M. Steel. The complexity of reconstructing trees from qualitative characters and subtrees. J. of Classification, 9:91--116, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a final solution to the overall problem based on the solutions to the quartets in a consistent manner. However, under the quartet paradigm, any attempt to assemble a solution optimally from the solutions of the subproblems (known as the Maximum Quartet Consistency (MQC) problem) is itself NP hard [49]. Many quartet based methods such as Quartet Puzzling (QP) 51] WO method [41] Short Quartet Method [15] solves the MQC problem in a heuristic sense. One method called HyperCleaning [3] will solve a slight relaxed version of the MQC problem exactly under the assump In a nutshell, the ....

Steel, M. The complexity of reconstructing trees from qualitative characters and subtrees. Journal of Classification 9 (1992), 91--116.

....A set of characters is compatible on n species iff the associated character state intersection graph G can be c triangulated. See Figure 3 for a spirit of the proof. The PP and TRIAN GULATED COLORED GRAPH (TCG) problems are equivalent and NP Complete. Kannan and Warnow [KW90] and Steel [St92], also F. McMorris, T. Warnow, and T. Wimer [McMWW93] Additionally they are parametrically hard by Bodlaender, Fellows and Hallett [BFH94] The 2 character case of the perfect phylogeny problem is solvable in polynomial time. For triangulating colored graphs, McMorris, Warnow and Wimer ....

M. Steel, "The Complexity of Reconstructing Trees from Qualitative Characters and Subtrees," J. Classification, Vol. 9 (1992), 91-116.

....of the recognition problem: Certain edges must definitely be included in the graph, and certain edges are disallowed, but there is freedom in deciding to include any subset of the (possibly many) other edges. Sandwich problems have been studied explicitly in [20] and implicitly in [21] 2] and [33]. Below we give several examples of important sandwich problems arising in practice. Definitions of the graph families mentioned in the examples are given in later sections. Physical Mapping of DNA [5] In molecular biology, information on intersection or non intersection of pairs of segments ....

....for the threshold sandwich problem. Phylogenetic Trees: Buneman [4] showed that the perfect phylogeny (PP) problem in evolution reduces to the graph theoretical problem of triangulating colored graphs (TCG) Kannan and Warnow [25] showed that TCG reduces to PP. Bodlaender et al. 2] and Steel [33] recently proved that TCG is NPcomplete. It is easy to see that TCG is a restriction of the chordal sandwich problem, which is therefore also NP complete. Sparse Systems of Linear Equations: Consider the system of equations Ax = b where A is a sparse, symmetric and positive definite. When ....

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M. Steel. The complexity of reconstructing trees from qualitative characters and subtrees. J. of Classification, 9:91--116, 1992. 19

....corresponding character states share a species in common. Thus, the number of colors of TCG corresponds to the number of characters in the Perfect Phylogeny problem. In 1990, Kannan and Warnow [40] showed that these two problems were polynomially equivalent. Though these problems are NP complete [48, 53], over the last few years, polynomial time algorithms for the various fixed parameter versions of these problems have been found. Since molecular data results in characters with few states, attention has particularly been given to the case where the parameter r is bounded. When the characters are ....

....characters with few states, attention has particularly been given to the case where the parameter r is bounded. When the characters are binary (i.e. r = 2) the problem can be solved in O(sk) time [36] For r = 3, character compatibility can be determined in O(s k) time [41] or O(sk ) time [48]. For r = 4 (the case for characters derived from DNA sequences) the problem can be solved in O(s k) time [41] An O(2 (sk k ) time algorithm has been found for the general case by Agarwala and Fernandez Baca [2] This has been improved by Kannan and Warnow [42] to an O(2 sk ....

M. A. Steel. The complexity of reconstructing trees from qualitative characters and subtrees. J. Classification, 9:91--116, 1992.

....In the example only binary characters are used, and on the edges the indix of the character is reported that has evolved from 0 to 1. Definition 2.2 (Perfect Phylogeny problem) Given a set S of N species decide whether or not S is compatible. The Perfect Phylogeny problem is NP complete ( [2, 26]) An O( m k 1 (k 1) k Nk 2 ) algorithm exists ( 22] which is polynomial when the number k of characters is bounded by a constant; in particular a lineartime algorithm is known when only three characters are used ( 14, 13] In the case of binary characters (m = 2) there exists an ....

M. A. STEEL. The complexity of reconstructing trees from qualitative characters and subtrees. Journal of Classification, (9):91--116, 1992.

....1 ; v 2 ; vng. Question: Does there exist a tree TQ with leaves labeled by points in S such that if q = v i v j ; v k v l ) 2 Q, then there is an edge e in TQ such that v i ; v j are on one side of e and v k ; v l are on the other side. The UQC problem was shown to be NP complete by Steel [11]. This is shown by reduction to betweenness problem. We shall use this problems to show the NP hardness of triangle ordinal clustering as well as total ordinal clustering. The counter example of lemma 5, imposes a quartet constraint and shows what edge expansion is needed. The main idea is to ....

M. A. Steel, The complexity of reconstructing trees from qualitative characters and subtrees, J. Classification, 9, 1992.

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M. Steel, The complexity of reconstructing trees from qualitative characters and subtrees, Z. J.Classification 9 1992 , 91]116.

