Greedy Algorithm compatibility and heavy-set structures

Abstract. Phylogenetic diversity (PD) is a measure of the extent to which different subsets of taxa span an evolutionary tree, and provides a quantitative tool for studying biodiversity conservation. Recently, it was shown that the problem of finding subsets of taxa of given size to maximize PD can be efficiently solved by a greedy algorithm. In this paper, we extend this earlier work, beginning with a more explicit description of the underlying combinatorial structure of the problem and its connection to greedoid theory. Next we show that an extension of the PD optimization problem to a phylogeographic setting is NP-hard, although a special case has a polynomial-time solution based on the greedy algorithm. We also show how the greedy algorithm can be used to solve some special cases of the PD optimization problem when the sets that are restricted to are ecologically 'viable'. Finally, we show that three measures related to PD fail to be optimized by a greedy algorithm.

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“…Bryant and Brooksbank, and Goecke derive a number of properties of this type of greedoid in [4] and [11], respectively.…”

Section: Some Facts From Greedoid Theorymentioning

confidence: 99%

Optimizing phylogenetic diversity under constraints

Moulton

Semple

Steel

2007

Journal of Theoretical Biology

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“…The next theorem shows that the ordered transversal structures are examples of strong greedoids which , as we showed in [2] , are precisely the structures which are fully compatible with the greedy algorithm . Indeed , it was the search for a large class of compatible set systems which led us to the ordered transversal structures .…”

Section: B Asic D Efinitions and R Esultsmentioning

confidence: 59%

“…In [2] , strong greedoids are defined as greedoids which satisfy the following stronger version of (M3) , known as the strong exchange property :…”

Section: B Asic D Efinitions and R Esultsmentioning

confidence: 99%

“…(2) ( E , Ᏹ ) is a strong greedoid with the following additional property : (M3) † if X Ᏹ and Z ϶ л , then there exists x X with X ‫گ‬ ͕ x ͖ Ᏹ and such that (2) . Let ( E , Ᏹ ) be a transversal greedoid and let Ꮽ ϭ ( A 1 , .…”

Section: C Haracterization Of the T Ransversal G Reedoidsmentioning

confidence: 99%

“…Examination of the structure of these systems reveals that they are in fact examples of strong greedoids (see [2]) , namely greedoids which are greedy algorithm compatible , and we call them transversal greedoids . In this paper , such set systems are shown to be characterized by a simple exchange property .…”

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confidence: 99%

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Transversal Greedoids

Brooksbank

1997

European Journal of Combinatorics

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In a natural way , a new class of set system is defined in terms of transversals of a family of sets . Examination of the structure of these systems reveals that they are in fact examples of strong greedoids (see [2]) , namely greedoids which are greedy algorithm compatible , and we call them transversal greedoids . In this paper , such set systems are shown to be characterized by a simple exchange property .

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Gaussian, Strong and Transversal Greedoids

Bryant

Sharpe

1999

European Journal of Combinatorics

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Various types of greedoids have been studied in relation to the Greedy Algorithm and therefore it is not surprising that these structures are inter-linked in various ways. However, it is quite surprising that the Gauss greedoids and the strong greedoids, approached from quite different points of view, turn out to be precisely the same. Although this can be seen very indirectly by combining the two sets of separate results, the main purpose of this short paper is to prove directly that the two structures are identical, thus giving a simple axiomatic characterisation of the Gauss greedoids. We also note that the transversal greedoids provide an interesting class of examples of such greedoids. c 1999 Academic Press THE GAUSS GREEDOIDSRecall that a greedoid (E, E) consists of a finite set E and a collection of subsets E of E which satisfy:In both cases, the members of E are called the independent sets, the maximal independent sets are the bases, and the common cardinality of the bases is the rank of the structure. More generally, the rank of a set is the cardinality of its maximal independent subsets, and a flat is a maximal set of any given rank. All the standard terminology and results concerning matroids can be found in [6] and the corresponding results for greedoids in [5].We now consider some greedoids motivated by the work of Goecke [3, 4] (and recalled in [5]).DEFINITION. An inclusion chain = (E 1 , E 2 , . . . , E n ) consists of a family of matroids on a ground set E, where each E i ⊂ E i+1 and where the rank of each E i equals i. Given such a define the associated Gaussian structure G( ) as the collection of subsets of E consisting of the empty set ∅ together with all sets of the form {x 1 , x 2 , . . . , x k } such that {x 1 , x 2 , . . . , x i } ∈ E i for 1 i k.It is straightforward to check that a Gaussian structures is a greedoid. Note also that each of its non-empty independent sets is a basis of one of the matroids in the inclusion chain, but that not all such bases are necessarily independent in the Gaussian structure. Now in order to consider a strengthened form of an inclusion chain based on Goecke's work using 'strong maps' we need the following result: KEY LEMMA. Let (E, E 1 ) be a matroid of rank one less than the rank of the matroid (E, E 2 ). Then the following properties are equivalent:(1) Every flat of E 1 is also a flat of E 2 .(2) E 1 ⊂ E 2 and, given bases B 1 and B 2 of (E, E 1 ) and (E, E 2 ), respectively, there is anx ∈ B 2 \ B 1 such that B 1 ∪ {x} ∈ E 2 and B 2 \ {x} ∈ E 1 .

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Greedy Algorithm compatibility and heavy-set structures

Cited by 6 publications

References 2 publications

Optimizing phylogenetic diversity under constraints

Optimizing phylogenetic diversity under constraints

Transversal Greedoids

Gaussian, Strong and Transversal Greedoids

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