Greedoid

 ( Order Theory ) 

In combinatorics, a greedoid is a type of set system. It arises from the notion of the matroid, which was originally introduced by Hassler Whitney in 1935 to study and was later used by Jack Edmonds to characterize a class of optimization problems that can be solved by . Around 1980, Bernhard 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.

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Definitions A set system is a collection of of a ground set (i.e. is a subset of the power set of ). When considering a greedoid, a member of is called a feasible set. When considering a matroid, a feasible set is also known as an independent set.

An accessible set system is a set system in which every nonempty feasible set contains an element such that $X\; \backslash setminus\; \backslash \{x\backslash \}$ is feasible. This implies that any nonempty, finite set, accessible set system necessarily contains the empty set ∅.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.

A greedoid is a finite accessible set system that satisfies the exchange property:

• for all $X,\; Y\; \backslash in\; F$ with $|X|>|Y|,$ there is some $x\; \backslash in\; X\; \backslash setminus\; Y$ such that $Y\; \backslash cup\; \backslash \{x\backslash \}\; \backslash in\; F.$

(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 of is a maximal feasible set contained in .

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 of is the size of a basis of . Just as with matroids, greedoids have a cryptomorphism in terms of rank functions. A function $r:2^E\; \backslash to\; \backslash Z$ is the rank function of a greedoid on the ground set if and only if is subcardinal, monotonic, and locally semimodular, that is, for any $X,Y\; \backslash subseteq\; E$ and any $e,f\; \backslash in\; E$ we have:

• subcardinality: $r(X)\backslash le|X|$
• monotonicity: $r(X)\backslash le\; r(Y)$ whenever $X\; \backslash subseteq\; Y\; \backslash subseteq\; E$
• local semimodularity: $r(X)\; =\; r(X\backslash cup\backslash \{e,f\backslash \})$ whenever $r(X)\; =\; r(X\; \backslash cup\; \backslash \{e\backslash \})\; =\; r(X\; \backslash cup\; \backslash \{f\backslash \})$

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Classes 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 is a greedoid that satisfies the Interval Property:

• if $A,B,C\; \backslash in\; F$ with $A\; \backslash subseteq\; B\; \backslash subseteq\; C,$ then, for all $x\; \backslash in\; E\; \backslash setminus\; C:$

$$\backslash begin\{matrix\}\; A\; \backslash cup\; \backslash \{x\backslash \}\; \backslash in\; F\; \backslash \backslash \; C\; \backslash cup\; \backslash \{x\backslash \}\; \backslash in\; F\; \backslash end\{matrix\}\; \backslash implies\; B\; \backslash cup\; \backslash \{x\backslash \}\; \backslash in\; F.$$

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 is a greedoid that satisfies the Interval Property without Upper Bounds:

• if with then, for all implies

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 is a non-empty greedoid that satisfies the Interval Property without Lower Bounds:

• if with then, for all implies

It is easy to see that a matroid is also an interval greedoid.

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Examples

• Consider an undirected graph . Let the ground set be the edges of and the feasible sets be the edge set of each forest (i.e. subgraph containing no cycle) of . 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 rooted graph at the vertex . Let the ground set be the vertices of and the feasible sets be the vertex subsets containing that induce connected subgraphs of . This is called the vertex search greedoid and is a kind of antimatroid.
• Consider a finite, directed graph rooted at . Let the ground set be the (directed) edges of D and the feasible sets be the edge sets of each directed subtree rooted at with all edges pointing away from . 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 matrix . Let the ground set be the indices of the columns from 1 to and the feasible sets be $$F\; =\; \backslash \{X\; \backslash subseteq\; E:\; \backslash text\{\; submatrix\; \}\; M\_\{\backslash \{1,\backslash ldots,|X|\backslash \},X\}\; \backslash text\{\; is\; an\; invertible\; matrix\}\backslash \}.$$ This is called the Gaussian elimination greedoid because this structure underlies the Gaussian elimination algorithm. It is a greedoid, but not an interval greedoid.

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Greedy algorithm 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 with finite.

A subset of is rank feasible if the largest intersection of with any feasible set has size equal to the rank of . In a matroid, every subset of is rank feasible. But the equality does not hold for greedoids in general.

A function $w:\; E\; \backslash to\; \backslash R$ is R-compatible if $\backslash \{x\; \backslash in\; E:\; w(x)\; \backslash geq\; c\backslash \}$ is rank feasible for all .

An objective function $f:\; 2^S\; \backslash to\; \backslash R$ is linear over a set $S$ if, for all $X\; \backslash subseteq\; S,$ we have $f(X)\; =\; \backslash sum\_\{x\; \backslash in\; X\}\; w(x)$ for some weight function $w:\; S\; \backslash to\; \backslash Re.$

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.

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See also

• Matroid
• Polymatroid

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External links

• Introduction to Greedoids
• Theory of Greedy Algorithms
• Submodular Functions and Optimization
• Matchings, Matroids and Submodular Functions

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