A mixed integer linear programming model to reconstruct phylogenies from single nucleotide polymorphism haplotypes under the maximum parsimony criterion
 Daniele Catanzaro^{1}Email author,
 Ramamoorthi Ravi^{2} and
 Russell Schwartz^{3}
https://doi.org/10.1186/1748718883
© Catanzaro et al; licensee BioMed Central Ltd. 2013
Received: 13 June 2012
Accepted: 2 January 2013
Published: 23 January 2013
Abstract
Background
Phylogeny estimation from aligned haplotype sequences has attracted more and more attention in the recent years due to its importance in analysis of many finescale genetic data. Its application fields range from medical research, to drug discovery, to epidemiology, to population dynamics. The literature on molecular phylogenetics proposes a number of criteria for selecting a phylogeny from among plausible alternatives. Usually, such criteria can be expressed by means of objective functions, and the phylogenies that optimize them are referred to as optimal. One of the most important estimation criteria is the parsimony which states that the optimal phylogeny T^{∗}for a set$\mathcal{H}$of n haplotype sequences over a common set of variable loci is the one that satisfies the following requirements: (i) it has the shortest length and (ii) it is such that, for each pair of distinct haplotypes${h}_{i},{h}_{j}\in \mathcal{H}$, the sum of the edge weights belonging to the path from h_{ i } to h_{ j } in T^{∗} is not smaller than the observed number of changes between h_{ i } and h_{ j }. Finding the most parsimonious phylogeny for$\mathcal{H}$involves solving an optimization problem, called the Most Parsimonious Phylogeny Estimation Problem (MPPEP), which is$\mathcal{N}\mathcal{P}$hard in many of its versions.
Results
In this article we investigate a recent version of the MPPEP that arises when input data consist of single nucleotide polymorphism haplotypes extracted from a population of individuals on a common genomic region. Specifically, we explore the prospects for improving on the implicit enumeration strategy of implicit enumeration strategy used in previous work using a novel problem formulation and a series of strengthening valid inequalities and preliminary symmetry breaking constraints to more precisely bound the solution space and accelerate implicit enumeration of possible optimal phylogenies. We present the basic formulation and then introduce a series of provable valid constraints to reduce the solution space. We then prove that these constraints can often lead to significant reductions in the gap between the optimal solution and its nonintegral linear programming bound relative to the prior art as well as often substantially faster processing of moderately hard problem instances.
Conclusion
We provide an indication of the conditions under which such an optimal enumeration approach is likely to be feasible, suggesting that these strategies are usable for relatively large numbers of taxa, although with stricter limits on numbers of variable sites. The work thus provides methodology suitable for provably optimal solution of some harder instances that resist all prior approaches.
Keywords
Combinatorial optimization Exact algorithms Mixed integer programming Phylogeny estimation Haplotype estimation Maximum parsimony Single nucleotide polymorphism Computational biologyBackground
Molecular phylogenetics studies the hierarchical evolutionary relationships among species, or taxa, by means of molecular data such as DNA, RNA, amino acid or codon sequences. These relationships are usually described through a weighted tree, called a phylogeny, whose leaves represent the observed taxa, internal vertices represent the intermediate ancestors, edges represent the estimated evolutionary relationships, and edge weights represent measures of the similarity between pairs of taxa.
Accurately characterizing evolutionary relationships between organisms and their genomes is the basis of comparative genomic methods for interpreting meaning in sequence data, and for this reason the use of molecular phylogenetics has become widely used (and sometimes indispensable) in a multitude of research fields such as systematics, medical research, drug discovery, epidemiology, and population dynamics[3]. For example, the use of molecular phylogenetics was of considerable assistance in predicting the evolution of human influenza A[4], understanding the relationships between the virulence and the genetic evolution of HIV[5, 6], identifying emerging viruses as SARS[7], recreating and investigating ancestral proteins[8], designing neuropeptides causing smooth muscle contraction[9], and relating geographic patterns to macroevolutionary processes[10].
The literature on molecular phylogenetics proposes a number of criteria for selecting the phylogeny of a set$\mathcal{H}$of haplotypes extracted from n taxa from among plausible alternatives. The criteria can usually be quantified and expressed in terms of objective functions, giving rise to families of optimization problems whose general paradigm can be stated as follows[11]:
Problem 1

The Phylogeny Estimation Problem (PEP)$\begin{array}{cc}\mathit{\text{optimize}}\hfill & f\left(T\right)\hfill \\ \phantom{\rule{2em}{0ex}}\phantom{\rule{.8em}{0ex}}\mathit{s.t.}\hfill & g(\mathcal{H},T)=1\hfill \\ T\in \mathcal{T},\hfill \end{array}$
where T a phylogeny of$\mathcal{H},$$\mathcal{T}$the set of all possible phylogenies of$\mathcal{H},$$f:\mathcal{T}\to \mathbb{R}$ a function modeling the selected criterion of phylogeny estimation, and$g:\mathcal{H}\times \mathcal{T}\to \mathbb{R}$ is a characteristic function equal to one if a phylogeny T is compatible (according to the selected criterion) for the set$\mathcal{\text{H.}}$ A specific optimization problem is completely characterized by defining the functions f and g, and the phylogeny T^{∗} that optimizes f and satisfies g is referred to as optimal.
