## An effective implementation of the linkernighan traveling salesman heuristic (2000)

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Venue: | European Journal of Operational Research |

Citations: | 120 - 1 self |

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@ARTICLE{Helsgaun00aneffective,

author = {Keld Helsgaun},

title = {An effective implementation of the linkernighan traveling salesman heuristic},

journal = {European Journal of Operational Research},

year = {2000},

pages = {106--130}

}

### Years of Citing Articles

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### Abstract

This report describes an implementation of the Lin-Kernighan heuristic, one of the most successful methods for generating optimal or nearoptimal solutions for the symmetric traveling salesman problem. Computational tests show that the implementation is highly effective. It has found optimal solutions for all solved problem instances we have been able to obtain, including a 7397-city problem (the largest nontrivial problem instance solved to optimality today). Furthermore, the algorithm has improved the best known solutions for a series of large-scale problems with unknown optima, among these an 85900-city problem. 1.

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Citation Context ...d, the hash table is consulted to see whether the new tour happens to be local optimum found earlier. If this is the case, fruitless checkout time is avoided. This technique is described in detail in =-=[46]. (3-=-) The concept of the don’t look bit, introduced by Bentley [44], is used. If for a given choice of t1 the algorithm previously failed to find an improvement, and if t1’s tour neighbors have not ch... |

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Citation Context ...nd cities, then the algorithm will have difficulties in obtaining the optimum. The inadequacy of this rule manifests itself particularly clearly in large problems. For example, for a 532-city problem =-=[19]-=- one of the links in the optimal solution is the 22nd nearest neighbor city for one of its end points. So in order to find the optimal solution to this problem, the number of nearest neigh17sbors to b... |

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Citation Context ...nted as an array of nodes, or as a doubly linked list of nodes, the reversal of the path takes time O(n). It turns out that data structures exist that allow logarithmic time complexity to be achieved =-=[13, 40, 41, 42, 43]-=-. These data structures, however, should not be selected without further notice. The time overhead of the corresponding update algorithms is usually large, and, unless the problem is large, typically ... |

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Citation Context ...usters. In contrast, a minimum spanning tree is (by definition) always connected. Other candidate sets may be considered. An interesting candidate set can be obtained by exploiting the Delaunay graph =-=[13, 33]-=-. The Delaunay graph is connected and may be computed in linear time, on the average. A disadvantage of this approach, however, is that candidate sets can only be computed for geometric problem instan... |

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Citation Context ...tained in a short time. 25sNo general methods to determine an optimum strategy for the choice of step size are known. However, many strategies have been suggested that are quite effective in practice =-=[25, 27, 28, 29, 30, 31]-=-. These strategies are heuristics, and different variations have different effects on different problems. In the present implementation of the modified Lin-Kernighan algorithm the following strategy w... |

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Citation Context ...le 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 o... |

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Citation Context ...ded for solving symmetric TSPs. However, any asymmetric problem may be transformed into a symmetric problem and therefore be solved by the algorithm. The transformation method of Jonker and Volgenant =-=[51] tra-=-nsforms a asymmetric problem with n nodes into a problem 2n nodes. Let C = (c ij ) denote the nxn cost matrix of the asymmetric problem. Then let C’ = (c’ ij ) be the 2nx2n symmetric matrix comput... |

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Citation Context ...top criterion is satisfied, then stop. 7. Choose a step size, t k . 8. Let π k+1 = π k + t k v k . 9. Let k = k + 1 and go to Step 2. Figure 4.5 Subgradient optimization algorithm. It has been prove=-=n [26] that W wil-=-l always converge to the maximum of w(π), if t k → 0 for k → ∞ and ∑t k = ∞ . These conditions are satisfied, for example, if t k is t 0 /k, where t 0 is some arbitrary initial step size. B... |

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Citation Context ...time on a large network of computers to determine the exact solution of the previously mentioned 7397-city problem [8]. Symmetric problems are usually more difficult to solve than asymmetric problems =-=[9]-=-. Today the 7397-city problem is the largest (nontrivial) symmetric problem that has been solved. In comparison, the optimal solution of a 500,000-city asymmetric problem has been reported [10]. 5s2.3... |

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Citation Context ...ated this way, the total time complexity becomes O(n 3 ), which is unsatisfactory. However, it is possible to obtain a total complexity of O(n 2 ) by exploiting a simple relation between the α-values=-= [23, 24]. L-=-et β(i,j) denote the length of the edge to be removed from the spanning tree when edge (i,j) is added. Thus α(i,j) = c(i,j) - β(i,j). Then the following fact may be exploited (see Figure 4.2). If (... |

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Citation Context ... improve upon a tour by performing various exchanges. The composite algorithms combine these two features. A simple example of a tour construction algorithm is the so-called nearestneighbor algorithm =-=[11]-=-: Start in an arbitrary city. As long as there are cities, that have not yet been visited, visit the nearest city that still has not appeared in the tour. Finally, return to the first city. This appro... |

