Degeneracy Graphs and the Neighbourhood Problem

by H.-J. Kruse

Overview

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.

Product Details

ISBN-13:	9783540160496
Publisher:	Springer Berlin Heidelberg
Publication date:	03/17/1986
Series:	Lecture Notes in Economics and Mathematical Systems , #260
Edition description:	1986
Pages:	132
Product dimensions:	6.10(w) x 9.25(h) x 0.01(d)

Table of Contents

1. Introduction.- 2. The Concept of Degeneracy.- 3. Degeneracy Graphs.- 3.1 The concept of degeneracy graphs.- 3.2 Properties of degeneracy graphs.- 3.3 Degeneracy tableaux.- 4. On the Number of Nodes of Degeneracy Graphs.- 4.1 The maximum number of nodes of degeneracy graphs.- 4.2 The density of degeneracy tableaux.- 4.3 The minimum number of nodes of degeneracy graphs.- 4.4 On the existence and uniqueness of—xn-degeneracy graphs.- 4.5 An algorithm for determining the number of nodes of degeneracy graphs.- 5. A Method to Solve the Neighbourhood Problem.- 5.1 Examples of the occurrence of the neighbourhood problem.- 5.2 Solution of the neighbourhood problem by means of degeneracy graphs.- Appendices.- A. Basic concepts of linear programming and of theory of convex polytopes.- B. Basic concepts of graph theory.- C. On 2xn-degeneracy graphs.- D. Flow-charts.- References.- Index of symbols.- Index of terms.