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## An Exact Characterization of Greedy Structures (1993)

Venue: | SIAM Journal on Discrete Mathematics |

Citations: | 12 - 1 self |

### Citations

149 |
Matroids and the greedy algorithm
- Edmonds
- 1971
(Show Context)
Citation Context ...dings. 1 Introduction Obtaining an exact characterization of the class of problems for which the greedy algorithm returns an optimal solution has been an open problem. Rado [9], Gale [3], and Edmonds =-=[1]-=- independently showed that matroids characterize a subclass of problems on which the greedy algorithm always optimizes linear objectives; their results are limited by the assumption that the greedy al... |

33 |
Optimal assignments in an ordered set: an application of matroid theory
- Gale
- 1968
(Show Context)
Citation Context ...han matroid embeddings. 1 Introduction Obtaining an exact characterization of the class of problems for which the greedy algorithm returns an optimal solution has been an open problem. Rado [9], Gale =-=[3]-=-, and Edmonds [1] independently showed that matroids characterize a subclass of problems on which the greedy algorithm always optimizes linear objectives; their results are limited by the assumption t... |

33 | Note on independence functions
- Rado
- 1957
(Show Context)
Citation Context ...strained than matroid embeddings. 1 Introduction Obtaining an exact characterization of the class of problems for which the greedy algorithm returns an optimal solution has been an open problem. Rado =-=[9]-=-, Gale [3], and Edmonds [1] independently showed that matroids characterize a subclass of problems on which the greedy algorithm always optimizes linear objectives; their results are limited by the as... |

26 |
Mathematical structures underlying greedy algorithms
- Korte, Lovasz
- 1981
(Show Context)
Citation Context ...s on which the greedy algorithm optimizes linear objectives, but the assumption of a partial order, which constrains the choices of the greedy algorithm, limits the characterization. Korte and Lovasz =-=[6; 7]-=- have defined greedoids, a generalization of matroids, and have provided necessary and sufficient conditions for the greedy algorithm to be optimal with respect to linear objectives when run on greedo... |

9 | Greedoids and linear objective functions - Korte, Lovász - 1984 |

8 |
The greedy algorithm for partially ordered sets
- Faigle
- 1979
(Show Context)
Citation Context ...s are limited by the assumption that the greedy algorithm operates on a hereditary set system, whereas most common greedy algorithms operate on set systems that do not obey the heredity axiom. Faigle =-=[2]-=- has provided an exact characterization of the partially ordered set systems on which the greedy algorithm optimizes linear objectives, but the assumption of a partial order, which constrains the choi... |

4 |
A greedy algorithm for hereditary set systems and a generalization of the Rado-Edmonds characterization of matroids, Discrete Applied mathematics 20
- Goecke
- 1988
(Show Context)
Citation Context ... need not return an optimal solution on a greedoid) and too constraining: there exist set systems on which the greedy algorithm always optimizes linear objectives, but which are not greedoids. Goecke =-=[4]-=- has given necessary and sufficient conditions for the optimization of linear objectives over set systems by a variant of the greedy algorithm, but his variant of the greedy algorithm (find any soluti... |

4 |
A theory of greedy structures based on k-ary dominance relations
- Helman
- 1989
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Citation Context ...l bases. 2 Proof: We prove the implications (1) =) (2) and (2) =) (3); the implication (3) =) (1) is trivial. (1) =) (2) We begin by showing that C must be a matroid, a result first derived by Helman =-=[5]-=-. Assume two sets, X; Y 2 C , with jX j = jY j + 1, between which augmentation fails in C . Since augmentation fails, no basis that contains all of Y can contain any element of X \Gamma Y . We design ... |

3 |
Unstructured path problems and the making of semirings
- Lengauer, Theune
- 1991
(Show Context)
Citation Context ...hen so too is any other consistent objective. This suggests a study of families of objective functions along much the same lines as classical complexity theory; in this direction, Lengauer and Theune =-=[8]-=- have demonstrated reductions among cost functions for path problems. Now that we have a proper setting for optimal greedy algorithms, we can investigate the complexity of such algorithms. This proble... |