S. Lin. Computer Solutions of the Traveling Salesman Problem. Bell System Technical Journal, 44:2245--2269, 1965. 149  Home/Search   Document Not in Database   Summary   Related Articles   Check

....iteration of the algorithm, once all the ants have built a solution, pheromone trail is added to the arcs used by the ant that found the best tour from the beginning of the trial. In ACS 3 opt the daemon rst activates a local search procedure based on a variant of the 3 opt local search procedure [71] to improve the solutions generated by the ants and then performs o ine pheromone trail update. The o ine pheromone trail update rule is: ij (t) 1 ) ij (t) ij (t) 7) where 2 (0; 1] is a parameter governing pheromone decay, ij (t) 1=L , and L is the length of T , the ....

S. Lin. Computer solutions of the traveling salesman problem. Bell Systems Journal, 44:2245-2269, 1965.

.... that local optimization algorithms seem to be the best t for our application ( 10] compares various algorithms) Thereby, in order to improve a feasible tour it is modi ed by deleting some edges, thus breaking the tour into paths, and then reconnection those paths in an other possible way [12]. Neural nets are too slow, and genetic algorithms are too complex and, therefore, also not fast enough. The migration optimization module calculates a suitable migration strategy i.e. how migration is done in Tracy (pull code, push code, etc. see [4] For this task mainly the line data on the ....

S. Lin. Computer Solutions of the Traveling Salesman Problem. Bell System Technical Journal, 44:2245-2269, 1965.

....available. The next chapter of this book presents an overview of the available ACO algorithms for the QAP. Results obtained by the application of ACO algorithms to the TSP are very encouraging (see [20] results obtained using a rather unsophisticated local search procedure based on the 3 opt [38] allow to obtain results which are very close to those obtainable by much more sophisticated methods. More research will be necessary to assess whether ACO algorithms can reach the performance of state of the art algorithms like Iterated Lin Kernighan [35] An ACO algorithm called AntNet [13, ....

S. Lin. Computer solutions of the traveling salesman problem. Bell Systems Journal, 44:2245-2269, 1965.

.... move of local search is the 2 opt [4] move which consists in removing two edges from a tour and reconstruct a new di#erent tour (there is always one and only one way to reconstruct a new valid tour that is di#erent from the original) The 3 opt proposed by Bock in 1958 exchanges up to 3 edges [21]. We can go on and define the k opt which exchanges k edges. In practice, values of k 4 are seldomly used. The LK move, defined by Lin and Kernighan in 1973 [22] is based on the exchange of edges, the number of edges to exchange being evaluated to get the best possible improvement. Since 1973, ....

S. Lin, Computer Solutions Of The Traveling Salesman Problem, Bell System Technical Journal, 44, 2245--2269, 1965

....path between N cities. This path should start and end in the same city, and 3 should pass only once through each city. An analogy can be made for the JSP where cities become tasks and paths become setup times. The TSP is a practical problem that has been studied widely in the OR literature [4, 8, 12, 10]. However this model imposes that we know at the start the duration of each task, but in our problem this is one of the questions. Therefore, we need a new model that can represent the problem correctly. 3 The Job Sequencing model for CLP In order to find a solution, we first need to formulate ....

S. Lin. Computer solutions of the traveling salesman problem. Bell Syst. Tech. J., 44:2245--2269, 1965.

....for the travelling salesman problem A multilevel TSP algorithm is perhaps not so intuitive as that for the GPP. Clearly the LK or CLK ILK algorithms should make a good refinement method although in principle any iterative refinement procedure including the well known 2 opt, 9] and 3 opt, [37], algorithms could be used. However, with no graph as such, how can the problem be coarsened In fact from [58] it seems that the crucial point in devising a coarsening algorithm is the requirement that the solution to each coarsened problem must contain a solution of the original problem (even ....

....the basis of the 2 opt algorithm due to Croes, 9] which is a steepest descent approach, repeatedly flipping pairs of edges if they improve the tour quality until it reaches a local minimum of the objective function and no more such flips exist. In a similar vein, the 3 opt algorithm of Lin, [37], 13 exchanges 3 edges at a time. The Lin Kernighan (LK) algorithm, 38] also referred to as variable opt, however incorporates a limited amount of hill climbing by searching for a sequence of exchanges, some of which may individually increase the tour length, but which combine to form a shorter ....

