Letchford, A. N. (2000) *Separating a superclass of comb inequalities in planar graphs.* Mathematics of Operations Research, 25 (3). pp. 443-454. ISSN 0364-765X

## Abstract

Many classes of valid and facet-inducing inequalities are known for the family of polytopes associated with the Symmetric Travelling Salesman Problem (STSP), including subtour elimination, 2-matching and comb inequalities. For a given class of inequalities, an exact separation algorithm is a procedure which, given an LP relaxation vector x*, finds one or more inequalities in the class which are violated by x*, or proves that none exist. Such algorithms are at the core of the highly successful branch-and-cut algorithms for the STSP. However, whereas polynomial time exact separation algorithms are known for subtour elimination and 2-matching inequalities, the complexity of comb separation is unknown. A partial answer to the comb problem is provided in this paper. We define a generalization of comb inequalities and show that the associated separation problem can be solved efficiently when the subgraph induced by the edges with x*_e > 0 is planar. The separation algorithm runs in O(n^3) time, where n is the number of vertices in the graph.

Item Type: | Article |
---|---|

Journal or Publication Title: | Mathematics of Operations Research |

Uncontrolled Keywords: | Symmetric travelling salesman problem ; branch-and-cut ; valid inequalities ; Separation algorithm |

Subjects: | |

Departments: | Lancaster University Management School > Management Science |

ID Code: | 43273 |

Deposited By: | ep_importer_pure |

Deposited On: | 11 Jul 2011 18:56 |

Refereed?: | Yes |

Published?: | Published |

Last Modified: | 17 Aug 2017 01:55 |

Identification Number: | |

URI: | http://eprints.lancs.ac.uk/id/eprint/43273 |

### Actions (login required)

View Item |