### Abstract

Consider a truck running along a road. It picks up a load L_{i} at point β_{i} and delivers it at α_{i}, carrying at most one load at a time. The speed on the various parts of the road in one direction is given by f(x) and that in the other direction is given by g(x). Minimizing the total time spent to deliver loads L_{1},...,L_{n} is equivalent to solving the Traveling Salesman Problem (TSP) where the cities correspond to the loads L_{i} with coordinates (α_{i}, β_{i}) and the distance from L_{i} to L_{j} is given by ∫_{αi}^{βj} f(x)dx if β_{j} ≥ α_{i} and by ∫_{βj}^{αi} g(x)dx if β_{j} < α_{i}. This case of TSP is polynomially solvable with significant real-world applications. Gilmore and Gomory obtained a polynomial time solution for this TSP [6]. However, the bottleneck version of the problem (BTSP) was left open. Recently, Vairaktarakis showed that BTSP with this distance metric is NP-complete [10]. We provide an approximation algorithm for this BTSP by exploiting the underlying geometry in a novel fashion. This also allows for an alternate analysis of Gilmore and Gomory's polynomial time algorithm for the TSP. We achieve an approximation ratio of (2 + γ) where γ ≥ f(x)/g(x) ≥ 1/γ ∀x. Note that when f(x) = g(x), the approximation ratio is 3.

Original language | English (US) |
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Title of host publication | Algorithms and Complexity - 6th Italian Conference, CIAC 2006, Proceedings |

Publisher | Springer Verlag |

Pages | 223-235 |

Number of pages | 13 |

ISBN (Print) | 354034375X, 9783540343752 |

DOIs | |

State | Published - Jan 1 2006 |

Event | 6th Italian Conference on Algorithms and Complexity, CIAC 2006 - Rome, Italy Duration: May 29 2006 → May 31 2006 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 3998 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 6th Italian Conference on Algorithms and Complexity, CIAC 2006 |
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Country | Italy |

City | Rome |

Period | 5/29/06 → 5/31/06 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Algorithms and Complexity - 6th Italian Conference, CIAC 2006, Proceedings*(pp. 223-235). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 3998 LNCS). Springer Verlag. https://doi.org/10.1007/11758471_23

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*Algorithms and Complexity - 6th Italian Conference, CIAC 2006, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 3998 LNCS, Springer Verlag, pp. 223-235, 6th Italian Conference on Algorithms and Complexity, CIAC 2006, Rome, Italy, 5/29/06. https://doi.org/10.1007/11758471_23

**An approximation algorithm for a bottleneck traveling salesman problem.** / Kao, Ming-Yang; Sanghi, Manan.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - An approximation algorithm for a bottleneck traveling salesman problem

AU - Kao, Ming-Yang

AU - Sanghi, Manan

PY - 2006/1/1

Y1 - 2006/1/1

N2 - Consider a truck running along a road. It picks up a load Li at point βi and delivers it at αi, carrying at most one load at a time. The speed on the various parts of the road in one direction is given by f(x) and that in the other direction is given by g(x). Minimizing the total time spent to deliver loads L1,...,Ln is equivalent to solving the Traveling Salesman Problem (TSP) where the cities correspond to the loads Li with coordinates (αi, βi) and the distance from Li to Lj is given by ∫αiβj f(x)dx if βj ≥ αi and by ∫βjαi g(x)dx if βj < αi. This case of TSP is polynomially solvable with significant real-world applications. Gilmore and Gomory obtained a polynomial time solution for this TSP [6]. However, the bottleneck version of the problem (BTSP) was left open. Recently, Vairaktarakis showed that BTSP with this distance metric is NP-complete [10]. We provide an approximation algorithm for this BTSP by exploiting the underlying geometry in a novel fashion. This also allows for an alternate analysis of Gilmore and Gomory's polynomial time algorithm for the TSP. We achieve an approximation ratio of (2 + γ) where γ ≥ f(x)/g(x) ≥ 1/γ ∀x. Note that when f(x) = g(x), the approximation ratio is 3.

AB - Consider a truck running along a road. It picks up a load Li at point βi and delivers it at αi, carrying at most one load at a time. The speed on the various parts of the road in one direction is given by f(x) and that in the other direction is given by g(x). Minimizing the total time spent to deliver loads L1,...,Ln is equivalent to solving the Traveling Salesman Problem (TSP) where the cities correspond to the loads Li with coordinates (αi, βi) and the distance from Li to Lj is given by ∫αiβj f(x)dx if βj ≥ αi and by ∫βjαi g(x)dx if βj < αi. This case of TSP is polynomially solvable with significant real-world applications. Gilmore and Gomory obtained a polynomial time solution for this TSP [6]. However, the bottleneck version of the problem (BTSP) was left open. Recently, Vairaktarakis showed that BTSP with this distance metric is NP-complete [10]. We provide an approximation algorithm for this BTSP by exploiting the underlying geometry in a novel fashion. This also allows for an alternate analysis of Gilmore and Gomory's polynomial time algorithm for the TSP. We achieve an approximation ratio of (2 + γ) where γ ≥ f(x)/g(x) ≥ 1/γ ∀x. Note that when f(x) = g(x), the approximation ratio is 3.

UR - http://www.scopus.com/inward/record.url?scp=33746078429&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33746078429&partnerID=8YFLogxK

U2 - 10.1007/11758471_23

DO - 10.1007/11758471_23

M3 - Conference contribution

AN - SCOPUS:33746078429

SN - 354034375X

SN - 9783540343752

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 223

EP - 235

BT - Algorithms and Complexity - 6th Italian Conference, CIAC 2006, Proceedings

PB - Springer Verlag

ER -