S. Lin. Computer solutions of the traveling salesman problem. Bell Syst. Tech. J., 44:2245--2269, 1965.

....the basis of the 2 opt algorithm due to Croes, 8] which is a steepest descent approach, repeatedly flipping pairs of edges if they improve the tour quality until it reaches a local minimum of the objective function and no more such flips exist. In a similar vein, the 3 opt algorithm of Lin, [23], exchanges 3 edges at a time. The Lin Kernighan (LK) algorithm, 24] also referred to as variable opt, however incorporates a limited amount of hill climbing by searching for a sequence of exchanges, some of which may individually increase the tour length, but which combine to form a shorter ....

S. Lin. Computer Solutions of the Traveling Salesman Problem. Bell Syst. Tech. J., 44:2245--2269, 1965.

....the approaches with respect to all the surveyed work is that our main goal at this level of abstraction is to develop a technique for creation of IP property at the higher level of abstraction. It appears that the first paper that introduced heuristic search is by Newell and Ernst [New65] and Lin [Lin65]. The first comprehensive experimental study of the heuristic procedure was done by Doran and Michie [Dor66] In 1968, Hart, Nilsson, and Raphael [Har68] introduced a generic heuristic search technique named A . Pohl [Poh77] and Pearl [Pea84] theoretically analyzed the procedure. Simultaneously, ....

S. Lin. Computer solutions of the traveling salesman problem. Bell Systems Journal, Vol. 44, pp. 2245-2269, 1965.

....called simply k Opt, and in this section we study various pure and restricted heuristics of this kind. Currently, 2 Opt and 3 Opt are the main k Opt heuristics used in practice, introduced respectively by Flood and Croes [27, 24] and by Bock [15] In Shen Lin s influential 1965 study of 3 Opt [49], he concluded that the extra time required for 4 Opt was not worth the small improvement in tour quality it yielded, and no results have appeared since then to contradict this conclusion. In contrast, there have been several attempts to trade tour quality for improved running time in 3Opt by ....

S. Lin. Computer solutions of the traveling salesman problem. Bell System Technical Journal, 44:2245--2269, 1965. REFERENCES 79

....to minimize the total motor cost required to reproduce the reference text. Our optimization scheme is based on earlier work done by our research team for the optimization of ambiguous keypads (like on a telephone) 2] which in turn was inspired by research in the field of operations research [3]. In this approach, characters are initially assigned to random keys. The total motor cost of this arrangement is computed as described above. We then try to decrease the motor cost by systematically re arranging the characters. This is accomplished by repeatedly selecting pairs of characters and ....

.... T U M Y R A S P B L O F X J Q A E Space Space Z K Y Q G N D B C I L U H T R M V S O P X W F J Optimization of Single Finger Keyboards Our character swapping algorithm is a specific instance of a more general methodology known as n optimization (or n opt) [3]. In this paradigm, one considers the effect of rearranging n characters at a time, rather than just swapping 2 characters at a time (a 2 opt approach) The larger the value of n, the more likely the final result is the true optimum the single best possible arrangement. We have subjected the ....

Lin, S. (1965). Computer solutions of the traveling salesman problem. Bell Systems Technical Journal, 44, 2245--2269.

....k 0 (h k 0 1 ) k 0 (h k 0 2 ) k 0 (h k 0 2 1) Figure 4. A cross exchange operation The 2 opt neighborhood was proposed in [21] which is a variant of the 2 opt neighborhood for the traveling salesman problem (TSP, a special case of VRP in which the number of vehicles is one) [15]. A 2 opt operation removes two edges from two di erent routes (one from each) to divide each route into two parts and exchanges the second parts of the two routes (see Fig. 5) Let N 2opt ( k; k 0 ) be the set of all solutions obtainable by 2 opt operations on two routes k and ....

....always 20 correspond to the valid cycles found during the above repeated calls to FVC, since k u 6= k v holds for any two vertices u and v in these valid cycles. 6 Metaheuristic algorithms In this section, we describe three metaheuristic algorithms, the multi start local search (MLS) [15, 16, 22], the iterated local search (ILS) 12, 17, 18] and the adaptive multi start local search (AMLS) 5, 26] All of these algorithms are based on the LS described so far. We describe how to generate initial solutions within their frameworks. 6.1 The multi start local search and iterated local ....

