## Cost Versus Distance In the Traveling Salesman Problem (1995)

Citations: | 36 - 0 self |

### BibTeX

@TECHREPORT{Boese95costversus,

author = {Kenneth D. Boese},

title = {Cost Versus Distance In the Traveling Salesman Problem},

institution = {},

year = {1995}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper studies the distribution of good solutions for the traveling salesman problem (TSP) on a well-known 532-city instance that has been solved optimally by Padberg and Rinaldi [16]. For each of five local search heuristics, solutions are obtained from 2,500 different random starting points. Comparisons of these solutions show that lower-cost solutions have a strong tendency to be both closer to the optimal tour and closer to other good solutions. (Distance between two solutions is defined in terms of the number of edges they have in common.) These results support the conjecture of Boese, Kahng and Muddu [3] that the solution spaces of TSP instances have a "globally convex" or "big valley" character. This observation was used by [3] to motivate a new multi-start strategy for global optimization called Adaptive Multi-Start (AMS). 1 Introduction Local search is probably the most successful approach to finding heuristic solutions to combinatorial global optimization problems. In gl...

### Citations

251 |
1965] “Computer Solutions of the Traveling Salesman Problem
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(Show Context)
Citation Context ...from\Omega\Gamma n 2 ) for Random 2-Opt to approximately \Theta(n log n) on average. 3. Fast 3-Opt. We follow Bentley's [2] efficient implementation of the 3-Opt heuristic originally described by Lin =-=[10]-=-. 2 4. Lin-Kernighan. We have implemented, as accurately and completely as possible, Lin and Kernighan's [11] variation of k-Opt that searches a small but effective subset of all k-Opt moves for 2sksn... |

147 |
Local optimization and the traveling salesman problem
- Johnson
- 1990
(Show Context)
Citation Context ...nal paper [3]. Interestingly, Random 2-Opt returns solutions with significantly higher cost than those obtained by Fast 2-Opt. Heuristics 2 through 4 have been compared to other heuristics by Johnson =-=[7]-=- and Bentley [1] and appear to be among the most effective TSP heuristics. For example, 3-Opt and Lin-Kernighan return tours even better than simulated annealing 1 The same result was proved independe... |

129 |
Fast algorithms for geometric traveling salesman problems
- Bentley
- 1992
(Show Context)
Citation Context ...ble 2-Opt moves in random order, until an improving move is found or the current tour is shown to be a local minimum. 2. Fast 2-Opt. At each iteration, we perform the 2-Opt search proposed by Bentley =-=[2]-=-. This reduces the time complexity of 2-Opt from\Omega\Gamma n 2 ) for Random 2-Opt to approximately \Theta(n log n) on average. 3. Fast 3-Opt. We follow Bentley's [2] efficient implementation of the ... |

125 |
An effective heuristic algorithm for the traveling-salesman problem
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(Show Context)
Citation Context ... Bentley's [2] efficient implementation of the 3-Opt heuristic originally described by Lin [10]. 2 4. Lin-Kernighan. We have implemented, as accurately and completely as possible, Lin and Kernighan's =-=[11]-=- variation of k-Opt that searches a small but effective subset of all k-Opt moves for 2sksn. 5. Large-Step Markov Chains (LSMC) Finally, we use the heuristic of Martin et al. [12] [13] which iterative... |

109 |
A new adaptive multi-start technique for combinatorial global optimizations
- Boese, Kahng, et al.
- 1994
(Show Context)
Citation Context ...our and closer to other good solutions. (Distance between two solutions is defined in terms of the number of edges they have in common.) These results support the conjecture of Boese, Kahng and Muddu =-=[3] that the solution s-=-paces of TSP instances have a "globally convex" or "big valley" character. This observation was used by [3] to motivate a new multi-start strategy for global optimization called Ad... |

92 | Large-step markov chains for the traveling salesman problem
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- 1991
(Show Context)
Citation Context ...ed in a This work was performed under support from the UCLA Dissertation Year Fellowship. 1 Figure 1: Intuitive picture of the "big valley" solution space structure. number of other studies =-=including [12, 13]-=- and was solved to optimality by Padberg and Rinaldi in 1987 [16]. We have chosen this instance because (i) it represents a real-world geometric TSP instance; (ii) it is large enough to prove difficul... |

74 |
Introduction to probability and statistics for engineers and scientists
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- 1987
(Show Context)
Citation Context ...a for ATT532. a value of approximately 2:0 or greater indicates a correlation significant at the 95% confidence level, and a value of 2:6 or greater indicates significance at the 99% confidence level =-=[17]-=-. With t-Statistics ranging from 19 to 54, the correlations between distance and cost are highly significant statistically. 6 Mean distance to other solutions (a) Distance to optimal (b) Cost x 10 3 2... |

58 | Large-step markov chains for the tsp incorporating local search heuristics
- Martin, Otto, et al.
- 1992
(Show Context)
Citation Context ...ed in a This work was performed under support from the UCLA Dissertation Year Fellowship. 1 Figure 1: Intuitive picture of the "big valley" solution space structure. number of other studies =-=including [12, 13]-=- and was solved to optimality by Padberg and Rinaldi in 1987 [16]. We have chosen this instance because (i) it represents a real-world geometric TSP instance; (ii) it is large enough to prove difficul... |