One of the classic criteria for phylogeny estimation is the parsimony criterion, which assumes that one taxon evolves from another by means of small changes and that the most plausible phylogeny will be that requiring the smallest number of changes. That evolution proceeds by small rather than smallest changes is due to the fact that the neighborhood of possible alleles that are selected at each instant of the life of a taxon is finite and, perhaps more important, that the selective forces acting on the taxon may not be constant throughout its evolution[12, 13]. Over the long term (periods of environmental change, including the intracellular environment), the accumulation of small changes will not generally correspond to the smallest possible set of changes consistent with the observed final sequences. Nevertheless, it is plausible to believe, at least for wellconserved molecular regions where mutations are reasonably rare and unlikely to have occurred repeatedly at any given variant locus, that the process of approximating small changes with smallest changes could provide a good approximation to the true evolutionary process of the observed set of taxa[14]. Such an assumption is likely to be reasonable, for example, in intraspecies phylogenetics, where few generations have elapsed since the observed taxa shared a common ancestor and thus the expected number of mutations per locus is much less than one. When such assumtions hold, a phylogeny of$\mathcal{H}$is defined to be optimal under the parsimony criterion if it satisfies the following requirements: (i) it has the shortest length, i.e., the minimum sum of the edge weights, and (ii) it is such that, for each pair of distinct haplotypes${h}_{i},{h}_{j}\in \mathcal{H},$ the sum of the edge weights belonging to the path from h_{ i } to h_{ j } in T^{∗} is not smaller than the observed number of changes between h_{ i } and h_{ j }[11]. The first condition imposes the assumption that the smallest number of mutations consistent with the observed sequences is a good approximation to the true accumulated set of mutations; the second condition correlates the edge weights to the observed data.
The parsimony assumption can be considered accurate in the limit of low mutation rates or short time scales, making it a reasonable model for situations such as analysis of intraspecies variation where little time is presumed to have elapsed since the existence of a common ancestor of all observed taxa. Maximum parsimony also remains valuable as a model for novel methodology development in phylogenetics because of its relative simplicity and amenity to theoretical analysis. Novel computational strategies, such as those developed in this paper, might therefore productively be developed and analyzed in the context of maximum parsimony before being extended to more complicated models of phylogenetics.
Finding the phylogeny that satisfies the parsimony criterion involves solving a specific version of the PEP, called the Most Parsimonious Phylogeny Estimation Problem (MPPEP). Some of the variants of the MPPEP, see e.g.,[15, 16], can be solved in polynomial time, however, in the most general case, the problem is$\mathcal{N}\mathcal{P}$hard[11, 17] and this fact has justified the development of a number of exact and approximate solution approaches, such those described in[11, 17, 18]. Some recent versions of the MPPEP, such as the Most Parsimonious Phylogeny Estimation Problem from SNP haplotypes (MPPEPSNP) investigated in this article, play a fundamental role in providing predictions of practical use in several medical bioinformatics applications, such as disease association studies[19] or reconstruction of tumor phylogenies[20, 21]. In these contexts, it would be highly desirable to have the most accurate inferences possible for specific applications, but this in turn would imply to have algorithms able to exactly solve instances of such versions. As regards the MPPEPSNP, the literature describes some (rare) circumstances for which it is possible to solve the problem in polynomial time (see Section Methods). Unfortunately, in the general case the MPPEPSNP is$\mathcal{N}\mathcal{P}$hard and solving provably to optimality therefore generally requires the use of exact approaches based on implicit enumeration algorithms, similar to the mixed integer programming strategies described in[1, 2, 22].
In this article, we explore the prospects for improving on the implicit enumeration strategy of[1, 2] using a novel problem formulation and a series of additional constraints to more precisely bound the solution space and accelerate implicit enumeration of possible optimal phylogenies. We present a formulation for the problem based on an adaptation of[23]’s mixed integer formulation for the Steiner tree problem extended with a number of preprocessing techniques and reduction rules to further decrease its size. We then show that it is possible to exploit the high symmetry inherent in most instances of the problem, through a series of strengthening valid inequalities, to eliminate redundant solutions and reduce the practical search space. We demonstrate through a series of empirical tests on real and artificial data that these novel insights into the symmetry of the problem often leads to significant reductions in the gap between the optimal solution and its nonintegral linear programming bound relative to the prior art as well as often substantially faster processing of moderately hard problem instances. More generally, the work provides an indication of the conditions under which such an optimal enumeration approach is likely to be feasible, suggesting that these strategies are usable for relatively large numbers of taxa, although with stricter limits on numbers of variable sites. The work thus provides methodology suitable for provably optimal solution of some harder instances that resist all prior approaches. In future work, it may provide useful guidance for strategies and prospects of similar optimization methods for other variants of phylogeny inference.
