## Abstract

Let G = (V, E) be a graph and let w be a weight function w:E →Z^{+}. Let {Mathematical expression} be an even subset of the vertices of G. A T-cut is an edge-cutset of the graph which divides T into two odd sets. A T-join is a minimal subset of edges that meets every T-cut (a generalization of solutions to the Chinese Postman problem). The main theorem of this paper gives a tight upper bound on the difference between the minimum weight T-join and the maximum weight integral packing of T-cuts. This difference is called the (T-join) integral duality gap. Let τ_{w} be the minimum weight of a T-join, and let v_{w} be the maximum weight of an integral packing of T-cuts. If F is a non-empty minimum weight T-join, and n_{F} is the number of components of F, then we prove that τ_{w}-v_{w}≤n_{F}-1. This result unifies and generalizes Fulkerson's result for |T|=2 and Seymour's result for |T|= 4. For a certain integral multicommodity flow problem in the plane, which was recently proved to be NP-complete, the above result gives a solution such that for every commodity the flow is less than the demand by at most one unit.

Original language | English |
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Pages (from-to) | 183-191 |

Number of pages | 9 |

Journal | Mathematical Programming |

Volume | 55 |

Issue number | 1-3 |

DOIs | |

State | Published - 1 Apr 1992 |

Externally published | Yes |

## Keywords

- Chinese Postman problem
- T-cuts
- integral LP duality
- integral packing
- plane integral multicommodity flows

## ASJC Scopus subject areas

- Software
- General Mathematics