E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, D. B. Shmoys. The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(3) The objective is to nd a new ordering of the test set that minimizes the sum of the Hamming distances between consecutive vectors so that the size of the blocks can likely be increased. Reordering can be mapped to the problem of nding a minimal tour for the Traveling Salesman Problem (TSP) [12]. To perform the mapping between compression by reordering and the TSP, each vector in the set is associated to a node in the graph; the distance between each pair of nodes is given by the weight of the arc connecting the nodes (i.e. the weight is their Hamming distance) Then an utilitity ....

E.L. Lawler, A. H. Rinnooy-Kan, \The traveling salesman problem : a guided tour of combinatorial optimization," John Wiley & Sons, 1985.

....start. They can be generated as needed by a separation algorithm: one can start with the formulation (2.1) then generate subtour inequalities that are violated by the current LP solution. The separation algorithm for subtour constraints is based on network ow techniques; for further details, see [4]. 3. The comparison. 3.1. The strength of the two formulations. Solving a reasonably large (with at least, say 50 cities) problem to optimality is only possible using the subtour formulation at least we are not aware of any published computational studies that use the pure MTZ formulation. ....

E. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shmoys eds. The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, Wiley, 1985

....in section 4 which is followed by the conclusion. 2 Conventional Approach The rst GA approach on TSP was by Brady [1] which was then followed by Grefenstette et al. 4] Goldberg and Linge [6] and Oliver et al. For a detailed discussion on TSP a good reference is the study of Lawler et al. [11]. A perfect review article on GA for TSP is by Larranaga et al. 10] In the conventional approach a chromosome which is devised to represent a solution constitutes of N (count of the cities) genes. Each gene holds a number which is a label of a city. So the n th gene holds the label of the city ....

Lawler, E.L., J.K. Lenstra, A.H.G. Rinnooy Kan, and D.B. Shmoys (Eds.). The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, Chichester, 1985.

....set of constraints. The best known problem of this type is the classical Travelling Salesman Problem (TSP) calling for a minimum cost Hamiltonian cycle on a given graph. This problem has been extensively studied in the last decades; see the book edited by Lawler, Lenstra, Rinnooy Kan, and Shmoys [10] for a comprehensive cover of the literature up to 1985. In several applications a feasible cycle is allowed to visit only a subset of the nodes of the graph, chosen according to some speci ed criterion. For example, in the Prize Collecting TSP each node has an associated prize, and a salesman ....

E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, D.B. Shmoys (Editors), The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, Wiley, 1985.

....a large number of cities. Iterative improvement algorithms can find a good approximation to an optimal solution, and naturally yield an interruptible anytime algorithm. The anytime traveling salesman algorithm is a randomized algorithm that repeatedly tries to perform a tour improvement step [20, 22]. In the general case of tour improvement procedures, r edges in a feasible tour are exchanged for r edges not in that solution as long as the result remains a tour and the cost of that tour is less than the cost of the previous tour. The simplest case is when r = 2. Figure 5 demonstrates one step ....

E. L. Lawler et al., eds., The traveling salesman problem: a guided tour of combinatorial optimization, New York (Wiley, 1987).

....with constant velocities. We propose approximate and exact algorithms for several natural variants of Moving Target TSP. 1 Introduction The classical Traveling Salesman Problem (TSP) has been studied extensively, and many TSP heuristics have been proposed over the years (see surveys such as [8] and [11] Previous works on TSP have assumed that the cities targets to be visited are stationary. However, several practical scenarios give rise to TSP instances where the targets to be visited are themselves in motion (e.g. when a supply ship resupplies patrolling boats, or when an aircraft ....

....as well as its quadratic runtime. 2. 2 Heuristics for Few Moving Targets In this subsection, we consider Moving Target TSP when only some of the targets are moving (while the majority are stationary) From among the many existing approximation algorithms for classical (stationary) TSP [8], choose one such heuristic, having performance bound ff. Using this algorithm for stationary targets, we now show how to construct an efficient algorithm having a performance bound of 1 ff when the number of moving targets is sufficiently small. Theorem 6 Moving Target TSP where at most O( ....

E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy, and D. B. Shmoys, The Traveling Salesman Problem: a Guided Tour of Combinatorial Optimization, John Wiley and Sons, Chichester, New York, 1985.