Methods
Notation and problem formulation
Before proceeding, we shall introduce some notation and definitions that will prove useful throughout the article. The human genome is divided in 23 pairs of chromosomes, i.e., organized structures of DNA that contain many genes, regulatory elements and other nucleotide sequences. When a nucleotide site of a specific chromosome region shows a variability within a population of individuals then it is called a Single Nucleotide Polymorphism (SNP). Specifically, a site is considered a SNP if for a minority of the population a certain nucleotide is observed (called the minor allele) while for the rest of the population a different nucleotide is observed (the major allele). The minor allele, or mutant type[24], is generally encoded as ‘1’, as opposed to the major allele, or wild type[24], generally encoded as ‘0’. A haplotype is a set of alleles, or more formally, a string of length m over an alphabet Σ = {0,1}[25].
Given a set$\mathcal{H}$of n haplotypes, denote$\mathcal{S}=\{1,2,\dots ,m\}$ as the set of alleles and h_{ i }(s),$s\in \mathcal{S}$, as the sth allele of haplotype${h}_{i}\in \mathcal{H}$. Given two distinct haplotypes${h}_{i},{h}_{j}\in \mathcal{H},$ we denote${\mathcal{S}}_{{h}_{i}{h}_{j}}$ as the subset of different alleles between h_{ i } and h_{ j },${d}_{{h}_{i}{h}_{j}}=\sum _{s\in {\mathcal{S}}_{{h}_{i}{h}_{j}}}\left{h}_{i}\right(s){h}_{j}(s\left)\right$ as the distance between h_{ i } and h_{ j }, and we say that h_{ i } and h_{ j } are adjacent if${d}_{{h}_{i}{h}_{j}}=1$. From a biological point of view, the adjacency between a pair of distinct haplotypes means that one of the two haplotypes evolved from the other by mutation of a specific SNP over time.
Consider a graph$G=(\mathcal{H},E)$ having a vertex for each haplotype in$\mathcal{H}$and an edge for each pair of adjacent haplotypes${h}_{i},{h}_{j}\in \mathcal{\text{H.}}$ Then, a phylogeny T of$\mathcal{H}$is a spanning tree of G, i.e., an acyclic subgraph of G in which a pair of vertices${h}_{i},{h}_{j}\in \mathcal{H}$ is adjacent in T if${d}_{{h}_{i}{h}_{j}}=1$. It is worth noting that, according to the definition of the edge set E, in general a phylogeny of$\mathcal{H}$may not exist as the graph$G=(\mathcal{H},E)$ might not be connected. To always ensure the existence of a phylogeny for$\mathcal{\text{H,}}$ we introduce the set${\mathcal{H}}^{\prime}$ which consists of the minimum number of haplotypes that should be added to$\mathcal{H}$in such a way that, defined$\overline{\mathcal{H}}=\mathcal{H}\cup {\mathcal{H}}^{\prime}$ and$\overline{E}=\left\{\right({h}_{i},{h}_{j}):{h}_{i},{h}_{j}\in \overline{\mathcal{H}}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{d}_{{h}_{i}{h}_{j}}=1\}$, the graph$\overline{G}=(\overline{\mathcal{H}},\overline{E})$ is connected. From a biological point of view, the set${\mathcal{H}}^{\prime}$ represents the set of haplotypes that are ancestors of the observed ones but either had gone extinct or just were not observed in that sample (also called Steiner nodes).
Denote$\overline{T}$ as a phylogeny of$\overline{H}$,$\overline{E}\left(\overline{T}\right)$ as the edge set of$\overline{T}$, and$L\left(\overline{T}\right)$ as the length of the phylogeny$\overline{T}$, i.e., the sum of the distances${d}_{{h}_{i}{h}_{j}}$, for all$({h}_{i},{h}_{j})\in \overline{E}\left(\overline{T}\right)$. Then, the problem of finding a phylogeny of$\mathcal{H}$that satisfies the parsimony criterion consists of solving the following optimization problem:
Problem 2
The Most Parsimonious Phylogeny Estimation Problem from SNP haplotypes (MPPEPSNP).Given a set$\mathcal{H}$of n haplotypes having m alleles each, find the minimum cardinality haplotype set${\mathcal{H}}^{\prime}$ to be added to$\mathcal{H}$so that the phylogeny${\overline{T}}^{\star}$ has minimum length.
If the haplotype set${\mathcal{H}}^{\prime}$is empty, i.e., if$G=(\mathcal{H},E)$ is connected, then MPPEPSNP can be solved in polynomial time as the minimum spanning tree is a (optimal) solution to the MPPEPSNP. Similarly, if the input haplotype set$\mathcal{H}$satisfies the perfect phylogeny condition i.e., the requirement that each allele changes only once throughout the optimal phylogeny (see[19]), then the MPPEPSNP can be still solved in polynomial time[26–28]. Unfortunately, it is possible to prove that in the general case the MPPEPSNP is$\mathcal{N}\mathcal{P}$hard (see[1, 22]). In fact, the binary nature of the SNP haplotypes allows us to interpret a generic haplotype${h}_{i}\in \mathcal{H}$ as a vertex of a mdimensional unit hypercube, its sth allele as the sth coordinate of the vertex h_{ i }, and the set${\mathcal{H}}^{\prime}$ as the set of Steiner vertices of the unit hypercube. Hence the MPPEPSNP can be seen as particular case of the Steiner tree problem in a graph, a notorious$\mathcal{N}\mathcal{P}$hard combinatorial optimization problem[29].