S. Lin, \Computer solutions of the traveling salesman problem," Bell System Technical Journal, 44 (1965) 2245-2269.

....achieved in much less time by their domainspecific local search counterparts. Domain specific heuristics such as 2 Opt [4] E.J.W. Boers et al. Eds. EvoWorkshop 2001, LNCS 2037, pp. 193 202, 2001. c # Springer Verlag Berlin Heidelberg 2001 194 R. Baraglia, J.I. Hidalgo, and R. Perego 3 Opt [5], and Lin Kernighan (LK) 6] are e#ective. In particular LK is considered to be the heuristic that leads to the best solutions. E#cient implementations have been devised for LK which take just a few seconds to compute a high quality solution for problems with hundreds of cities [7,8] Several ....

S. Lin. Computer solution of the traveling salesman problem. Bell System Technical Journal, 44:2245--2269, 1965.

....CSP problems show very encouraging performances. 1 Introduction Heuristic (i.e. non complete) methods have been used in Combinatorial Optimization for nding optimal or near optimal solutions since a few decades, originating with the pioneering work of Lin on the Traveling Salesman Problem [10]. In the last few years, the interest for the family of Local Search methods for solving large combinatorial problems has been revived, and they have attracted much attention from both the Operations Research and the Arti cial Intelligence communities, see for instance the collected papers in [1] ....

S. Lin. Computer solutions of the traveling salesman problem. Bell System Technical Journal, vol. 44 (1965), pp 2245-2269.

.... from a given tour and attempt to reduce its length by exchanging edges chosen according to some heuristic rule until a local optimum is found (i.e. until no further improvement is possible using the heuristic rule) The most used and well known tour improvement heuristics are 2 opt and 3 opt [30], and Lin Kernighan [31] in which respectively two, three, and a variable number of edges are exchanged. It has been experimentally shown [35] that, in general, tour improvement heuristics produce better quality results than tour constructive heuristics. A general approach is to use tour ....

S. Lin., "Computer solutions of the traveling salesman problem," Bell Systems Journal, vol. 44, pp. 2245--2269, 1965.

....CSP problems show very encouraging performances. Introduction Heuristic (i.e. non complete) methods have been used in Combinatorial Optimization for finding optimal or near optimal solutions since a few decades, originating with the pioneering work of Lin on the Traveling Salesman Problem [10]. In the last few years, the interest for the family of Local Search methods for solving large combinatorial problems has been revived, and they have attracted much attention from both the Operations Research and the Artificial Intelligence communities, see for instance the collected papers in [1] ....

S. Lin. Computer solutions of the traveling salesman problem. Bell System Technical Journal, vol. 44 (1965), pp 2245-2269.

....have shown to be a very illustrative tool for the graphical presentation of the cost distance relationship [3,24,34] Here, we exemplify results on the FDC analysis using some instances which are larger than previously studied ones. For our investigation we use a 3 opt local search algorithm [27]. This local search algorithm proceeds by systematically testing whether the current tour can be improved by replacing at most three arcs. Straightforward 3 opt implementations require O(n 3 ) exchanges to be examined. Since this is too time consuming in practice, we use a number of standard ....

S. Lin. Computer solutions for the traveling salesman problem. Bell Systems Technology Journal, 44:2245--2269, 1965.

....(s; V; H(p) C) 13) 8q 2 P (V; V ) and 8y 2 P (V; V ) These Lemmas tell us that we can get the solution set of KMCSP by expanding the trivial path p(s; s) and all its extended feasible head paths step by step. This gives the basic ideas of the A Prune Algorithm. III. A PRUNE ALGORITHM A search[15][16] 17] 18] as well as uniform search, breadth first search and depth first search are well known searching strategies in Artificial Intelligence [19] We combine the A search with a proper pruning technique to get the A Prune algorithm, which can be used to solve the KMCSP problem. Using the ....

S. Lin. Computer solutions of the traveling salesman problem. Bell Systems Technical Journal, 44(10):2245-2269, 1965.

....A solution for the TSP is simply a permutation of the list of cities, each appearing exactly once and in the order specified by the current tour. A common generation mechanism for a SA approach to the TSP is the 2 opt transition first introduced independently by Croes [20] and later by Lin [67]. A 2 opt transition consists of choosing two cities at random from the current tour and reversing the order of the cities between them. This generation mechanism defines a neighborhood structure of size n(n 1) 2, or O(n 2 ) The cost of a solution is the sum of the distances between the cities ....