57 |
Optimization of a 532 City Symmetric Traveling Salesman Problem by Branch and Cut
- Padberg, Rinaldi
- 1987
(Show Context)
Citation Context ...596 USA Abstract This paper studies the distribution of good solutions for the traveling salesman problem (TSP) on a well-known 532-city instance that has been solved optimally by Padberg and Rinaldi =-=[16]-=-. For each of five local search heuristics, solutions are obtained from 2,500 different random starting points. Comparisons of these solutions show that lower-cost solutions have a strong tendency to ... |

48 |
Configuration space analysis of the traveling salesman problem
- Kirkpatrick, Toulouse
- 1985
(Show Context)
Citation Context ...efine the distance d(t 1 ; t 2 ) to be n minus the number of edges contained in both t 1 and t 2 . This measure of distance has been used in a number of previous studies of TSP solution spaces (e.g., =-=[9, 14, 18]-=-). In [3], we showed that this distance approximates 2 the number of 2-Opt operations required to transform one tour into another, to within a factor of two. 1 Each of the heuristics used in this repo... |

42 |
Mattheyses, "A linear-time heuristic for improving network partitions
- Fiduccia, M
- 1982
(Show Context)
Citation Context ... and Kahng [5] have shown for circuit partitioning at least, that this relationship deteriorates for lower-cost solutions (i.e., those produced by more powerful heuristics such as Fiduccia-Mattheyses =-=[4]-=-). In other problem formulations we also find weaker cost-distance relationships than in the TSP, although in some of them (e.g., job shop scheduling) the relationship becomes more apparent when we us... |

42 |
Exact and approximation algorithms for the inversion distance between two permutations
- Kececioglu, Sankoff
- 1993
(Show Context)
Citation Context ...be among the most effective TSP heuristics. For example, 3-Opt and Lin-Kernighan return tours even better than simulated annealing 1 The same result was proved independently by Kececioglu and Sankoff =-=[8]-=- in the context of computing the number of chromosome inversions required to evolve one organism into another. 2 Note that our implementations of Fast 2-Opt and Fast 3-Opt differ slightly from Bentley... |

34 |
A replica analysis of the travelling salesman problem
- Mézard, Parisi
- 1986
(Show Context)
Citation Context ...efine the distance d(t 1 ; t 2 ) to be n minus the number of edges contained in both t 1 and t 2 . This measure of distance has been used in a number of previous studies of TSP solution spaces (e.g., =-=[9, 14, 18]-=-). In [3], we showed that this distance approximates 2 the number of 2-Opt operations required to transform one tour into another, to within a factor of two. 1 Each of the heuristics used in this repo... |

33 | Genetic algorithms
- Mühlenbein
- 1997
(Show Context)
Citation Context ...euristic. The current study augments [3] by using four additional local search heuristics for an instance with a known globally optimal tour. We also note that other authors such as Muhlenbein et al. =-=[15]-=- and Sourlas [18] have used similar plots to justify their heuristics. However, our results in [3] and in this report (i) involve more solutions and use better local search heuristics; (ii) compare me... |

32 |
Experiments on traveling salesman heuristics
- Bentley
- 1990
(Show Context)
Citation Context ...nterestingly, Random 2-Opt returns solutions with significantly higher cost than those obtained by Fast 2-Opt. Heuristics 2 through 4 have been compared to other heuristics by Johnson [7] and Bentley =-=[1]-=- and appear to be among the most effective TSP heuristics. For example, 3-Opt and Lin-Kernighan return tours even better than simulated annealing 1 The same result was proved independently by Kececiog... |

18 | Combining problem reduction and adaptive multistart: a new technique for superior iterative partitioning
- Hagen, Kahng
- 1997
(Show Context)
Citation Context ...ions between distance and cost for the five heuristics applied to ATT532. (Based on the unique minima resulting from 2,500 runs of each heuristic.) between cost and distance. However, Hagen and Kahng =-=[5]-=- have shown for circuit partitioning at least, that this relationship deteriorates for lower-cost solutions (i.e., those produced by more powerful heuristics such as Fiduccia-Mattheyses [4]). In other... |

12 |
Optimization of Globally Convex Functions
- Hu, Klee, et al.
- 1989
(Show Context)
Citation Context ...w, more effective heuristics (e.g., multi-start strategies or better annealing schedules). Our results indicate that cost surfaces for the traveling salesman problem (TSP) exhibit a "globally con=-=vex" [6] or what w-=-e call a "big valley" structure. Figure 1 gives an intuitive picture of the big valley, in which the set of local minima appears convex with one central global minimum. In this paper, we dis... |

12 |
Statistical mechanics and the travelling salesman problem, Europhys
- Sourlas
- 1986
(Show Context)
Citation Context ...rent study augments [3] by using four additional local search heuristics for an instance with a known globally optimal tour. We also note that other authors such as Muhlenbein et al. [15] and Sourlas =-=[18]-=- have used similar plots to justify their heuristics. However, our results in [3] and in this report (i) involve more solutions and use better local search heuristics; (ii) compare mean distances to o... |