Finding the optimal solutions to the MPPEPSNP is fundamental to validating the parsimony criterion, i.e., to verify whether, for a given instance of MPPEPSNP, the results predicted by the criterion match the experimental ones. Unfortunately, the$\mathcal{N}\mathcal{P}$hardness of the MPPEPSNP limits the size of the instances analyzable to the optimum, which in turn affects the ability to validate the parsimony criterion, hence the practical utility of the predictions themselves. In order to address this concern, in the following section we shall develop an integer programming model able to provide optimal solutions to real instances of the MPPEPSNP.
A mixed integer programming model for the MPPEPSNP
Denote u_{ i }, i ∈ V, as a decision variable equal to 1 if the ith vertex of V is considered in the optimal solution to the MPEPPSNP and 0 otherwise;${x}_{i}^{s}$ as a decision variable equal to 1 if in the optimal solution to the MPPEPSNP the sth coordinate of the vertex u_{ i }, i ∈ V, is 1 and 0 otherwise;${z}_{\mathit{\text{ij}}}^{s}$ as a decision variable equal to 1 if in the optimal solution to the MPPEPSNP the pair of distinct vertices i,j ∈ V has a change at sth coordinate, and 0 otherwise; and y_{ ij } as a decision variable equal to 1 if the pair of distinct vertices i,j∈V is adjacent in the optimal solution to the MPPEPSNP and 0 otherwise. Finally, let${V}_{\mathcal{H}}=\{1,2,\dots ,n\}$,${V}_{{\mathcal{H}}^{\prime}}=\{n+1,n+2,\dots ,n+\mathit{UB}\}$, and Q = {1,2,…,n + LB}. Then, a valid formulation for the MPPEPSNP is the following:
Formulation 1
The objective function (1a) aims at minimizing the length of the optimal phylogeny. Constraints (1b) impose that the coordinates of the first n vertices in V are exactly the ones of the input haplotype set$\mathcal{\text{H.}}$ Constraints (1c) impose that the sth coordinate of vertex u_{ i }, i ∈ V, can assume value 1 only if vertex u_{ i } is considered in the optimal solution to the problem. Constraints (1d)(1e) force variables${z}_{\mathit{\text{ij}}}^{s}$ to be one if in the optimal solution to the problem there exist a pair of adjacent vertices i j ∈ V having a different value at the sth coordinate. Constraints (1f) impose that in an optimal solution to the problem two distinct vertices i j ∈ V can be adjacent only if${d}_{{h}_{i}{h}_{j}}=1$. Constraints (1g)(1h) impose that in the optimal solution to the problem variable y_{ ij } may assume value 1 only if both vertices i and j are considered. Vice versa, constraints (1i) impose that if in the optimal solution to the problem a vertex u_{ i }, i ∈ V, is considered then at least one variable y_{ ij } must assume value 1. Constraints (1j) and (1k) impose the Generalized Subtour Elimination Constraints (GSEC)[23]. Finally, constraints (1l) impose that the first n + LB vertices in V have to be considered in the optimal solution to the problem.
Note that Formulation 1 can be easily extended to the case in which the haplotypes are represented by multicharacter data, i.e., sequences over an alphabet Σ = {0,1,2,…,γ}. In fact, in such a case it is sufficient to replace each character c in the haplotype by a binary γ vector ν such that the sth coordinate of ν is equal to 1 if the character c is equal to s, s ∈ Σ, and 0 otherwise. For example, if the generic haplotype were, for example, the string 〈AACGT〉, then it could be represented as 〈1000 1000 0100 0010 0001〉.
Reducing model size
Formulation 1 is characterized by a polynomial number of variables and an exponential number of constraints. Its implementation can be performed by means of standard branchandcut approaches that use GSEC separation oracles such as those described in[32].
