imported>Spamakin: Grammatical Fix (embedding -> embeddings)

2025-11-24T16:18:40Z

Grammatical Fix (embedding -> embeddings)

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	# The collection of all subsets of feasible sets forms a [[matroid]].		# The collection of all subsets of feasible sets forms a [[matroid]].

	Matroid ~~embedding was~~ introduced by {{harvtxt\|Helman\|Moret\|Shapiro\|1993}} to characterize problems that can be optimized by a [[greedy algorithm]].		Matroid embeddings were introduced by {{harvtxt\|Helman\|Moret\|Shapiro\|1993}} to characterize problems that can be optimized by a [[greedy algorithm]].

	== References ==		== References ==

imported>Bkell: formatting touchups

2022-10-31T19:28:52Z

formatting touchups

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{{Short description|Set system related to matroids}}
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'', where ext(''X'') is 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 ''e'' or no ''e'' in ext(''X'').
# The collection of all subsets of feasible sets forms a [[matroid]].

Matroid embedding was introduced by {{harvtxt|Helman|Moret|Shapiro|1993}} to characterize problems that can be optimized by a [[greedy algorithm]].

== References ==
*{{citation
| last1 = Helman | first1 = Paul
| last2 = Moret | first2 = Bernard M. E. | author2-link = Bernard Moret
| last3 = Shapiro | first3 = Henry D.
| doi = 10.1137/0406021
| issue = 2
| journal = [[SIAM Journal on Discrete Mathematics]]
| mr = 1215233
| pages = 274–283
| title = An exact characterization of greedy structures
| volume = 6
| year = 1993| citeseerx = 10.1.1.37.1825}}

[[Category:Matroid theory|Embedding]]

Matroid embedding - Revision history

imported>Spamakin: Grammatical Fix (embedding -> embeddings)

imported>Bkell: formatting touchups