VLSI (very large scale integration) is a magic acronym in modern science and technology. Amongst other strik- ing aspects this field has turned out to be an excellent arena for applications of discrete mathematics and combinatorial optimization. The contributions to this volume, covering a wide range of research in this important field, develop from lectures delivered at a scientific meeting focused on paths, flows and VLSI-layout, held in Bonn in the summer 1988. The speakers, including several of the most prominent researchers in the field, were asked to formalize and extend their lectures into written surveys, giving a complete account of their topics. The volume will serve as a compendium demonstrating the liveliness of the area, as well as indicating directions for future work.

Path, Flows and Vlsi-Layout (Algorithms & Combinatorics)

Science & Math

Korte, Bernhard

Lovász, László

Prömel, Hans Jürgen

Schrijver, A.

Technology

Springer, Verlag

Discrete Mathematics: Elementary and Beyond (Undergraduate Texts in Mathematics)

Aimed at undergraduate mathematics and computer science students, this book is an excellent introduction to a lot of problems of discrete mathematics. It discusses a number of selected results and methods, mostly from areas of combinatorics and graph theory, and it uses proofs and problem solving to help students understand the solutions to problems. Numerous examples, figures, and exercises are spread throughout the book.

Pelikán, József

Vesztergombi, Katalin

Computers & Technology

Springer

The aim of this book is to introduce a range of combinatorial methods for those who want to apply these methods in the solution of practical and theoretical problems. Various tricks and techniques are taught by means of exercises. Hints are given in a separate section and a third section contains all solutions in detail. A dictionary section gives definitions of the combinatorial notions occurring in the book.Combinatorial Problems and Exercises was first published in 1979. This revised edition has the same basic structure but has been brought up to date with a series of exercises on random walks on graphs and their relations to eigenvalues, expansion properties and electrical resistance. In various chapters the author found lines of thought that have been extended in a natural and significant way in recent years. About 60 new exercises (more counting sub-problems) have been added and several solutions have been simplified.

Combinatorial Problems and Exercises

Mathematics

Elsevier Science Ltd.

This book grew out of the fourth Special Year at DIMACS, which was devoted to the subject of combinatorial optimization. During the special year, a number of workshops, small and large, dealt with various aspects of this theme. Organizers of the workshops and selected participants were asked to write surveys about the hottest results and ideas in their fields. Therefore, this book is not a set of conference proceedings but rather a carefully refereed collection of invited survey articles written by outstanding researchers. Aimed at researchers in discrete mathematics, operations research, and the theory of computing, this book offers an in-depth look at many topics not treated in textbooks.

Combinatorial Optimization: Papers from the Dimacs Special Year (Dimacs Series in Discrete Mathematics and Theoretical Computer Science)

Cook,William

Amer Mathematical Society

A study of how complexity questions in computing interact with classical mathematics in the numerical analysis of issues in algorithm design. Algorithmic designers concerned with linear and nonlinear combinatorial optimization will find this volume especially useful. Two algorithms are studied in detail: the ellipsoid method and the simultaneous diophantine approximation method. Although both were developed to study, on a theoretical level, the feasibility of computing some specialized problems in polynomial time, they appear to have practical applications. The book first describes use of the simultaneous diophantine method to develop sophisticated rounding procedures. Then a model is described to compute upper and lower bounds on various measures of convex bodies. Use of the two algorithms is brought together by the author in a study of polyhedra with rational vertices. The book closes with some applications of the results to combinatorial optimization.

An Algorithmic Theory of Numbers, Graphs and Convexity (CBMS-NSF Regional Conference Series in Applied Mathematics, Series Number 50)

Literature & Fiction

Classics

Society for Industrial and Applied Mathematics

With the advent of computers, algorithmic principles play an ever increasing role in mathematics. Algorithms have to exploit the structure of the underlying mathematical object, and properties exploited by algorithms are often closely tied to classical structural analysis in mathematics. This connection between algorithms and structure is in particular apparent in discrete mathematics, where proofs are often constructive, and can be turned into algorithms more directly. The principle of greediness plays a fundamental role both in the design of continuous algorithms (where it is called the steepest descent or gradient method) and of discrete algorithms. The discrete structure most closely related to greediness is a matroid; in fact, matroids may be characterized axiomatically as those independence systems for which the greedy solution is optimal for certain optimization problems (e.g. linear objective functions, bottleneck functions). This book is an attempt to unify different approaches and to lead the reader from fundamental results in matroid theory to the current borderline of open research problems. The monograph begins by reviewing classical concepts from matroid theory and extending them to greedoids. It then proceeds to the discussion of subclasses like interval greedoids, antimatroids or convex geometries, greedoids on partially ordered sets and greedoid intersections. Emphasis is placed on optimization problems in greedois. An algorithmic characterization of greedoids in terms of the greedy algorithm is derived, the behaviour with respect to linear functions is investigated, the shortest path problem for graphs is extended to a class of greedoids, linear descriptions of antimatroid polyhedra and complexity results are given and the Rado-Hall theorem on transversals is generalized. The self-contained volume which assumes only a basic familarity with combinatorial optimization ends with a chapter on topological results in connection with greedoids.