6 |
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Citation Context ...≤ 4 are made in case an improvement of the tour is possible). In order simplify tour updating, the following fact may be used: Any r-opt move (r ≥ 2) is equivalent to a finite sequence of 2-opt mo=-=ves [16, 39]-=-. In the case of 5-opt moves it can be shown that any 5-opt move is equivalent to a sequence of at most five 2-opt moves. Any 4-opt move as well as any 3-opt move is equivalent to a sequence of at mos... |

6 |
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Citation Context ...high quality solutions, preferably optimal solutions, in a reasonable short time. Achieving simplicity was of minor concern here. In comparison, the modified Lin-Kernighan algorithm of Mak and Morton =-=[54]-=- has a very simple algorithmic structure. However, this simplicity has been achieved with the expense of a reduced ability to find optimal solutions. Their algorithm does not even guarantee 2-opt opti... |

5 |
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Citation Context ... ). Furthermore, there is no nontrivial upper bound of the number of λ–exchanges. As a result, the values λ = 2 and λ = 3 are the most commonly used. In one study the values λ = 4 and λ = 5 wer=-=e used [16]. -=-However, it is a drawback that λ must be specified in advance. It is difficult to know what λ to use to achieve the best compromise between running time and quality of solution. Lin and Kernighan re... |

5 |
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Citation Context ...nted as an array of nodes, or as a doubly linked list of nodes, the reversal of the path takes time O(n). It turns out that data structures exist that allow logarithmic time complexity to be achieved =-=[13, 40, 41, 42, 43]-=-. These data structures, however, should not be selected without further notice. The time overhead of the corresponding update algorithms is usually large, and, unless the problem is large, typically ... |

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Citation Context ...ated this way, the total time complexity becomes O(n 3 ), which is unsatisfactory. However, it is possible to obtain a total complexity of O(n 2 ) by exploiting a simple relation between the α-values=-= [23, 24]. L-=-et β(i,j) denote the length of the edge to be removed from the spanning tree when edge (i,j) is added. Thus α(i,j) = c(i,j) - β(i,j). Then the following fact may be exploited (see Figure 4.2). If (... |

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Citation Context ...mplementation is almost unaffected. 6.4 Pathological problems The Lin-Kernighan algorithm is not always as effective as it seems to be with “random” or “typical” problems. Papadimitriou and St=-=eiglitz [53]-=- have constructed a special class of instances of the TSP for which local search algorithms, such as the Lin-Kernighan algorithm, appears to be very ineffective. Papadimitriou and Steiglitz denote thi... |

3 |
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Citation Context ...r tour is found, all edges shared by this new tour and the previous shortest tour become the first two candidate edges for their end nodes. This method of selecting candidates was inspired by Stewart =-=[32]-=-, who demonstrated how minimum spanning trees could be used to accelerate 3-opt heuristics. Even when subgradient optimization is not used, candidate sets based on minimum spanning trees usually produ... |

3 |
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Citation Context ...ple to answer. Adrabinsky and Syslo [34], for instance, found that the farthest insertion construction heuristic was capable of producing good initial tours for the Lin-Kernighan algorithm. Perttunen =-=[35]-=- found that the 30sClarke and Wright savings heuristic [36] in general improved the performance of the algorithm. Reinelt [13] also found that is better not to start with a random tour. He proposed us... |

3 |
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Citation Context ...For non-metric problems it can proven that it is impossible to construct an algorithm of polynomial complexity which find tours whose length is bound by a constant multiple of the optimal tour length =-=[47]-=-. The purpose of the second method, probabilistic analysis, is to evaluate average behavior of the algorithms. For example, for an approximate TSP algorithm probability analysis can used be to estimat... |

2 |
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Citation Context ...ems with a few hundred cities, although there have been reports on the solution of problems with thousands of cities. The most effective exact algorithms are cutting-plane or facet-finding algorithms =-=[6, 7, 8]-=-. These algorithms are quite complex, with codes on the order of 10,000 lines. In addition, the algorithms are very demanding of computer power. For example, the exact solution of a symmetric problem ... |

2 |
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Citation Context ...ally deterministic, so it may not be possible to get more than one solution. However, the question of whether or not to use a construction heuristic is not that simple to answer. Adrabinsky and Syslo =-=[34]-=-, for instance, found that the farthest insertion construction heuristic was capable of producing good initial tours for the Lin-Kernighan algorithm. Perttunen [35] found that the 30sClarke and Wright... |

1 |
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Citation Context ...ems with a few hundred cities, although there have been reports on the solution of problems with thousands of cities. The most effective exact algorithms are cutting-plane or facet-finding algorithms =-=[6, 7, 8]-=-. These algorithms are quite complex, with codes on the order of 10,000 lines. In addition, the algorithms are very demanding of computer power. For example, the exact solution of a symmetric problem ... |

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Citation Context ...nted as an array of nodes, or as a doubly linked list of nodes, the reversal of the path takes time O(n). It turns out that data structures exist that allow logarithmic time complexity to be achieved =-=[13, 40, 41, 42, 43]-=-. These data structures, however, should not be selected without further notice. The time overhead of the corresponding update algorithms is usually large, and, unless the problem is large, typically ... |