Geometry of Cuts and Metrics

Cuts and metrics are well-known objects that arise - independently, but with many deep and fascinating connections - in diverse fields: in graph theory, combinatorial optimization, geometry of numbers, distance geometry, combinatorial matrix theory, statistical physics, VLSI design etc. A main feature of this book is its interdisciplinarity. The book contains a wealth of results, from different mathematical disciplines, which are presented here in a unified and comprehensive manner. Geometric representations and methods turn out to be the linking theme. This book will provide a unique and invaluable source for researchers and graduate students.

From the Reviews:

"This book is definitely a milestone in the literature of integer programming and combinatorial optimization. It draws from the interdisciplinarity of these fields as it gathers methods and results from polytope theory, geometry of numbers, probability theory, design and graph theory around two objects, cuts and metrics. [... ] The book is very nicely written [... ] The book is also very well structured. With knowledge about the relevant terms, one can enjoy special subsections without being entirely familiar with the rest of the chapter. This makes it not only an interesting research book but even a dictionary. [... ] In my opinion, the book is a beautiful piece of work. The longer one works with it, the more beautiful it becomes." Robert Weismantel, Optima 56 (1997)

"... In short, this is a very interesting book which is nice to have." Alexander I. Barvinok, MR 1460488 (98g:52001)

"... This is a large and fascinating book. As befits a book which contains material relevant to so many areas of mathematics (and related disciplines such as statistics, physics, computing science, and economics), it is self-contained and written in a readable style. Moreover, the index, bibliography, and table of contents are all that they should be in such awork; it is easy to find as much or as little introductory material as needed." R.Dawson, Zentralblatt MATH Database 0885.52001

Contenu

Outline of the Book.- I.Measure Aspects: El-Embeddability and Probability.- Preliminaries on Distances.- The Cut Cone and #x2113;-Metrics.- The Correlation Cone and {0. 1}-Covariances.- Conditions for -Embeddability.- Operations.- -Metrics from Lattices, Semigroups and Normed Spaces.- Metric Transforms of -Spaces.- Lipschitz Embeddings.- Dimensionality Questions for -Embeddings.- Examples of the Use of the -Metric.- Basic Definitions.- I1.Hypermetric Spaces: an Approach via Geometry of Numbers.- Preliminaries on Lattices.- Hypermetrics and Delaunay Polytopes.- Delaunay Polytopes: Rank and Hypermetric Faces.- Extreme Delaunay Polytopes.- Hypermetric Graphs.- I11.Isometric Embeddings of Graphs.- Preliminaries on Graphs.- Isometric Embeddings of Graphs into Hypercubes.- Isometric Embeddings of Graphs into Cartesian Products.- -Graphs.- IV.Hypercube Embeddings and Designs.- Rigidity of the Equidistant Metric.- Hypercube Embeddings of the Equidistant Metric.- Recognition of Hypercube Embeddable Metrics.- Cut Lattices, Quasi -Distances and Hilbert Bases.- V.Facets of the Cut Cone and Polytope.- Operations on Valid Inequalities and Facets.- Triangle Inequalities.- Hypermetric Inequalities.- Clique-Web Inequalities.- Other Valid Inequalities and Facets.- Geometric Properties.