....S of partial splits with T #S#=fTg for some binary X tree T =#V; E; f# must contain at least one partial split for every inner edge that specifically fits this edge and, hence, S must contain at least # E = #X ,3 distinct nontrivial splits in view of inequality (1. 2) It is easily shown [1, 14] that, for S # S part #X# and Q #S# : fQ 2 Q #X# : Q # S for some S 2 Sg; 2.2) the relation T #S#=T #Q #S## must hold. Thus, there is no loss of generality in restricting one s attention to quartet splits when reconstructing (phylogenetic) trees from partial splits. Similarly, we have T #F ....

....: fq#e# : e 2 Fg#Q #X#; and q#F# : # e2F q#e#: We will say that q defines T if T is the only phylogenetic X tree concordant with q# E#. A quartet encoding q is called tight if, for each edge e 2 E, there exists no other edge in E separating the two subsets in q#e#. It is easy to see (cf. [14]) that a quartet encoding that defines a tree T is tight; furthermore, given a binary X tree T and a tight quartet encoding q of T,thenq# E#=X and #q# E#=#X ,3 holds. It is also easy to see that T #Q #=fTg holds for some set of quartet splits Q with #Q = #X ,3 and some (necessarily binary) ....

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M. Steel, The complexity of reconstructing trees from qualitative characters and subtrees, J. Classification 9 (1992) 91--116.

....wish to amalgamate these input trees into a single supertree (parent tree) in such a way that each input tree is displayed by that supertree. Clearly, it may be impossible to amalgamate the input trees in this way, and just determining whether this is the case is known to be an NP hard problem [16]. Furthermore, even when the trees can be amalgamated, there may be exponentially many supertrees. For example, there may be a supertree that has internal vertices of high degree, in which case any refinement of this tree also gives a supertree. Yet, even if every supertree is binary, an ....

....results that also deal with special cases where one can easily determine whether or not F is compatible, and, if so, definitive. For example, if # T#F L(T ) #= # then one can determine in polynomial time (in L(F) whether or not F is compatible [1] and if so whether F is definitive [16]. Alternatively, if the number of trees in F is bounded, then there is also an algorithm that runs in polynomial time in L(F) for answering these last two questions, see [16] Some heuristic and approximation based approaches to tree amalgamation have also been proposed, particularly for ....

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M. Steel, The complexity of reconstructing trees from qualitative characters and subtrees, J. Classification 9: 91--116 (1992).

....of X . In the more general setting of X trees, the first problem is to determine if there exists an X tree that displays # and, if there is such an X tree, determine 4 CHARLES SEMPLE AND MIKE STEEL whether it is unique up to isomorphism. Deciding the first part is an NP complete problem [3, 11]. However, Theorem 1.1 (indicated in [5] and [10] and formally proved in [11] is a graph theoretic characterization for when there exists such an X tree. Theorem 1.1. Let # be a set of partial partitions of X. Then there exists an X tree that displays # if and only if there exists a ....

....if there exists an X tree that displays # and, if there is such an X tree, determine 4 CHARLES SEMPLE AND MIKE STEEL whether it is unique up to isomorphism. Deciding the first part is an NP complete problem [3, 11] However, Theorem 1. 1 (indicated in [5] and [10] and formally proved in [11]) is a graph theoretic characterization for when there exists such an X tree. Theorem 1.1. Let # be a set of partial partitions of X. Then there exists an X tree that displays # if and only if there exists a restricted chordal completion of int(#) Our first main result, Theorem 1.2, is the ....

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M. Steel, The complexity of reconstructing trees from qualitative characters and subtrees, J. Classif. 9(1) (1992) 91--116.

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M. Steel. The complexity of reconstructing trees from qualitative characters and subtrees. J. of Classification, 9:91--116, 1992.

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M. Steel. The complexity of reconstructing trees from qualitative characters and subtrees. J. of Classification, 9:91--116, 1992.

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Steel, M., 1992. The complexity of reconstructing trees from qualitative characters and subtrees. J. Classi#cation 9, 91--116.

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M. Steel. The complexity of reconstructing trees from qualitative characters and subtrees. Journal of Classification, 9:91--116, 1992.

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M. Steel, The complexity of reconstructing trees from qualitative characters and subtrees, J. Classi#cation 9 (1992) 91--116.

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M. Steel, The complexity of reconstructing trees from qualitative characters and subtrees, Journal of Classification, 9 (1992), pp. 91--116.

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M. Steel. The complexity of reconstructing trees from qualitative characters and subtrees. Journal of Classification, 9:91--116, 1992.

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M. Steel. (1992): The complexity of reconstructing trees from qualitative characters and subtrees. Journal of Classification, 9, 91-116.

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M. Steel. The complexity of reconstructing trees from qualitative characters and subtree. Journal of Classification, 9:91--116, 1992.

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M. A. Steel. The complexity of reconstructing trees from qualitative characters and subtrees. Journal of Classi cation, 9:91-116, 1992.

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M. Steel. The complexity of reconstructing trees from qualitative characters and subtrees. Journal of Classification, 9:91--116, 1992.

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Steel, M. The complexity of reconstructing trees from qualitative characters and subtrees. Journal of Classi cation 9 (1992), 91-116.

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M. Steel. The complexity of reconstructing trees from qualitative characters and subtrees. J. of Classification, 9:91--116, 1992.

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M. Steel, The complexity of reconstructing trees from qualitative characters and subtrees, J. Classifications 9(1992), 91-116.

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M. Steel. The complexity of reconstructing trees from qualitative characters and subtrees. J. Classification, Vol. 9 (1992), 91-116.

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