....algorithm a b c d f e a b c d f e (b) b c d e f Figure 4.3: The operation of randomized tour improvement is described below. The anytime traveling salesman algorithm is a randomized algorithm that repeatedly tries to perform a tour improvement step [Lin and Kernighan, 1973; Lawler et al. 1987] In the general case of tour improvement procedures, edges in a feasible tour are exchanged for edges not in that solution as long as the result remains a tour and the cost of that tour is less than the cost of the previous tour. I have implemented the algorithm for the case where . ....

E. L. Lawler et al. (eds.). The traveling salesman problem: a guided tour of combinatorial optimization. New York: Wiley, 1987.

....The Minimum Spanning Tree Problem (MSTP) see e.g. Magnanti and Wolsey [45] The MSTP is to determine a minimum cost tree on G that includes all the vertices of V . This problem is polynomially solvable. 2. The Traveling Salesman Problem (TSP) see e.g. Lawler, Lenstra, Rinnooy Kan and Shmoys [42]) The TSP consists of finding a minimum cost cycle that passes through each vertex exactly once. This problem is NP hard. 3. The Minimum Perfect Matching Problem (MPMP) see e.g. Cook, Cunningham, Pulleyblank and Schrijver [8] A matching M E is a subset of edges such that each vertex of M ....

E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, D.B. Shmoys (Editors). The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, New York (1985).

....assumed to be known by the salesman. Distance can be replaced by another notion, such as time or money. In the following the term cost is used to represent any such notion. This problem, the traveling salesman problem (TSP) is one of the most widely studied problems in combinatorial optimization [2]. The problem is easy to state, but hard to solve. Mathematically, the problem may be stated as follows: Given a cost matrix C = c ij ) where c ij represents the cost of going from city i to city j, i, j = 1, n) find a permutation (i 1 , i 2 , i 3 , i n ) of the integers from 1 ....

....This approach is simple, but often too greedy. The first distances in the construction process are reasonable short, whereas the distances at the end of the process usually will be rather long. A lot of other construction algorithms have been developed to remedy this problem (see for example [2], 12] and [13] The tour improvement algorithms, however, have achieved the greatest success. A simple example of this type of algorithm is the so called 2 opt algorithm: Start with a given tour. Replace 2 links of the tour with 2 other links in such a way that the new tour length is shorter. ....

E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan & D. B. Shmoys (eds.), The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization , Wiley, New York (1985).

....ranged widely from 7.7 to 93 seconds running on test environment workstations. 5.2. 6 Genetic algorithm for solving a traveling salesman problem The ga tsp application is an implementation of a distributed genetic algorithm [70] approach to solving the well known traveling salesman problem (TSP) [51]. In a genetic algorithm, a population of candidate solutions to a problem are encoded as a sequence of characters, much like a genetic sequence. These sequences are then adapted in stages called generations to produce new and hopefully better solutions in each passing generation. For each new ....

Lawler, E. L., Lenstra, J. K., Rinnooy-Kan, A. H. G., and Shmoys, D. B., Eds. The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. John Wiley and Sons, Chichester, West Sussex; New York, New York, Sept. 1985.

....algorithm exists for 3 solving any NP complete problem, many algorithms have been developed to solve special cases of NP complete problems. For example many algorithms exist for solving special cases of the traveling salesman problem. For an overview of these special cases, see Lawler et al. [30] or Burkard et al. 18] Researchers also design algorithms that exploit the structure of a metric to speed up polynomial time algorithms. For example, consider the development of single source shortest path algorithms. The Bellman Ford algorithm developed from Bellman s 1958 paper [12] This ....

E.L. Lawler, J.K. Lenstra, A.H.G. Rinooy Kan, and D.B. Shmoys, The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, Wiley (1985).

....of all inter city distances, the symmetric traveling salesman problem (TSP) seeks a shortest tour which visits each city exactly once. The symmetric TSP is NP hard [10] and has been extensively studied both in terms of its combinatorial structure and as a testbed for exact and heuristic methods [14] [13] Studies such as [12] point to greedy local search (e.g. using the fast 3 Opt [5] or the Lin Kernighan (LK) 15] neighborhood structure) as the most effective approach for practical instances. Over the past decade, iterated descent [3] 4] has been shown to be an effective means of applying ....

E. L. Lawler, J. K. Lenstra, A. Rinnooy-Kan, and D. Shmoys. The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, Chichester, 1985.