It is worth noting that variables${x}_{i}^{s}$ and${z}_{\mathit{\text{ij}}}^{s}$ can be relaxed in Formulation1c)(1e) and the convexity constraint (1f) suffice to guarantee their integrality in any optimal solution to the problem. Moreover, Formulation 1 can be reduced in size by removing those variables that are redundant or whose value is known in the optimal solution to the problem. For example, variables y_{ ij } can be removed from Formulation 1 as it is easy to realize that they are redundant. Similarly, all variables${z}_{\mathit{\text{ij}}}^{s}$ such that$i,j\in {V}_{\mathcal{H}}$ and d_{ ij } > 1 do not need to be defined as their value will be always zero for any$s\in \mathcal{S}$ and in any feasible solution to the problem. Variables u_{ i }, i ∈ Q, do not need to be declared as their value will be always 1 any feasible solution to the problem. Finally, variables${x}_{i}^{s}$,$i\in {V}_{\mathcal{H}}$, can be removed as their value is univocally assigned by the input haplotype set$\mathcal{\text{H.}}$ The reduction process can be further combined with the preprocessing strategies described in[1] to obtain even smaller formulations. Such strategies allow one to remove alleles from the input haplotype set$\mathcal{H}$without altering the optimal solution to the problem. For example, suppose that the haplotype set$\mathcal{H}$is such that there exists an allele$\u015d\in \mathcal{S}$ such that${h}_{i}\left(\u015d\right)=1$, for all${h}_{i}\in \mathcal{H}$; then it is easy to realize that$\u015d$can be removed from$\mathcal{S}$as in any feasible solution to the problem the$\text{\u015dth}$ coordinate of any vertex in the phylogeny would be characterized by having x i ŝ = 1. A similar situation occurs when there exists an allele$\u015d\in \mathcal{S}$ such that${h}_{i}\left(\u015d\right)=0$, for all${h}_{i}\in \mathcal{H}$. Analogously, suppose that the input haplotype set$\mathcal{H}$is characterized by equal alleles, i.e., by the existence of two alleles, say${\u015d}_{1}$ and${\u015d}_{2}$, such that${h}_{i}\left({\u015d}_{1}\right)={h}_{i}\left({\u015d}_{2}\right)$, for all$i\in \mathcal{S}$. Then it is easy to realize that if one removes one of the two alleles from$\mathcal{\text{S,}}$ say${\u015d}_{2}$, and multiplies the${\u015d}_{1}$th coordinate by 2 does not alter neither the structure nor the value of the optimal solution to the problem. Describing all the preprocessing techniques for shrinking the input haplotype set$\mathcal{H}$is beyond the scope of the present article. The interested reader will find a systematic discussion of this topic in[1].
By applying the previously cited reduction strategies to Formulation 1 and denoting$\u015c$as the set of alleles resulting from the application of the preprocessing strategies described in[1], w^{ s } as the number of alleles in$\mathcal{S}$equal to the sth,$s\in \u015c$, Z as the set for which variables${z}_{\mathit{\text{ij}}}^{s}$ are defined, R = {n + LB + 1,n + LB + 2,…,n + UB}, and${C}_{\mathcal{H}}=\{i\in C:i\in {V}_{\mathcal{H}}\}$, for any C ⊂ V, the following reduced formulation for the MPPEPSNP can be obtained:
Formulation 2
Note that in Formulation 2 variables${x}_{i}^{s}$ and${z}_{\mathit{\text{ij}}}^{s}$ cannot be relaxed anymore.
Strengthening valid inequalities
By exploiting the integrality of variables u_{ i },${x}_{i}^{s}$, and${z}_{\mathit{\text{ij}}}^{s}$, a number of valid inequalities can be developed to strengthen Formulation 2.
Proposition 1
are valid for Formulation 2.
Proof
In a feasible solution to the problem variable u_{ i }, i ∈ V∖(Q ∪ {n + UB}), can assume only value 0 or 1. If u_{ i } = 0, constraint (3) reduces to u_{i + 1} ≤ 0 which is trivially valid for Formulation 2. If u_{ i } = 1, constraint (3) reduces to u_{i + 1} ≤ 1 which is again valid. □
Constraints (3) impose an ordering on the variables u_{ i }, i ∈ R, so that vertex u_{i + 1} can be considered in the optimal solution to the problem only if vertex u_{ i } has been already considered.
Proposition 2
are valid for Formulation 2.
Proof
which is again valid for the above arguments. □
Proposition 3
are valid for Formulation 2.
Proof
As observed in the previous proposition, in a feasible solution to the problem$\sum _{s\in \u015c}{z}_{\mathit{\text{ij}}}^{s}$,$i,j\in {V}_{{\mathcal{H}}^{\prime}}$, i,j∈Z, can assume only value 0 or 1. If$\sum _{s\in \u015c}{z}_{\mathit{\text{ij}}}^{s}=0$, then constraint (5) (respectively constraint (6)) reduces to$+{x}_{i}^{{s}_{2}}{x}_{j}^{{s}_{2}}\le 2$ (respectively${x}_{i}^{{s}_{2}}+{x}_{j}^{{s}_{2}}\le 2$), which is trivially valid due to the integrality of variables${x}_{i}^{s}$. If$\sum _{s\in \u015c}{z}_{\mathit{\text{ij}}}^{s}=1$, then either$\sum _{\begin{array}{c}s\in \u015c:\\ s\ne {s}_{1}\end{array}}{z}_{\mathit{\text{ij}}}^{s}=1$ or${z}_{\mathit{\text{ij}}}^{{s}_{1}}=1$. If$\sum _{\begin{array}{c}s\in \u015c:\\ s\ne {s}_{1}\end{array}}{z}_{\mathit{\text{ij}}}^{s}=1$ then constraint (5), (respectively constraint (6)) reduces to$+{x}_{i}^{{s}_{2}}{x}_{j}^{{s}_{2}}\le 1$ (respectively${x}_{i}^{{s}_{2}}+{x}_{j}^{{s}_{2}}\le 1$), which is trivially valid. If${z}_{\mathit{\text{ij}}}^{{s}_{1}}=1$ then constraint (5) (respectively constraint (6)) reduces to$+{x}_{i}^{{s}_{2}}{x}_{j}^{{s}_{2}}\le 0$ (respectively${x}_{i}^{{s}_{2}}+{x}_{j}^{{s}_{2}}\le 0$), which is again valid. □
Similar arguments can be used to prove the following proposition:
Proposition 4
are valid for Formulation 2.
