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Greedoids. (English) Zbl 0733.05023
Algorithms and Combinatorics, 4. Berlin etc.: Springer-Verlag. viii, 211 p. 17 fig. DM 128.00 (1991).

Matroids can be characterized by the greedy algorithms. The maximum weight feasible sets of a set system F (provided $\varnothing \in F$; and $X\in F$ and $Y\subseteq X$ implies $Y\in F\right)$ are found by the greedy algorithm for every weight function if and only if F is the collection of independent sets of a matroid. A more general concept, greedoids, can be obtained if only “reasonable” weight functions are considered (where for ${x}_{0}\notin A\subset B$ if ${x}_{0}$ is optimal among every x with $A\cup \left\{x\right\}$ feasible then ${x}_{0}$ is also optimal among every x with $B\cup \left\{x\right\}$ feasible).

Alternatively, a greedoid is a set system where every subset of a feasible set need not be feasible but the empty set must be reachable from every feasible set through feasible sets by single element deletions.

Thus greedoids are generalizations of matroids, and, at the same time, of “antimatroids” (arising from ideals of posets, from scheduling problems with alternative posets, from scheduling problems with alternative precedence constraints) and many other combinatorial structures related to algorithms and decomposition procedures (like Gaussian elimination, matching algorithms, ear-decompositions of graphs, perfect elimination schemes etc).

The present volume - the first monograph of greedoids - introduces matroids, antimatroids and greedoids in the first 4 chapters. The next 2 chapters are devoted to the structural properties of greedoids. Chapters VII through X deal with special classes of greedoids while optimization and topological aspects are covered in the last two chapters.

The bibliography contains some 140 titles. About 50 of them explicitly deal with greedoids, more than 20 are authored or coauthored by one or more of the presents authors, who introduced the concept and performed most of the research on greedoids. This self-contained volume can warmly be recommended to all scientists interested in algorithmic aspects of mathematics and computer science.

Reviewer: A.Recski

##### MSC:
 05B35 Matroids, geometric lattices (combinatorics) 05-02 Research monographs (combinatorics)
##### Keywords:
greedoids; set system; matroids; antimatroids; structural properties