....crossover operator for permutation problems [45] and the other GA parameter settings discussed in the previous section. These two methods are used due to the superior solution quality of each. Johnson [51] presents an extensive empirical study comparing SA, the 3 opt TSP heuristic of Lin [67], and the k opt TSP heuristic of Lin and Kernighan (L K) The Johnson study validates SA as being significantly more effective than the 3 opt heuristic and nearly as effective as the L K heuristic for the TSP. Additionally, Johnson provides evidence indicating that GAs are capable of producing ....

S. Lin, "Computer Solutions of the Traveling Salesman Problem," Bell System Tech. J., vol. 44, 2245-2269, 1965.

....Competitive and Cooperative approach using the three moves described and performing a uniform decrement of the temperature according with a geometric schedule. The moves used have been chosen due to its reported efficiency in the TSP among the literature of iterative edge exchange heuristics [57] [126] [127] 143] 124] I would like to remark again that a MA does not need to start from scratch in a given optimization problem. Usually there are good iterative improvements procedures which can be used to do the local search and to reach local optima. The interactions between individuals, as in ....

S. Lin, "Computer Solutions of the Traveling Salesman Problem", Bell System Tech. J. 44, pp. 2245 (1965).

....= p(1) p(2) p(n) where p(i) denotes the city that is the successor city of p(i 1) in the tour represented by S ; the successor of p(n) is p(1) ffl The cost function to be minimised is. f(S) X i=2;n d p(i Gamma1) p(i) d p(n) p(1) ffl New solutions can be generated by k change [LIn65] which chooses k cities and swaps their positions in S, e.g. the 2 change neighbourhood operator. In Chapter 3 we mentioned that this has to be recast as a CSOP in which each solution to the problem has a unique representation. We adopt the CSOP representation and the repair procedure described ....

S. LIn. Computer solutions of the traveling salesman problem. Bell System Technical Journal, 44:2245--2269, 1965.

....and local search (abbreviated as LS; also called as hill climbing method or neighborhood search) there is a recent trend of metaheuristics which combine such heuristic tools in more sophisticated frameworks. 1 Those metaheuristics include random multi start local search (abbreviated as MLS) [90, 101, 102, 126, 136, 135], genetic algorithm (abbreviated as GA; also called as evolutionary computation 1 ) 34, 68, 76, 94, 115, 149] simulated annealing (abbreviated as SA) 1, 3, 24, 86, 92] tabu search (abbreviated as TS) 55, 59, 62, 64, 75] and so on. Among variants of these are genetic local search ....

....using about 70 hours. 2.3 Multi start local search In the multi start local search (MLS) LS is repeated from a number of initial solutions and the best solution found during the entire search is output. This is one of the most commonly used techniques for combinatorial optimization problems [90, 101, 102, 136, 135]. The initial solution may be generated randomly or by using greedy methods. Some important ideas of using previous search history (1) to generate new initial solutions [135] or (2) to speed up the search [101, 102] appeared in early literature. 2.4 Genetic algorithm The genetic algorithm (GA) ....

[Article contains additional citation context not shown here]

S. Lin, \Computer solutions of the traveling salesman problem," Bel l System Technical J., vol.44, pp.2245-2269, 1965.

....allocated resources. In AS all the ants deposit pheromone and no problem specific daemon actions are performed. The triggering of pheromone evaporation happens after all ants have completed their tours. Of course, it would be easy to add a local optimization daemon action, like a 3 opt procedure (Lin, 1965); this has been done in most of the ACO algorithms for the TSP that have been developed as extensions of AS (see for example (Dorigo Gambardella, 1997; St utzle Hoos, 1998a) The amount of pheromone trail # ij (t) maintained on connection l ij is intended to represent the learned ....

....for an overview of applications of ACO algorithms to the TSP) they are often better than those obtained using other general purpose heuristics like evolutionary computation or simulated annealing. Also, when adding to ACO algorithms rather unsophisticated local search procedures based on 3opt (Lin, 1965), the quality of the results obtained (Dorigo Gambardella, 1997) is close to that obtainable by more sophisticated methods. More research will be necessary to assess whether ACO algorithms can reach the performance of state of the art algorithms like Iterated Lin Kernighan (Johnson McGeoch, ....