Proposition 5
are valid for Formulation 2.
Proof
The statement trivially follows from the integrality of variables${x}_{i}^{s}$ and from constraints (2b). □
Constraints (9)(10) impose an ordering on the coordinates of the vertices in${V}_{{\mathcal{H}}^{\prime}}$ by means of the smallest bigM possible, i.e., a power of 2. Note that the distinction between constraints (9) and (10) is necessary, as in principle vertices in R may not be needed in the optimal solution to the problem.
Proposition 6
are valid for Formulation 2.
Proof
In a feasible solution to the problem, the sum$\sum _{s\in \u015c}{z}_{\mathit{\text{ij}}}^{s}$,$i,j\in {V}_{{\mathcal{H}}^{\prime}}$, i,j∈Z, can assume only value 0 or 1. If$\sum _{\begin{array}{c}j\in V:\\ j\in Z\end{array}}\phantom{\rule{0.3em}{0ex}}\sum _{s\in \u015c}{z}_{(i+1)j}^{s}=0$, constraint (11) reduces to$\sum _{\begin{array}{c}j\in V:\\ j\in Z\end{array}}\phantom{\rule{0.3em}{0ex}}\sum _{s\in \u015c}{z}_{\mathit{\text{ij}}}^{s}\ge 0$ which is trivially valid. Vice versa, If$\sum _{\begin{array}{c}j\in V:\\ j\in Z\end{array}}\phantom{\rule{0.3em}{0ex}}\sum _{s\in \u015c}{z}_{(i+1)j}^{s}=1$, constraint (11) reduces to$\sum _{\begin{array}{c}j\in V:\\ j\in Z\end{array}}\phantom{\rule{0.3em}{0ex}}\sum _{s\in \u015c}{z}_{\mathit{\text{ij}}}^{s}\ge 1$ which is again valid due to Propositions 1 and 2. □
Let${Q}_{3}=\{i,j\in {V}_{\mathcal{H}}:{d}_{\mathit{\text{ij}}}\ge 3\}$ and k ∈ V, k ∉ Q_{3}. Then the following proposition holds:
Proposition 7
are valid for Formulation 2.
Proof
In a feasible solution to the problem the path between two distinct haplotypes${h}_{i},{h}_{j}\in \mathcal{H}$ cannot be shorter than the distance${d}_{{h}_{i}{h}_{j}}$. Hence, if the distance between h_{ i }and h_{ j }is greater or equal to three, vertices i and j cannot be adjacent to a same vertex k, i.e., only one of the two sums$\sum _{s\in \mathcal{S}}{z}_{\mathit{\text{ik}}}^{s}$ or$\sum _{s\in \mathcal{S}}{z}_{\mathit{\text{jk}}}^{s}$ can be equal to 1. □
Note that if k ∈ R then (12) can be strengthened by replacing the righthandside by u_{ k }. Moreover, Proposition 7 can be generalized as follows. Consider the sets${Q}_{d}=\{i,j\in {V}_{\mathcal{H}}:{d}_{\mathit{\text{ij}}}\ge d\}$, d ∈ {3,4,…,m}, C ⊂ V such that 2 ≤ C ≤ d − 1 and C ∩ Q_{ d } = ∅, and a path p that involves only vertices in C. Denote p_{ k } the kth vertex in p. Then the following proposition holds:
Proposition 8
called forbidden path constraints, are valid for Formulation 2.
Proof
Similarly to Proposition 7, in a feasible solution to the problem the path p between two distinct haplotypes${h}_{i},{h}_{j}\in \mathcal{H}$ cannot be shorter than the distance${d}_{{h}_{i}{h}_{j}}$. Hence, if the distance between h_{ i } and h_{ j } is greater or equal to d, at most C vertices can belong to p. □
Experiments
In this section we analyze the performance of our model to solve the MPPEPSNP. Our experiments were motivated by a twofold reason, namely: to evaluate, with respect to Formulation 1, the benefits obtained by the removal of the redundant variables and by the inclusion of the valid inequalities previously described; and to allow the analysis of larger datasets with respect to the ones analyzable by means of[1]’s algorithm, currently the best known exact approach to solution of the MPPEPSNP.
Similar to[1], we emphasize that the experiments aim simply to evaluate the runtime performance of our model for solving MPPEPSNP. We neither attempt to study the efficiency of MPPEPSNP for phylogeny estimation nor compare the accuracy of our algorithm to phylogeny estimation solvers that do not use the parsimony criterion. The reader interested in a systematic discussion about such issues is referred to[19, 33].
Implementation
We implemented Formulations 1 and 2 by means of Mosel 64 bit 3.2.0 of XpressMP, Optimizer version 22, running on a Pentium 4, 3.2 GHz, equipped with 2 GByte RAM and operating system Gentoo release 7 (kernel linux 2.6.17). In both formulations, we computed the overall solution time to solve a generic instance of the problem as the sum of the preprocessing time due to the implementation of[22]’s reduction rules plus the solution time taken by the Optimizer to find the optimal solution to the instance. In preliminary experiments, we observed that Formulation 2 has two main advantages with respect to Formulation 1, namely: it requires much less memory to load the model (at least 27% RAM less in the analyzed instances) and it is characterized by faster linear relaxations at each node of the search tree. As result, Formulation 2 allows potentially the analysis of larger instances than Formulation 1 and may be characterized by faster solution times. Hence, we preferred to use Formulation 2 in our experiments.
