## Large-Step Markov Chains for the Traveling Salesman Problem (1991)

Venue: | Complex Systems |

Citations: | 98 - 6 self |

### BibTeX

@ARTICLE{Martin91large-stepmarkov,

author = {Olivier Martin and Steve W. Otto and Edward W. Felten},

title = {Large-Step Markov Chains for the Traveling Salesman Problem},

journal = {Complex Systems},

year = {1991},

volume = {5},

pages = {299--326}

}

### Years of Citing Articles

### OpenURL

### Abstract

We introduce a new class of Markov chain Monte Carlo search procedures, leading to more powerful optimization methods than simulated annealing. The main idea is to embed deterministic local search techniques into stochastic algorithms. The Monte Carlo explores only local optima, and it is able to make large, global changes, even at low temperatures, thus overcoming large barriers in configuration space. We test these procedures in the case of the Traveling Salesman Problem. The embedded local searches we use are 3-opt and Lin-Kernighan. The large change or step consists of a special kind of 4-change followed by local-opt minimization. We test this algorithm on a number of instances. The power of the method is illustrated by solving to optimality some large problems such as the LIN318, the AT&T532, and the RAT783 problems. For even larger instances with randomly distributed cities, the Markov chain procedure improves 3-opt by over 1.6%, and Lin-Kernighan by 1.3%, leading to a new best h...

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Citation Context ...o achieve in practice. See Bentley and Johnson for an an extensive comparison of the above heuristics [12]. Another type of iterative sampling algorithm is the class of so-called "genetic algorit=-=hms" [13, 14, 15]. Here one starts wi-=-th an ensemble of tours which "compete:" the best tours replicate and the worst tours are eliminated. To create new kinds of tours, one applies "mutations" such as random k-changes... |

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346 |
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Citation Context ...t the tour strictly improve as one goes from one tour to the next --- that is, the tours are constructed so as to decrease the length at each step. The most effective such algorithms are those of Lin =-=[8]-=- and Lin and Kernighan [9]. Lin starts with the idea of a k-change: take the current tour and remove k different links from it. Now re-connect the dangling sections in a new way so as to again achieve... |

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Citation Context ...number of exact methods (i.e., which are guaranteed to find the exact optimum in a bounded number of steps) for solving the TSP. One family consists of the Branch and Bound algorithm of Held and Karp =-=[2, 3]-=- and its derivatives [4]. These algorithms attempt to prove that sets of links belong or do not belong to the optimal tour, using bounds from, for example, minimal spanning trees. There exist transfor... |

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Citation Context ...o achieve in practice. See Bentley and Johnson for an an extensive comparison of the above heuristics [12]. Another type of iterative sampling algorithm is the class of so-called "genetic algorit=-=hms" [13, 14, 15]. Here one starts wi-=-th an ensemble of tours which "compete:" the best tours replicate and the worst tours are eliminated. To create new kinds of tours, one applies "mutations" such as random k-changes... |

74 |
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Citation Context ...s not strictly enforced during the computation. (Note that this also occurs in cutting-plane algorithms.) In practice, the method has not yet been successful at solving problems of size 40 or greater =-=[17]. In Secti-=-on 3, we introduce a class of Markov chains in which each step is produced by a "kick" followed by a local search optimization. The local search method is described in section 4 and a number... |

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Citation Context ...and they are still to some extent an art form. In the last ten years, these exact methods have been pursued so vigorously that it is now possible to exactly solve problems with several hundred cities =-=[6, 7]-=-. The state of the art algorithms are quite complex, with codes on the order of 9000 lines. There are also many approximate or heuristic algorithms. These obtain good solutions in a (relatively) small... |

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Citation Context ...ne most rapidly. Though the pruning is dramatic, branch and bound is still an exponential (in N ) algorithm. To date, the most effective exact methods are the cutting-plane or facetfinding algorithms =-=[5, 6]-=-. These use an integer linear programing formulation of the TSP. Roughly speaking, various constraints are added to a linear programming problem until the solution found is a legal tour. The performan... |

30 |
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Citation Context ...her indicator is the fact that the Monte Carlo visits large numbers of tours just above the optimum and in fact, visits them multiple times. For instance, it visits the tour of length 41,349 given in =-=[24]-=- and the evidence from the Monte Carlo is that this is the first sub-optimal tour; the Monte Carlo found no tours between 41,345 and 41,349. In terms of speed, when the local search was L-K, the avera... |

28 |
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Citation Context ...ne most rapidly. Though the pruning is dramatic, branch and bound is still an exponential (in N ) algorithm. To date, the most effective exact methods are the cutting-plane or facetfinding algorithms =-=[5, 6]-=-. These use an integer linear programing formulation of the TSP. Roughly speaking, various constraints are added to a linear programming problem until the solution found is a legal tour. The performan... |

27 |
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Citation Context ... number of tours of length between l and l + dl, divided by dl. The main question is: how fast does this grow with l and with N? The answer is not known, but several distributions have been suggested =-=[27, 28]-=-. Here, we present a model which is very simple, but which seems to describe well the data for randomly scattered Euclidean TSP's. Let us first consider the set of all tours. The (N \Gamma 1)!=2 tours... |

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Searching for Optimal Configurations by Simulated Tunneling ", Zeitschrift fur Physik
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Citation Context ...o achieve in practice. See Bentley and Johnson for an an extensive comparison of the above heuristics [12]. Another type of iterative sampling algorithm is the class of so-called "genetic algorit=-=hms" [13, 14, 15]. Here one starts wi-=-th an ensemble of tours which "compete:" the best tours replicate and the worst tours are eliminated. To create new kinds of tours, one applies "mutations" such as random k-changes... |

5 |
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Citation Context ... algorithms begin by eliminating links from consideration. For instance, for a random city N = 400 problem, typically 75% of the links are eliminated by the first pass of our branch and bound program =-=[22]-=-. Since we know these links cannot appear in the optimal tour, we can set the corresponding distances d ij to infinity in the Monte Carlo, effectively removing them from consideration. In practical te... |

3 |
Accelerated Branch Exchange Heuristics for Symmetric Traveling Salesman Problems, Networks 17,423-437
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Citation Context ...er of N operations. This is to be contrasted with the k-opt algorithms introduced by Lin which require O(N k ) steps. More recently, an O(N ) check-out time approximation to 3-local-opt was presented =-=[20]-=- as a heuristic way of getting close to 3-opt quality tours. The purpose of this section is to show that 3-opt can, with no approximation, be realized so that check-out time is O(N ) rather than O(N 3... |

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Citation Context ...e is guaranteed to find the true optimum if one waits long enough, but this is almost impossible to achieve in practice. See Bentley and Johnson for an an extensive comparison of the above heuristics =-=[12]. Another type of it-=-erative sampling algorithm is the class of so-called "genetic algorithms" [13, 14, 15]. Here one starts with an ensemble of tours which "compete:" the best tours replicate and the ... |

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Citation Context ... number of tours of length between l and l + dl, divided by dl. The main question is: how fast does this grow with l and with N? The answer is not known, but several distributions have been suggested =-=[27, 28]-=-. Here, we present a model which is very simple, but which seems to describe well the data for randomly scattered Euclidean TSP's. Let us first consider the set of all tours. The (N \Gamma 1)!=2 tours... |