....set of constraints. The best known problem of this type is the classical Travelling Salesman Problem (TSP) calling for a minimum cost Hamiltonian cycle on a given graph. This problem has been extensively studied in the last decades; see the book edited by Lawler, Lenstra, Rinnooy Kan, and Shmoys [15] for a comprehensive cover of the literature up to 1985. In several applications a feasible cycle is allowed to visit only a subset of the nodes of the graph, chosen according to a speci ed criterion. For example, in the Prize Collecting TSP each node has an associated prize, and a salesman calls ....

E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, D.B. Shmoys (Editors), The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, Wiley, 1985.

....(n 1) of which are cyclic. So for a matrix W whose entries are drawn independently and uniformly at random from some interval there is a 1=n chance of solving the TSP by computing a solution to the AP. In the book on the traveling salesman problem edited by Lawler, Lenstra, Rinnooy Kan and Shmoys [19], Balas and Toth report of 400 computational experiments where they solved both the TSP and its assignment relaxation for problems with 50 n 250 vertices and edge weights drawn independently from a uniform distribution of the integers over the intervals [1; 100] and [1; 1000] As the outcome ....

E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, D.B. Shmoys (editors), The Traveling Salesman Problem { A Guided Tour of Combinatorial Optimization. John Wiley & Sons (1985).

....any constant factor approximation in polynomial time, unless P = NP . However, this hardness result applies in the worst case only. There is a very simple polynomially solvable relaxation called the assignment problem which can be exploited to obtain near optimal tours on average. Balas and Toth [5] carried out 400 computational experiments where they solved both the TSP and its assignment relaxation for problems with 50 n 250 vertices and edge weights drawn independently from a uniform distribution of the integers over the intervals [1; 100] and [1; 1000] As the outcome of these ....

E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, D.B. Shmoys (editors), The Traveling Salesman Problem { A Guided Tour of Combinatorial Optimization, John Wiley & Sons, 1985.

....The only condition is that the path should be long enough to be unique. 1 The second question must be answered by the user of the robot or by the planning system. And the third one of these questions represents the problem of nding optimum path, which can be solved, for example, using some TSP [15] solution. We de ne the optimum path as the shortest sequence of actions (shortest path) to reach the destination. There has been success in representing environmental maps using hypergraphs [4] geometrical, topological and object oriented maps. Also there has been use of Voronoi diagrams, ....

E. L. Lawler, J. K. Lenstra, A. K. G. Rinnooy Kan, D. B. Shmoys. The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, 1985.

....(2) after termination at any point, they will return an answer, and (3) answers returned improve in some well behaved manner as a function of time. Many conventional algorithms satisfy these characteristics. Zilberstein [Zil96] for example, shows how the solution of randomised tour improvement [Law85] to the 9 1 2 c2 now c1 now c1 c2 (i) d d (ii) iv) iii) Figure 1: A simple example of continuous deliberation scheduling (from [BD94] Assume that an agent has two future events, c 1 and c 2 , to respond to, and aims to maximise the value of both responses. The agent has at its ....

E. L. Lawler. The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, New York, 1985.

....it has been found that many of the activities that need to be performed can be formulated as combinatorial optimization problems. A combinatorial optimization problem involves the search for an optimum either maximum or minimum of a function of potentially many independent discrete variables [19, 65, 79]. This function is typically known as the cost function, c( The variables comprising the cost function define a set of possible solutions S. The cost function assigns a real value to a given solution, which is a quantitative measure of its quality. Since most VLSI DA problems involve the search ....

.... spaces that grow exponentially with the size of the problem, and many have indeed been proven to be NP hard [29, 88] NP hard problems have no known algorithms with worst case complexity bounded by a polynomial function of the size of the problem that will always find a globally optimal solution [19, 65, 79]. As a result, most of the NP hard problems present in the field of VLSI DA are solved approximately by heuristic algorithms. A heuristic algorithm designed to solve an NP hard problem is essentially a compromise, trading off lower computational cost for an approximation to the globally optimal ....

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E.L Lawler, J.K. Lenstra, A.H.G Rinnooy Kan, and D.B. Shmoys, Ed., The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, John Wiley and Sons, New York, NY, 1985.

