### Abstract

In multi-criteria optimization, several objective functions are to be optimized. Since the different objective functions are usually in conflict with each other, one cannot consider only one particular solution as optimal. Instead, the aim is to compute so-called Pareto curves. Since Pareto curves cannot be computed efficiently in general, we have to be content with approximations to them. We are concerned with approximating Pareto curves of multi-criteria traveling salesman problems (TSP). We provide algorithms for computing approximate Pareto curves for the symmetric TSP with triangle inequality (Δ-STSP), symmetric and asymmetric TSP with strengthened triangle inequality (Δ(γ)-STSP and Δ(γ) -ATSP), and symmetric and asymmetric TSP with weights one and two (STSP(1,2) and ATSP(1,2)).

We design a deterministic polynomial-time algorithm that computes (1+γ+ε)-approximate Pareto curves for multi-criteria Δ(γ)-STSP for γ∈[1/2, 1]. We also present two randomized approximation algorithms for multi-criteria Δ(γ)-STSP achieving approximation ratios of 2γ^{3}+γ^{2}+2γ-1/2γ^{2}+ ε and 1+γ/1+3γ-4γ^{2} + ε, respectively. Moreover, we design randomized approximation algorithms for multi-criteria Δ(γ) -ATSP (ratio 1/2 + γ^{3}/1-3γ^{2} + ε for γ<1/√3), STSP(1,2) (ratio 4/3) and ATSP(1,2) (ratio 3/2). The algorithms for Δ;(γ)-ATSP, STSP(1,2), and ATSP(1,2) as well as one algorithm for Δ(γ) -STSP are based on cycle covers. Therefore, we design randomized approximation schemes for multi-criteria cycle cover problems by showing that multi-criteria graph factor problems admit fully polynomial-time randomized approximation schemes.

Original language | English |
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Title of host publication | Approximation and Online Algorithms |

Subtitle of host publication | 4th International Workshop, WAOA 2006, Revised Papers |

Editors | Thomas Erlebach, Christos Kaklamanis |

Place of Publication | Berlin, Heidelberg |

Publisher | Springer |

Pages | 302-315 |

Number of pages | 14 |

ISBN (Electronic) | 978-3-540-69514-1 |

ISBN (Print) | 978-3-540-69513-4 |

DOIs | |

Publication status | Published - 1 Dec 2007 |

Externally published | Yes |

Event | 4th Workshop on Approximation and Online Algorithms, WAOA 2006 - ETH Zürich, Zurich, Switzerland Duration: 14 Sep 2006 → 15 Sep 2006 Conference number: 4 |

### Publication series

Name | Lecture Notes in Computer Science |
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Publisher | Springer |

Volume | 4368 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 4th Workshop on Approximation and Online Algorithms, WAOA 2006 |
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Abbreviated title | WAOA 2006 |

Country | Switzerland |

City | Zurich |

Period | 14/09/06 → 15/09/06 |

Other | part of ALGO 2006 |

### Keywords

- Approximation algorithms
- Approximation ratio
- Travel salesman problem
- Hamiltonian cycle

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## Cite this

*Approximation and Online Algorithms: 4th International Workshop, WAOA 2006, Revised Papers*(pp. 302-315). (Lecture Notes in Computer Science; Vol. 4368). Berlin, Heidelberg: Springer. https://doi.org/10.1007/11970125_24