12,000

We have over 12,000 students, from over 100 countries, within one of the safest campuses in the UK

97%

97% of Lancaster students go into work or further study within six months of graduating

View graph of relations
### Links

### Text available via DOI:

Research output: Contribution to journal › Journal article

Published

<mark>Journal publication date</mark> | 2000 |
---|---|

<mark>Journal</mark> | Mathematical Programming |

Issue | 1 |

Volume | 87 |

Number of pages | 20 |

Pages | 37-56 |

<mark>Original language</mark> | English |

Separation is of fundamental importance in cutting-plane based techniques for Integer Linear Programming (ILP). In recent decades, a considerable research effort has been devoted to the definition of effective separation procedures for families of well-structured cuts. In this paper we address the separation of Chvátal rank-1 inequalities in the context of general ILP’s of the form min {c^Tx: Ax≤b,x integer}, where A is an m×n integer matrix and b an m-dimensional integer vector. In particular, for any given integer k we study mod-k cuts of the form λ^T A x ≤ ⌊λ T b⌋ for any λ ∈ {0,1/k,...,(k−1)/k}^m such that λ^T A is integer. Following the line of research recently proposed for mod-2 cuts by Applegate, Bixby, Chvátal and Cook [1] and Fleischer and Tardos [19], we restrict to maximally violated cuts, i.e., to inequalities which are violated by (k−1)/k by the given fractional point. We show that, for any given k, such a separation requires O(mn min{m,n}) time. Applications to both the symmetric and asymmetric TSP are discussed. In particular, for any given k, we propose an O(|V|^2|E*|)-time exact separation algorithm for mod-k cuts which are maximally violated by a given fractional (symmetric or asymmetric) TSP solution with support graph G *=(V,E*). This implies that we can identify a maximally violated cut for the symmetric TSP whenever a maximally violated (extended) comb inequality exists. Finally, facet-defining mod-k cuts for the symmetric and asymmetric TSP are studied.

## Algorithms for the Travelling Salesman Problem

Project: Non-funded Project › Research

## Cutting-Plane Approaches to Integer Programming

Project: Non-funded Project › Research