Matroid embedding
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In combinatorics, a matroid embedding is a set system (F, E), where F is a collection of feasible sets, that satisfies the following properties:
- (Accessibility Property) Every non-empty feasible set X contains an element x such that X\{x} is feasible;
- (Extensibility Property) For every feasible subset X of a basis (i.e., maximal feasible set) B, some element in B but not in X belongs to the extension ext(X) of X, or the set of all elements e not in X such that X∪{e} is feasible;
- (Closure-Congruence Property) For every superset A of a feasible set X disjoint from ext(X), A∪{e} is contained in some feasible set for either all or no e in ext(X);
- The collection of all subsets of every feasible set forms a matroid.
Matroid embedding was introduced by Helman et al. in 1993 to characterize problems that can be optimized by a greedy algorithm.
References
- Helman, Paul; Moret, Bernard M. E.; Shapiro, Henry D. (1993). An exact characterization of greedy structures. SIAM J. Discrete Math. 6 (2), 274–283.