G. B., R. Fulkerson, S. M. Johnson. 1954. Solution of a large-scale traveling salesman problem. Operations Research 2 393--410.

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Dantzig, G.B., R. Fulkerson and S.M. Johnson (1954). Solution of a LargeScale Traveling Salesman Problem. Operations Research 2, 393-410.

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--509. G. B. Dantzig, R. Fulkerson, and S. M. Johnson (1954). "Solution of a large-scale traveling salesman problem", Operations Research 2,

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George B. Dantzig, D. Ray Fulkerson, and Selmer M. Johnson, `Solution of a large scale traveling salesman problem', Operations Research, 2, 393--410 (1954).

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G.B. Dantzig, D.R. Fulkerson and S.M. Johnson (1954) "Solution of a large-scale travelingsalesman problem", Operations Research 2, 393--410.

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G.B. Dantzig, D.R. Fulkerson and S.M. Johnson (1954) "Solution of a large-scale travelingsalesman problem", Operations Research 2, 393--410.

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G.B. Dantzig, D.R. Fulkerson and S.M. Johnson (1954) "Solution of a large-scale traveling-salesman problem", Operations Research 2, 393--410.

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G.B. Dantzig, D.R. Fulkerson and S.M. Johnson (1954), Solution of a large-scale traveling salesman problem, Operations Research 2, 393--410.

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