In combinatorics, a greedoid is a type of set system. It arises from the notion of the matroid, which was originally introduced by Whitney in 1935 to study planar graphs and was later used by Edmonds to characterize a class of optimization problems that can be solved by greedy algorithms. Around 1980, Korte and Lovász introduced the greedoid to further generalize this characterization of greedy algorithms; hence the name greedoid. Besides mathematical optimization, greedoids have also been connected to graph theory, language theory, order theory, and other areas of mathematics.
DefinitionsEdit
A set system Template:Math is a collection Template:Mvar of subsets of a ground set Template:Mvar (i.e. Template:Mvar is a subset of the power set of Template:Mvar). When considering a greedoid, a member of Template:Mvar is called a feasible set. When considering a matroid, a feasible set is also known as an independent set.
An accessible set system Template:Math is a set system in which every nonempty feasible set Template:Mvar contains an element Template:Mvar such that <math>X \setminus \{x\}</math> is feasible. This implies that any nonempty, finite, accessible set system necessarily contains the empty set ∅.<ref>Note that the accessibility property is strictly weaker than the hereditary property of a matroid, which requires that every subset of an independent set be independent.</ref>
A greedoid Template:Math is a finite accessible set system that satisfies the exchange property:
- for all <math>X, Y \in F</math> with <math>|X|>|Y|,</math> there is some <math>x \in X \setminus Y</math> such that <math>Y \cup \{x\} \in F.</math>
(Note: Some people reserve the term exchange property for a condition on the bases of a greedoid, and prefer to call the above condition the “augmentation property”.)
A basis of a greedoid is a maximal feasible set, meaning it is a feasible set but not contained in any other one. A basis of a subset Template:Mvar of Template:Mvar is a maximal feasible set contained in Template:Mvar.
The rank of a greedoid is the size of a basis. By the exchange property, all bases have the same size. Thus, the rank function is well defined. The rank of a subset Template:Mvar of Template:Mvar is the size of a basis of Template:Mvar. Just as with matroids, greedoids have a cryptomorphism in terms of rank functions.<ref>Template:Citation </ref> A function <math>r:2^E \to \Z</math> is the rank function of a greedoid on the ground set Template:Mvar if and only if Template:Mvar is subcardinal, monotonic, and locally semimodular, that is, for any <math>X,Y \subseteq E</math> and any <math>e,f \in E</math> we have:
- subcardinality: <math>r(X)\le|X|</math>
- monotonicity: <math>r(X)\le r(Y)</math> whenever <math>X \subseteq Y \subseteq E</math>
- local semimodularity: <math>r(X) = r(X\cup\{e,f\})</math> whenever <math>r(X) = r(X \cup \{e\}) = r(X \cup \{f\})</math>
ClassesEdit
Most classes of greedoids have many equivalent definitions in terms of set system, language, poset, simplicial complex, and so on. The following description takes the traditional route of listing only a couple of the more well-known characterizations.
An interval greedoid Template:Math is a greedoid that satisfies the Interval Property:
- if <math>A,B,C \in F</math> with <math>A \subseteq B \subseteq C,</math> then, for all <math>x \in E \setminus C:</math>
<math display=block> \begin{matrix} A \cup \{x\} \in F \\ C \cup \{x\} \in F \end{matrix} \implies B \cup \{x\} \in F.</math>
Equivalently, an interval greedoid is a greedoid such that the union of any two feasible sets is feasible if it is contained in another feasible set.
An antimatroid Template:Math is a greedoid that satisfies the Interval Property without Upper Bounds:
- if Template:Tmath with Template:Tmath then, for all Template:Tmath Template:Tmath implies Template:Tmath
Equivalently, an antimatroid is (i) a greedoid with a unique basis; or (ii) an accessible set system closed under union. It is easy to see that an antimatroid is also an interval greedoid.
A matroid Template:Math is a non-empty greedoid that satisfies the Interval Property without Lower Bounds:
- if Template:Tmath with Template:Tmath then, for all Template:Tmath Template:Tmath implies Template:Tmath
It is easy to see that a matroid is also an interval greedoid.
ExamplesEdit
- Consider an undirected graph Template:Mvar. Let the ground set be the edges of Template:Mvar and the feasible sets be the edge set of each forest (i.e. subgraph containing no cycle) of Template:Mvar. This set system is called the cycle matroid. A set system is said to be a graphic matroid if it is the cycle matroid of some graph. (Originally cycle matroid was defined on circuits, or minimal dependent sets. Hence the name cycle.)
- Consider a finite, undirected graph Template:Mvar rooted at the vertex Template:Mvar. Let the ground set be the vertices of Template:Mvar and the feasible sets be the vertex subsets containing Template:Mvar that induce connected subgraphs of Template:Mvar. This is called the vertex search greedoid and is a kind of antimatroid.
- Consider a finite, directed graph Template:Mvar rooted at Template:Mvar. Let the ground set be the (directed) edges of D and the feasible sets be the edge sets of each directed subtree rooted at Template:Mvar with all edges pointing away from Template:Mvar. This is called the line search greedoid, or directed branching greedoid. It is an interval greedoid, but neither an antimatroid nor a matroid.
- Consider an Template:Math matrix Template:Mvar. Let the ground set Template:Mvar be the indices of the columns from 1 to Template:Mvar and the feasible sets be <math display=block>F = \{X \subseteq E: \text{ submatrix } M_{\{1,\ldots,|X|\},X} \text{ is an invertible matrix}\}.</math> This is called the Gaussian elimination greedoid because this structure underlies the Gaussian elimination algorithm. It is a greedoid, but not an interval greedoid.
Greedy algorithmEdit
In general, a greedy algorithm is just an iterative process in which a locally best choice, usually an input of maximum weight, is chosen each round until all available choices have been exhausted. In order to describe a greedoid-based condition in which a greedy algorithm is optimal (i.e., obtains a basis of maximum value), we need some more common terminologies in greedoid theory. Without loss of generality, we consider a greedoid Template:Math with Template:Mvar finite.
A subset Template:Mvar of Template:Mvar is rank feasible if the largest intersection of Template:Mvar with any feasible set has size equal to the rank of Template:Mvar. In a matroid, every subset of Template:Mvar is rank feasible. But the equality does not hold for greedoids in general.
A function <math>w: E \to \R</math> is R-compatible if <math>\{x \in E: w(x) \geq c\}</math> is rank feasible for all real numbers Template:Mvar.
An objective function <math>f: 2^S \to \R</math> is linear over a set <math>S</math> if, for all <math>X \subseteq S,</math> we have <math displaystyle=inline>f(X) = \sum_{x \in X} w(x)</math> for some weight function <math>w: S \to \Re.</math>
Proposition. A greedy algorithm is optimal for every R-compatible linear objective function over a greedoid.
The intuition behind this proposition is that, during the iterative process, each optimal exchange of minimum weight is made possible by the exchange property, and optimal results are obtainable from the feasible sets in the underlying greedoid. This result guarantees the optimality of many well-known algorithms. For example, a minimum spanning tree of a weighted graph may be obtained using Kruskal's algorithm, which is a greedy algorithm for the cycle matroid. Prim's algorithm can be explained by taking the line search greedoid instead.
See alsoEdit
ReferencesEdit
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