Rainer Schrader

Bernhard Korte

Laszlo Lovasz

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

Jens Vygen

This well-written textbook on combinatorial optimization puts special emphasis on theoretical results and algorithms with provably good performance, in contrast to heuristics. The book contains complete (but concise) proofs, as well as many deep results, some of which have not appeared in any previous books.

Theory and Algorithms

Combinatorial Optimization

Questo libro di testo di ottimizzazione combinatoria pone in particolare risalto i risultati teorici e gli algoritmi che, al contrario delle euristiche, hanno una garanzia di avere buone prestazioni. Comprende una vasta scelta di argomenti e nasce come riferimento di diversi corsi di ottimizzazione combinatoria sia di base che di livello avanzato. Il libro contiene dimostrazioni complete (ma concise) anche di molti risultati avanzati, alcuni dei quali non sono mai apparsi prima in un libro. Vengono anche trattati molti dei temi di ricerca più attuali e sono riportati molti riferimenti alla letteratura. Quindi questo libro, traduzione della quarta edizione in lingua originale, rappresenta lo stato dell’arte dell’ottimizzazione combinatoria.

Teoria e Algoritmi

Ottimizzazione Combinatoria

Now fully updated in a third edition, this is a comprehensive textbook on combinatorial optimization. It puts special emphasis on theoretical results and algorithms with provably good performance, in contrast to heuristics. The book contains complete but concise proofs, also for many deep results, some of which have not appeared in print before. Recent topics are covered as well, and numerous references are provided. This third edition contains a new chapter on facility location problems, an area which has been extremely active in the past few years. Furthermore there are several new sections and further material on various topics. New exercises and updates in the bibliography were added.

Bernhard Korte

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