# Degeneracy Graphs and the Neighbourhood Problem

Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 260)

Advertisement

Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 260)

A few years ago nobody would have anticipated that in connection with degeneracy in Linear Programming quite a new field. could originate. In 1976 a very simple question has been posed: in the case an extreme pOint (EP) of a polytope is degenerate and the task is to find all neighbouring EP's of the degenerate EP, is it necessary to determine all basic solutions of the corresponding equalities system associated with the degenerate EP -in order to be certain to determine all neighbours of this EP? This question implied another one: Does there exists a subset of the mentioned set of basic solutions such that it suffices to find such a subset in order to determine all neighbours? The first step to solve these questions (which are motivated in the first Chapter of this book) was to define a graph (called degeneracy graph) the nodes of which correspond to the basic solutions. It turned out that such a graph has some special properties and in order to solve the above questions firstly these properties had to be investigated. Also the structure of degeneracy graphs playes hereby an important role. Because the theory of degeneracy graphs was quite new, it was necessary to elaborate first a completely new terminology and to define new notions. Dr.

algorithms graph theory linear optimization sensitivity analysis

- DOI https://doi.org/10.1007/978-3-642-49270-9
- Copyright Information Springer-Verlag Berlin Heidelberg 1986
- Publisher Name Springer, Berlin, Heidelberg
- eBook Packages Springer Book Archive
- Print ISBN 978-3-540-16049-6
- Online ISBN 978-3-642-49270-9
- Series Print ISSN 0075-8442
- Buy this book on publisher's site