....Simultaneously, two pairs of connecting cities between adjacent clusters are also determined using CNNs. Through computer simulations, it is confirmed that the proposed method is e#ective to solve the large scale problems. 1 Introduction It is well known that TSP (traveling salesman problem) [1] is one of the combinatorial optimization problems and is categorized as NP hard problem. The TSP is a problem to find a closed tour which visits each city once, returns to the starting city and has a shortest total path length. A lot of e#orts have been devoted to solve TSP because its solutions ....

....5. Connect every two cities selected in step 3 to obtain a total tour of the large scale TSP. The algorithm will be illustrated using an example, which is 101 city problem, eil101.tsp, from TSPLIB 1 . The optimal tour is shown in figure 3. The coordinates of cities are normalized in the plane [0, 1] [0, 1] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 3: The optimal tour for 101 city problem (eil101.tsp) At first, 101 cities are divided into 9 clusters, which have 20 to 30 cities as shown in figure 4. In this figure, the same symbol ....

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Lawler E. L., Lenstra J. K., Rinnooy-Kan A. H. G., and Shmoys D. B., editors. The Traveling Salesman Problem --- A Guided Tour of Combinatorial Optimization. John Wiley & Sons, 1997.

....Figure 7 shows the average error of predictions made by our algorithm as a function of the time for the 10 hardest instances. These required at least 400,000 node expansions for their optimal solution. None of them were misclassified by the algorithm as too easy to be worth predicting. 4 See [11] for an overview of the field. 5 This simplifying assumption may appear rather drastic, but as a matter of fact, with only 20 cities, it gives tolerably accurate results and is therefore legitimate in this context. 2 4 6 8 10 time 2.5 5 7.5 10 12.5 15 17.5 20 error Figure 8. ....

E. L. Lawler, J. K. Lenstra, A. H. G. Rinooy Kan, and D. B. Shmoys, The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, J. Wiley, Chichester, 1985.

....it into another position. opt neighborhood is the set of solutions obtainable by changing at most edges of the tour, where ( 2) is a given constant. Among these, opt neighborhood with = 2; 3 and Or opt neighborhood are known to be e ective, and swap neighborhood is not competitive with others [98, 87]. This is mainly because opt and Or opt neighborhoods are defined on tour edges, which are tied directly with the objective function of TSP. In many cases, more than one definition of the neighborhood is possible and the performance of the resulting algorithm is strongly a ected by the adopted ....

E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan and D.B. Shmoys, eds., The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, John Wiley & Sons, 1985.

.... problem is that we cannot practically solve instances on dense graphs of more than a couple hundred nodes [16] The most successful approach to the exact solution of combinatorial optimization problems is probably Integer Linear Programming, which has been applied pro tably in very many cases [19, 7, 17]. The Integer Programming approach consists in formulating a problem as the maximization of a linear function of some integer variables and then solving it via branch and bound, where the upper bound comes from the linear programming relaxation. The LP relaxation is the same question, only that ....

E. L. Lawler, J. K. Lenstra, A. H. G. Rinnoy Kan and D. B. Shmoys eds., The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, Wiley, 1985.

....Although provably good heuristics are frequently outperformed by local optimization methods, the output of provably good heuristics can serve as a good starting point for local improvement post processing schemes. For example, it was shown in [20] that Christofides traveling salesperson heuristic [17] provides excellent initial traveling salesperson tours for further local rearrangements. 20 7 An Improved Bound for Instances with Degenerate Groups Additional solution quality and runtime improvements may be realized when an instance of the Group Steiner Problem contains degenerate groups ....

E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shmoys. The Traveling Salesman Problem: a Guided Tour of Combinatorial Optimization. John Wiley and Sons, Chichester, New York, 1985.

....Hamiltonian Circuit Problem and its weighted version known as the Traveling Salesman Problem are probably the most famous problems of Combinatorial Optimization. They represent perfect examples of so called NP hard problems and serve as a test for various computational methods (see, for example, [3,6]) In this paper we prove two new algorithmic results about these problems. We use the model of computation based on RAM (see, for example, 1] As a measure of complexity we consider the number of performed arithmetic operations (addition, subtraction, multiplication, division, and comparison of ....