In preliminary experiments we observed that the FlowRM outperforms the GESCRM in terms of solution time. This fact is mainly due to the computational overhead introduced by the GSEC separation oracle which seems to be not compensated by the size of the analyzed instances. Hence, we did not consider the GESCRM any further in our experiments.
During the runtime, we enabled the XpressMP automatic cuts and the XpressMP presolving strategy. Moreover, we also tested a number of generic primal heuristics for the Steiner tree problem to generate a first primal bound to the MPPEPSNP (see, e.g.,[34]). Unfortunately, in preliminary experiments we observed that the use of such heuristics interferes negatively with the Xpress Optimizer, by delaying the solution time of the analyzed instances. Hence, we disabled the used of the generic primal heuristics and enabled the use of the XpressMP primal heuristic instead. The source code of the algorithm can be downloaded athttp://homepages.ulb.ac.be/~dacatanz/Site/Software_files/iMPPEP.zip.
Separation oracle for the forbidden path constraints
to the formulation; otherwise, we select a different pair of vertices in V∖Q_{ d } and iterate this procedure until either 10 distinct paths have been generated or all possible pairs of vertices in V∖Q_{ d } have been selected. If all vertices have been selected but less than 10 distinct paths have been generated, then we select a larger subset of V∖Q_{ d } (e.g., a triplet of vertices) and we iterate again the previous steps. It is easy to realize that this procedure may potentially generate all the possible paths violating (13). However, we stop the procedure after generating 10 paths or after considering subset C containing more than 5 vertices as we observed in preliminary experiments that this strategy provides the best tradeoff between speed and tightness of the cut.
Branching strategies
We observed that even better runtime performance can be obtained by sorting the coordinates of the input haplotypes in decreasing way according to the weights w^{ s } and by branching first on variables${\beta}_{\alpha}^{s}$, then on variables u_{ i }, and subsequently on variables${x}_{i}^{s}$ and finally on variables${z}_{\mathit{ij}}^{s}$.
Performance analysis
In order to obtain a measure of the performance of the FlowRM, we compared[1]’s polynomialsize formulation versus the FlowRM on[1]’s real instances of the MPPEPSNP, namely: Human chromosome Y constituted by 150 haplotypes having 49 SNPs each; bacterial DNA constituted by 17 haplotypes having 1510 SNPs each; Chimpanzee mitochondrial DNA constituted by 24 haplotypes having 1041 SNPs each; Chimpanzee chromosome Y constituted by 24 haplotypes having 1041 SNPs each; and a set of four Human mitochondrial DNA from HapMap[35] constituted by 40 haplotypes having 52 SNPs each, 395 haplotypes having 830 SNPs each, 13 haplotypes having 390 SNPs each, and 44 haplotypes having 405 SNPs each, respectively. Such instances consist only of nonrecombining data (Y chromosome, mitochondrial, and bacterial DNA) and can be downloaded athttp://homepages.ulb.ac.be/~dacatanz/Site/Software_files/iMPPEP.zip.
Comparison between the gap of [[1]]’s polynomial size integer programming model for the MPPEPSNP versus the gap of the flowbased reduced model and its strengthening valid inequalities
Dataset  Haplotypes  SNPs  GAP (%)  GAP (%)  Optimum  MST 

[[1]]  FlowRM  Solution  
Human chromosome Y  150  49  0.00  0.00  16  yes 
Bacterial mtDNA  17  1510  26.04  20.83  96  no 
Chimpanzee mtDNA  24  1041  20.63  20.63  63  yes 
Chimpanzee chromosome Y  15  98  0.00  0.00  99  yes 
Human mtDNA  40  52  24.66  1.37  73  no 
Human mtDNA  395  830  22.64  7.55  53  no 
Human mtDNA  13  390  12.50  6.25  48  no 
Human mtDNA  44  405  6.98  4.65  43  no 
Interestingly, sometimes in real applications the number of haplotypes can be much bigger than the number of SNPs. Hence, it is important to test the ability of an exact algorithm to tackle instances of the MPPEPSNP containing e.g., hundreds haplotypes.[1] observed that their brute force enumeration algorithm is able to tackle instances of the problem containing up to 270 haplotypes having up to 9 SNPs each within 12 hours computing time. Unfortunately, the authors also observed that their algorithm is unable to solve larger instances of the MPPEPSNP, no matter the maximum runtime considered. In this context, the FlowRM makes the difference, being able to tackle instances of the MPPEPSNP having up to 300 haplotypes and 10 SNPs within 3 hours computing time. To show this result, we considered a number of random instances of the problem containing 100, 150, 200, 250, and 300 haplotypes, respectively. Fixing the number of haplotypes n∈{100,150,200,250,300}, we created an instance of the problem by generating at random n strings of length 10 over the alphabet Σ={0,1}. During the generation process, we randomly selected the number of SNPs equal to 1 in a given haplotype, and subsequently we randomly chose the sites of the haplotype to be set to 1. We iterated the instance generation process 10 times for a fixed value of n, obtaining eventually an overall number of 50 random instances of the MPPEPSNP downloadable athttp://homepages.ulb.ac.be/~dacatanz/Site/Software_files/iMPPEP.zip.
