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.

lgrsnf/A:\compressed\10.1007%2F978-3-642-58191-5.pdf

nexusstc/Greedoids/30124f0d6576b582cd46232dcf5de4f4.pdf

zlib/Mathematics/Bernhard Korte, Rainer Schrader, László Lovász (auth.)/Greedoids_2146789.pdf

Bernhard Korte; László Lovász; Rainer Schrader

Bernhard Korte; Laszlo Lovasz; Rainer Schrader

B H Korte; Rainer Schrader; László Lovász

Spektrum Akademischer Verlag. in Springer-Verlag GmbH

Steinkopff. in Springer-Verlag GmbH

Algorithms and combinatorics, 4, Berlin, Heidelberg, 1991

Algorithms and Combinatorics, 4, Berlin, 2013, [2013?

Springer Nature, Berlin, Heidelberg, 2012

Place of publication not identified, 2013

Germany, Germany

Oct 18, 2012

lg2757860

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Source title: Greedoids (Algorithms and Combinatorics (4))

Front Matter....Pages I-VIII
Introduction....Pages 1-8
Abstract Linear Dependence — Matroids....Pages 9-18
Abstract Convexity — Antimatroids....Pages 19-43
General Exchange Structures — Greedoids....Pages 45-55
Structural Properties....Pages 57-75
Further Structural Properties....Pages 77-88
Local Poset Greedoids....Pages 89-106
Greedoids on Partially Ordered Sets....Pages 107-118
Intersection, Slimming and Trimming....Pages 119-139
Transposition Greedoids....Pages 141-151
Optimization in Greedoids....Pages 153-181
Topological Results for Greedoids....Pages 183-195
Back Matter....Pages 197-214

This monograph attempts to unify different mathematical approaches and to lead the reader from fundamental results in matroid theory to the current state-of-the-art in open research problems. It reviews classical concepts from matroid theory and extends them to greedoids ("greedy" algorithms).

2013-08-01