### Abstract

We consider the metric Traveling Salesman Problem (Δ-TSP for short) and study how stability (as defined by Bilu and Linial [3]) influences the complexity of the problem. On an intuitive level, an instance of Δ-TSP is γ-stable (γ>1), if there is a unique optimum Hamiltonian tour and any perturbation of arbitrary edge weights by at most γ does not change the edge set of the optimal solution (i.e., there is a significant gap between the optimum tour and all other tours). We show that for γ ≥ 1.8 a simple greedy algorithm (resembling Prim’s algorithm for constructing a minimum spanning tree) computes the optimum Hamiltonian tour for every γ-stable instance of the Δ-TSP, whereas a simple local search algorithm can fail to find the optimum even if γ is arbitrary. We further show that there are γ-stable instances of Δ-TSP for every 1

Original language | English |
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Title of host publication | Proceedings of the 37th Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM) |

Pages | 382-393 |

Number of pages | 12 |

DOIs | |

Publication status | Published - 2011 |

Externally published | Yes |

## Cite this

Mihalák, M., Schöngens, M., Šrámek, R., & Widmayer, P. (2011). On the Complexity of the Metric TSP under Stability Considerations. In

*Proceedings of the 37th Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM)*(pp. 382-393) https://doi.org/10.1007/978-3-642-18381-2_32