S. Lin. 1965. Computer Solutions of the Traveling Salesman Problem. Bell Systems Journal, 44, 2245--2269.

.... Problem Having motivated the approach the next question is: how can the multilevel paradigm be applied to the TSP Clearly the LK or CLK ILK algorithms will make a good refinement method although in principle any iterative refinement procedure including the well known 2 opt, 8] and 3 opt, [24], algorithms could be used. However, with no graph as such, how can the problem be coarsened In fact it seems that the crucial point in devising a coarsening algorithm is the above requirement (C1) that the solution to each coarsened problem must contain a solution of the original problem ....

....the basis of the 2 opt algorithm due to Croes, 8] which is a steepest descent approach, repeatedly flipping pairs of edges if they improve the tour quality until it reaches a local minimum of the objective function and no more such flips exist. In a similar vein, the 3 opt algorithm of Lin, [24], exchanges 3 edges at a time. The Lin Kernighan (LK) algorithm, 25] also referred to as variable opt, however incorporates a limited amount of hill climbing by searching for a sequence of exchanges, some of which may individually increase the tour length, but which combine to form a shorter ....

S. Lin. Computer solutions of the traveling salesman problem. Bell Syst. Tech. J., 44:2245--2269, 1965.

....of such algorithms; rather, the properties are most likely generic to dynamics that are local. 5.1. Kernighan Lin algorithm. In simple local search, one performs elementary transformations to a feasible solution of the COP as long as they decrease the cost, a procedure sometimes called # opting [22]. A more sophisticated version consists in using variable depth search: one builds a sequence of p elementary transformations, usually according to a greedy criterion. p is not set ahead of time and depends on the sequence of costs found. The elementary transformations are not imposed to ....

S. Lin, Computer solutions of the traveling salesman problem, Bell System Tech. J., 44 (1965), pp. 2245--2269.

....of the algorithm, once all the ants have built a solution, pheromone trail is added to the arcs used by the ant that found the best tour from the beginning of the trial. In ACS 3 opt the daemon first activates a local search procedure based on a variant of the 3 opt local search procedure [71] to improve the solutions generated by the ants and then performs o#ine pheromone trail update. The o#ine pheromone trail update rule is: # ij (t) # (1 #)# ij (t) ### ij (t) 7) where # # (0, 1] is a parameter governing pheromone decay, ## ij (t) 1 L , and L is the length of T ....

S. Lin. Computer solutions of the traveling salesman problem. Bell Systems Journal, 44:2245--2269, 1965.

....have shown to be a very illustrative tool for the graphical presentation of the cost distance relationship [3,24,34] Here, we exemplify results on the FDC analysis using some instances which are larger than previously studied ones. For our investigation we use a 3 opt local search algorithm [27]. This local search algorithm proceeds by systematically testing whether the current tour can be improved by replacing at most three arcs. Straightforward 3 opt implementations require O(n 3 ) exchanges to be examined. Since this is too time consuming in practice, we use a number of standard ....

S. Lin. Computer solutions for the traveling salesman problem. Bell Systems Technology Journal, 44 (1965) 2245--2269.

....k produces solutions of increasing quality but the computational effort to test completely the k exchange set for a given solution usually restricts our attention to k exchange with k3. The most widely used edge exchange procedures set k to 2 or 3 (2 opt and 3 opt edge exchange procedures, Lin 1965), or to a variable value (Lin and Kernighan 1973) in which case a variable depth edge exchange search is performed. In this section we first make some observations about edge exchange techniques for TSP ATSP problems. Then, we concentrate our attention on path preserving edge exchanges for ATSPs, ....

Lin, S. 1965. Computer solutions of the traveling salesman problem. Bell Systems Journal 44 2245--2269.

....and examine its implementation when time constraints are added to the model. 2 G.A.P. Kindervater, J.K. Lenstra, M.W.P. Savelsbergh 2.3. Local search for the TSP Like so many other approaches in combinatorial optimization, local search was first seriously investigated in the context of the TSP. Lin [1965] calls a traveling salesman tour k optimal when it cannot be improved by replacing j of its edges by j other edges, for any j k. It is not known whether, for any fixed value of k 2, a k optimal tour can be generated in polynomial time. However, it is trivial to observe that the k optimality of ....