....in a polynomially bounded space, however it would give us O(n ) time complexity. The method of dynamic programming allows one to test a graph with the time complexity O(n 2 Delta 2 n ) however it uses O(2 n ) memory (since the basic idea is a reduction to subgraphs, see, for example, [6]) The algorithm from the paper [5] see the HPA3 algorithm from [5] uses O(n) memory but has 2 2n n O(1) time complexity. The second result deals with a special version of the Traveling Salesman Problem. Assume that we have a complete graph Gamma whose vertices P 1 ; Pn are ....

[Article contains additional citation context not shown here]

E. Lawler, et al., eds. The Traveling Salesman Problem: a guided tour of combinatorial optimization, Chichester, New York: Wiley, 1985.

....The traveling salesman problem (TSP) is one of the most important combinatorial problems discussed in the OR literature. Since its formulation by Dantzig, Fulkerson, and Johnson [3] DFJ) literally hundreds of papers have been written about it. The reader is referred to works by Lawler et al. [13] and Laporte [11] for an excellent expose on the TSP and related algorithmic procedures. In addition to solution approaches, several formulations for the TSP have been proposed. Several of these are discussed in Langevin et al. [12] and Padberg and Sung [16] The most widely referred to is the ....

E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, and D.B. Shmoys, "The traveling salesman problem. A Guided tour of combinatorial optimization," Wiley, Chester. (1985).

....variable at any time point, and then considering each of the resulting state partitions separately. In Graphplan and in heuristic search approaches, the splitting is done by applying all possible actions. Yet al..ternative branching schemes, are common in heuristic branch and bound search procedures [LRK85], in particular, in scheduling applications [CP89] Work on parallel planning, in particular involving actions of different durations, would most likely require such alternative branching schemes. ffl Modeling Languages: all the planners discussed in this paper are Strips planners. Yet few real ....

E. Lawler and A. Rinnooy-Kan, editors. The Traveling Salesman Problem : A Guided Tour of Combinatorial Optimization. Wiley, 1985.

.... only scheme we have been unable to find again the result given in [7] we give the value computed with our implementation (italic font) 4 A CASE OF NON BINARY REPRESENTATION: THE TSP The traveling salesperson problem (TSP) is a classic of combinatorial optimization (see for example [13] as a reference book) It is given as a set of N cities coordinates (or equivalently the inter cities distance matrix) and the objective is to find a tour as short as possible, visiting every cities. It has been studied intensively, notably in the EA community. Evolutionary inspired methods seem ....

E. C. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan, editors. The traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. John Wiley and Sons, 1990. ISBN: 0-471-90413-9.

....of different COP 3.1. The Traveling Salesman Problem (TSP) The search for the shortest hamiltonian cycle of a graph, or, less formally speaking, the shortest round trip through a given number of cities, has been one of the most popular combinatorial optimization problems for half a century [LAWLER 85] For symmetric TSPs, the so called 2 change is often chosen as the most simple local search operator defining a neighborhood N on the solution space by neighboring two solutions s i and s j iff they share all edges except for two of them [e.g. 1 2 3 4 5) and (1 4 3 2 5) are 2 change neighbors ....

Lawler E.L.; Lenstra J.K.; Rinnooy Kan A.H.G.; Shmoys D.B.: The Traveling Salesman Problem - A Guided Tour of Combinatorial Optimization; Chichester (Wiley).

....cutting plane algorithm have not proven to be very effective. However, recent work in problem specific cutting plane algorithms and algorithms that combine branch and bound with cutting planes has been somewhat more successful. Some recent examples of cutting plane techniques include [Jun85, Law85, PR91, Sto92] Some theoretical material on cutting planes can be found in [GLS88, Sch87] Heuristics for combinatorial optimization problems are algorithms that find good solutions without any guarantee that the solutions are optimal. In recent years, there has been a lot of interest in ....

Eugene Lawler. The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, New York, 1985.

....can serve as a good starting point for local improvement post processing schemes. For example, it was shown that Christofides heuristic (i.e. the best known heuristic for traveling salesperson in graphs) also provides excellent initial traveling salesperson tours for further local rearrangements [17] [20] 5 Instances with Degenerate Groups We now show how to more effectively handle instances of the group Steiner problem with some degenerate groups, i.e. groups of size 1. We will see that treating degenerate groups differently will yield 15 improvements in solution quality as well as in ....

E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy, and D. B. Shmoys. The Traveling Salesman Problem: a Guided Tour of Combinatorial Optimization. John Wiley and Sons, Chichester, New York, 1985.