Greedoids (Algorithms and Combinatorics)

Schrader, Rainer

Die diskrete Mathematik ist im Begriff, zu einem der wichtigsten Gebiete der mathematischen Forschung zu werden mit Anwendungen in der Kryptographie, der linearen Programmierung, der Kodierungstheorie und Informatik. Dieses Buch richtet sich an Studenten der Mathematik und Informatik, die ein Gefühl dafür entwickeln möchten, worum es in der Mathematik geht, wobei Mathematik hilfreich sein kann, und mit welcher Art Fragen sich Mathematiker auseinandersetzen. Die Autoren stellen eine Anzahl ausgewählter Ergebnisse und Methoden der diskreten Mathematik vor, hauptsächlich aus den Bereichen Kombinatorik und Graphentheorie, teilweise aber auch aus der Zahlentheorie, der Wahrscheinlichkeitsrechnung und der kombinatorischen Geometrie.Wo immer es möglich war, haben die Autoren Beweise und Problemlösungen verwendet, um den Studenten zu helfen, die Lösungen der Fragestellungen zu verstehen. Zusätzlich ist eine Vielzahl von Beispielen, Bildern und Übungsaufgaben über das Buch verteilt. László Lovász ist einer der Leiter der theoretischen Forschungsabteilung der Microsoft Corporation. Er hat 1999 den Wolf-Preis sowie den Gödel-Preis für die beste wissenschaftliche Veröffentlichung in der Informatik erhalten. József Pelikán ist Professor am Institut für Algebra und Zahlentheorie der Eötvös Loránd Universität in Budapest. Katalin Vesztergombi ist Senior Lecturer am Fachbereich Mathematik der Universität von Washington in Seattle.

Diskrete Mathematik (Springer-Lehrbuch) (German Edition)

Computer Science

Oh cieca cupidigia, oh ira folie, Che si ci sproni nella vita corta, E nell' eterna poi si mal c'immolle! o blind greediness and foolish rage, That in our fleeting life so goads us on And plunges us in boiling blood for ever! Dante, The Divine Comedy Inferno, XII, 17, 49/51. On an afternoon hike during the second Oberwolfach conference on Mathematical Programming in January 1981, two of the authors of this book discussed a paper by another two of the authors (Korte and Schrader [1981]) on approximation schemes for optimization problems over independence systems and matroids. They had noticed that in many proofs the hereditary property of independence systems and matroids is not needed: it is not required that every subset of a feasible set is again feasible. A much weaker property is sufficient, namely that every feasible set of cardinality k contains (at least) one feasible subset of cardinality k - 1. We called this property accessibility, and that was the starting point of our investigations on greedoids.

Greedoids (Algorithms and Combinatorics, 4)

Since the publication of the first edition of our book, geometric algorithms and combinatorial optimization have kept growing at the same fast pace as before. Nevertheless, we do not feel that the ongoing research has made this book outdated. Rather, it seems that many of the new results build on the models, algorithms, and theorems presented here. For instance, the celebrated Dyer-Frieze-Kannan algorithm for approximating the volume of a convex body is based on the oracle model of convex bodies and uses the ellipsoid method as a preprocessing technique. The polynomial time equivalence of optimization, separation, and membership has become a commonly employed tool in the study of the complexity of combinatorial optimization problems and in the newly developing field of computational convexity. Implementations of the basis reduction algorithm can be found in various computer algebra software systems. On the other hand, several of the open problems discussed in the first edition are still unsolved. For example, there are still no combinatorial polynomial time algorithms known for minimizing a submodular function or finding a maximum clique in a perfect graph. Moreover, despite the success of the interior point methods for the solution of explicitly given linear programs there is still no method known that solves implicitly given linear programs, such as those described in this book, and that is both practically and theoretically efficient. In particular, it is not known how to adapt interior point methods to such linear programs.

Geometric Algorithms and Combinatorial Optimization (Algorithms and Combinatorics)

Schrijver, Alexander

Grötschel, Martin

Reference

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