S. Lin (1965). Computer solutions of the traveling salesman problem. Bell System Tech. J. 44, 2245-2269.

.... search algorithms to the TSP, it has been shown that the best performance is obtained using the sophisticated Lin Kernighan local search algorithm [26] Yet, an efficient implementation of the Lin Kernighan heuristic is rather involved, thus, for simplicity we use 3 opt as a local search procedure [14]. Our implementation of 3 opt is sped up using standard techniques as described in [2, 12] In particular, we perform a fixed radius nearest neighbor search and use don t look bits for the outer loop optimization, see [2] for details on these techniques. In Table 1 we present results for the ....

S. Lin. Computer Solutions of the Traveling Salesman Problem. Bell Systems Technology Journal, 44:2245--2269, 1965.

....specific and will become more clear in connection with the batching problem later on) Thus one step in the k interchange neighborhood consists of k steps in the simple interchange neighborhood. Examples of such neighborhoods are the k interchange neighborhoods of Kernighan Lin [1970] and Lin [1965] for the uniform graph partitioning problem and the traveling salesman problem, respectively. The advantage of these neighborhoods is that they focus not only 4 on one simple step but try to find a good series of simple moves which together lead to a good result. With an increasing number of k ....

S. Lin [1965] Computer solutions of the traveling salesman problem, Bell Systems Technical Journal 44, 2245-2269.

....The next chapter of this book presents an overview of the available ACO algorithms for the QAP. ffl Results obtained by the application of ACO algorithms to the TSP are very encouraging (see [20] results obtained using a rather unsophisticated local search procedure based on the 3 opt [38] allow to obtain results which are very close to those obtainable by much more sophisticated methods. More research will be necessary to assess whether ACO algorithms can reach the performance of state of the art algorithms like Iterated Lin Kernighan [35] ffl An ACO algorithm called AntNet [13, ....

S. Lin. Computer solutions of the traveling salesman problem. Bell Systems Journal, 44:2245--2269, 1965.

....problem specific and will become more clear in connection with the batching problem later on) Thus one step in the k interchange neighborhood consists of k steps in the simple interchange neighborhood. Examples of such neighborhoods are the k interchange neighborhoods of Kernighan Lin [1970] Lin [1965] for the uniform graph partitioning problem and the traveling salesman problem, respectively. The advantage of these neighborhoods is that they focus not only on one simple step but try to find a good series of simple moves which together lead to a good result. With an increasing number of k this ....

S. Lin [1965] Computer solutions of the traveling salesman problem, Bell Systems Technical Journal 44, 2245-2269.

....nodes from which the k tour is generated consists of more than k nodes, the farthest insert algorithm has to be slightly modified because the farthest node in general is not a good candidate for building a short k tour. 4 Tour improvement heuristics for the TSSP For the TSP the r opt heuristic [12] and its generalization, the Lin Kernighan heuristic [11] are known as tour improvement algorithms. Both heuristics can be applied to the TSSP without any modification. However it is desirable to have an improvement heuristic that could realize a node exchange which inserts new nodes into the ....

Shen Lin, Computer Solutions of the Traveling Salesman Problem, The Bell System Technical Journal (Dezember 1965), pp 2245--2269

....heuristics, with the purpose of guiding the basic heuristic beyond local optimality. Ideas similar to those proposed in TS can be found in the denial strategy of [56] once common features are detected in many suboptimal solutions, they are forbidden) or in the opposite reduction strategy of [40] (in an application to the Traveling Salesman Problem, all edges that are common to a set of local optima are fixed) In the context of graph partitioning a related heuristic is the mentioned KL algorithm [35] The KL algorithm can be denoted as a variable depth search and it is briefly summarized ....

S. Lin, "Computer Solutions of the Traveling Salesman Problems," BSTJ, vol. 44, no. 10, pp. 2245--2269, 1965.

....parameters. As a representative example, we show results on one instance of a 200 city TSP, averaged over 5 runs each starting from a random initial tour. The city coordinates were chosen from a uniform random distribution over a 10 Theta 10 grid. The move generation rule used was uniform 2 opt [11], or segment reversal. A paper by Dueck and Scheuer [7] using an algorithm resembling the annealed demon algorithm contained the results of detailed testing on Grotschel s 442 city problem [12] This paper also quotes results on these two TSPs by Rossier et al. [13] using exhaustive Lin 2 opt and ....