....heaps suggested by Driscoll et al. as well as minor variations of Fibonacci heaps of our own design. 3. Experimental Design Experimental work in the area of combinatorial algorithms has been largely con ned to heuristics for NP hard problems such as the Travelling Salesperson Problem (see [10]) or the Minimum Test Set problem [13] or to exact algorithms for complex tractable problems such as the the Planar Graph Coloring problem (see [16] In the case of NP hard optimization problems, moreover, much of the experimental work revolves around the quality of the approximation, not the ....

Lawler, E.L., J.K. Lenstra, A.H.G. Rinnooy Kan, and D.B. Shmoys (eds.) The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. John Wiley & Sons, New York, 1985.

....of a new and a standard approach. All of these comparisons are done using a common framework. This is a summary of the research presented in [9] The traveling salesman problem (TSP####) is used as the basis of this case study. The TSP is a well known NPhard combinatorial optimization problem [5]. There are C## cities and the distance from city i to city j is d ij ## . A tour#### is a path that starts from a city, visits each city exactly once, and goes back to the starting city. A tour is represented by a vector t # of C cities. The tour starts from t(0) visits cities in the order ....

E. L. Lawler, J. K. Lenstra, and D. B. Shmoys, The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. New York, NY: WileyInterscience, 1985.

.... book by Wolsey [Wol] 1 Few optimization problems are as fascinating though as the traveling salesman problem (TSP) given n cities, and pairwise travel costs between them, find the shortest tour i.e. a directed cycle containing all cities; a wealth of information is contained in the survey [LLRS]; and on the website [Cook] Comparing a bad formulation to the only good one presently known the subtour formulation is less obvious, due to the 1 Somewhat to the dismay of OR teachers, many commercial software packages now automatically impose implied bounds, and generate cutting planes, ....

E. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shmoys (1985) eds., The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, Wiley

....variable at any time point, and then considering each of the resulting state partitions separately. In heuristic search approaches, the splitting is commonly done by applying all possible actions. Alternative branching schemes, however, are common in heuristic branch and bound search procedures (Lawler Rinnooy Kan 1985), and they may prove relevant in planning. We hope to explore some of these ideas in the future. Acknowledgments We thank Blai Bonet for useful discussions on the topic of this paper. Part of this work was done while H. Geffner was visiting Linkoping University. He thanks E. Sandewall and P. ....

Lawler, E., and Rinnooy-Kan, A., eds. 1985. The Traveling Salesman Problem : A Guided Tour of Combinatorial Optimization. Wiley.

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E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, D.B. Shmoys (eds.) (1985). The Traveling Salesman Problem: a Guided Tour of Combinatorial Optimization, Wiley, Chichester.

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E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, D. B. Shmoys. The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, 1985.

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E.L. Lawler, J.K. Lenstra, A.H.G. Rinooy Kan, and D.B. Shmoys (eds.). The traveling salesman problem: A guided tour of combinatorial optimization. Wiley, New York, 1985.

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E.L. Lawler, J.K. Lenstra, A.H.G. Rinooy Kan, and D.B. Shmoys (eds.). The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, New York, 1985.

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E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, and D.B. Shmoys (editors), The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, John Wiley and Sons, New York 1985.

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E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shmoys, editors. The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. John Wiley & Sons, 1985.

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E. L. Lawler, J. K. Lenstra, R.-K. A. H. G., and D. B. Shmoys. Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. John Wiley & Sons, 1990.

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E. Lawler, J. Lenstra, A. Rinnooy Kan, and D. Shmoys. The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, 1985. 174

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E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shmoys, The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. New York: Wiley and Sons, 1985.

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E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shmoys, The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, John Wiley & Sons, New York, NY, 1985.

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E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, and D.B. (eds.) Shmoys. The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, New York, 1985.

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E.L. Lawler, J.K. Lenstra, A.H.G. Rinooy Kan, and D.B. Shmoys, The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, Wiley (1985).

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E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan and D. B. Shmoys, The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, Wiley, 1985.

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Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., and Shmoys, D.B.: The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, New York, 1985

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E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy, and D. B. Shmoys, eds., The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, John Wiley & Sons, New York, 1985.

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E.L. Lawler, J.K. Lenstra, A.H.G. Rinooy Kan, and D.B. Shmoys, editors. The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. John Wiley & Sons, 1985. 12

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