Matroid embedding
Encyclopedia
In combinatorics
, a matroid embedding is a set system (F, E), where F is a collection of feasible sets, that satisfies the following properties:
Matroid embedding was introduced by Helman et al. in 1993 to characterize problems that can be optimized by a greedy algorithm
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Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
, 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 supersetSuperSetSuperSet Software was a group founded by friends and former Eyring Research Institute co-workers Drew Major, Dale Neibaur, Kyle Powell and later joined by Mark Hurst...
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 feasible sets forms a matroidMatroidIn combinatorics, a branch of mathematics, a matroid or independence structure is a structure that captures the essence of a notion of "independence" that generalizes linear independence in vector spaces....
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Matroid embedding was introduced by Helman et al. in 1993 to characterize problems that can be optimized by a greedy algorithm
Greedy algorithm
A greedy algorithm is any algorithm that follows the problem solving heuristic of making the locally optimal choice at each stagewith the hope of finding the global optimum....
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