S. Lin, "Computer solutions of the traveling salesman problem," The Bell System Technical Journal, vol. 44, pp. 2245--2269, 1965.

....a number of visits and improves before continuing. This resembles the dynamic procedure quite strongly and has been shown to be an effective technique. For the problems presented in section 5 we insert 10 visits at each iteration, and improve using a first accept local search incorporating 2 Opt [10], Or opt [11] relocate [14] exchange [15] and cross [12] We have found that the order of insertion has a large influence on the effectiveness of this and other insertion based heuristics. We use insertion in the order given by a 2 optimal Travelling Salesman Tour around all visits. That is, we ....

S. Lin. Computer solutions of the traveling salesman problem. Bell Systems Technical Journal, 44:2245--2269, 1965.

....and many heuristic approaches are a real challenge to new methods [117] 87] Among them we must remark: a) L. Few introduced a heuristic solution for the Euclidean problem with a guaranteed worst case performance of p 2N 1:75 [36] b) S. Lin s iterative improvement edge exchange methods [89] (known as r opt procedures where r stands for the number of exchanged edges in each iteration) and the later variable r opt [90] often produces near optimal tour configurations (for an early reference see also [26] c) N. Christofides presented in 1976 a heuristic which uses the concept of ....

....which is the one actually being optimized using the heuristic assigned to that agent. In one of the computer experiments all agents perform the same heuristic ( Greedy 2 Opt) and in the other we will use approximate 2 Opt, and One City and Two City insertions (subsets of the 3 Opt procedure) [89] [90] The method is thus designed for move set parallelization so different agents can use different moves. We will return to this point later. The best tour of each agent is actualized each time we get a new current tour of a smaller length which is the only criteria that governs selection ....

[Article contains additional citation context not shown here]

S. Lin, Computer Solutions of the Traveling Salesman Problem, Bell System Tech. Journal 44 (1965) 2245-2269.

No context found.

S. Lin. Computer solutions of the traveling salesman problem. Bell Syst. Tech. J., 44:2245--2269, 1965.

....the basis of the 2 opt algorithm due to Croes, 1] which is a steepest descent approach, repeatedly flipping pairs of edges if they improve the tour quality until it reaches a local minimum of the objective function and no more such flips exist. In a similar vein, the 3 opt algorithm of Lin, [8], exchanges 3 edges at a time. The Lin Kernighan (LK) algorithm, 9] also referred to as variable opt, however incorporates a limited amount of hill climbing by searching for a sequence of exchanges, some of which may individually increase the tour length, but which combine to form a shorter ....

S. Lin. Computer solutions of the traveling salesman problem. Bell Syst. Tech. J., 44:2245--2269, 1965.

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S. Lin. Computer Solutions of the Traveling Salesman Problem. Bell System Technical Journal, 44:2245--2269, 1965. 149

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S. Lin. Computer solutions of the traveling salesman problem. Bell Syst. Tech. J., 44:2245--2269, 1965.

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S. Lin. Computer Solutions of the Traveling Salesman Problem. Bell Syst. Tech. J., 44:2245--2269, 1965.

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S. Lin. Computer solutions of the traveling salesman problem. Bell Syst. Tech. J., 44:2245--2269, 1965.

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S. Lin, Computer Solutions of the Traveling Salesman problems, BSTJ 44(10) (1965) 2245-- 69.

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S. Lin. Computer solutions for the traveling salesman problem. Bell System Tech. J., 44:2245--2269, 1965.

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S. Lin, [1965] "Computer Solutions of the Traveling Salesman Problem", Bell System Tech. J. 44, pp. 2245-2269.

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S. Lin. Computer solutions for the traveling salesman problem. Bell System Tech. J., 44, 2245--2269.

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S. Lin. Computer solutions for the traveling salesman problem. Bell System Tech. J., 44:2245--2269, 1965.

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Lin, S. (1965). Computer solutions of the traveling salesman problem. Bell Systems Technical Journal, 44, 2245-2269.

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Lin, S., Computer solutions of the traveling salesman problem, Bell System Tech. J. 44, (1965), 2245-2269.

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