Performances of the FlowRM on a set of random instances of the MPPEPSNP
$\left\mathcal{H}\right$  Instance  $\left\mathcal{H}\right$post  Time  GAP (%)  Nodes  $\left\mathcal{H}\right$  Instance  $\left\mathcal{H}\right$post  Time  GAP (%)  Nodes 

reduction  (sec.)  reduction  (sec.)  
100  1  57  520.05  0  1807  150  1  82  284.51  0  424 
2  60  59.74  0  174  2  83  314.27  0.76  56  
3  63  377.75  1.45  110  3  81  799.01  0  67  
4  61  2491.62  3.81  3351  4  67  1809.26  2.66  6617  
5  60  2918.09  4.63  2062  5  79  1001.14  2.29  187  
6  57  349.54  1.59  264  6  74  1976.73  2.41  1071  
7  65  258.53  1.90  85  7  73  8719.65  3.92  4814  
8  58  293.97  0  1299  8  83  3497.73  2.17  421  
9  62  862.48  2.85  540  9  72  1154.77  2.51  410  
10  64  87.19  0  92  10  80  399.89  1.56  256  
200  1  99  614.86  0  72  250  1  117  1155.41  0  197 
2  99  1353.16  1.28  149  2  109  7757.98  1.72  1596  
3  96  896.68  0.67  226  3  117  387.141  0.84  180  
4  104  652.44  0.47  150  4  126  1267.77  0.51  114  
5  96  382.83  0  56  5  116  188.188  0.84  162  
6  106  2535.09  0.60  71  6  116  2311.61  1.14  685  
7  100  233.50  0  21  7  116  1256.24  0  265  
8  99  1650.17  0.96  79  8  124  67.556  0  528  
9  87  4600.69  2.10  954  9  122  2000.77  0.53  107  
10  102  2554.84  1.23  1965  10  111  1200.89  0.87  272  
300  1  133  297.19  0  15  
2  123  2753.53  0.39  68  
3  142  5371.05  0  941  
4  133  420.72  0  43  
5  126  388.99  0  433  
6  134  397.01  0  61  
7  138  1173.65  0  1788  
8  126  666.21  0  186  
8  127  449.30  0.77  42  
10  145  201.87  0  876 
The results showed that the integrality gaps are usually very low, ranging from 0% to 4.63% and assuming in average a value about 1%, confirming once again the tightness of the FlowRM and the efficiency of the strengthening valid inequalities.
Finally, we also tested the performance of the FlowRM on a set of 5 HapMap Human mitochondrial DNA instances of the MPPEPSNP that were not solvable by using[1]’s bruteforce enumeration algorithm, namely: f1 constituted by 63 haplotypes having 16569 SNPs each, i2 constituted by 40 haplotypes having 977 SNPs each, k3 constituted by 100 haplotypes having 757 SNPs each, m4 constituted by 26 haplotypes having 48 SNPs each, and p5 constituted by 21 haplotypes having 16548 SNPs each. Such instances can be downloaded at the same address and consist only of nonrecombining data (Y chromosome, mitochondrial, and bacterial DNA).
A part from m4, all the remaining instances gave rise to too large formulations (several hundreds Mbytes RAM) to be handled by the Xpress Optimizer. Hence, instead of analyzing entirely each instance we decomposed it into contiguous SNP blocks and analyzed the most difficult block separately. In more in detail, we define${\mathcal{H}}_{r}$ to be the haplotype matrix obtained by the application of[1]’s reduction rules, we sorted the columns of${\mathcal{H}}_{r}$ according to an increasing ordering of the weights w^{ s },$s\in \u015c$; subsequently, we considered the submatrices obtained by taking k contiguous SNPs (or kblock) in$\text{\u015c,}$k ∈ {10,13,15}. We did not consider greater values for k as we observed that k = 15 was already a threshold after which the haplotype submatrix gave rise to too large formulations. For each kblock$\mathcal{B}$in${\mathcal{H}}_{r}$ we considered the hamming distance${d}_{{h}_{i}{h}_{j}}=\sum _{s\in \mathcal{B}}\left{h}_{i}\right(s){h}_{j}(s\left)\right$ between each pair of distinct haplotypes in${\mathcal{H}}_{r}$, and chose the kblock maximizing the sum$\sum _{{h}_{i},{h}_{j}\in {\mathcal{H}}_{r},{h}_{i}<{h}_{j}}{d}_{{h}_{i}{h}_{j}}$. Finally, we assumed three hours as maximum runtime per instance.
Performances of the FlowRM on a set of real instances of the MPPEPSNP
Dataset  Haplotypes  SNPs  Block Size  Time (sec.)  GAP (%)  Nodes  MST Solution 

10  1      yes  
f1  63  16569  13  56  3.11  1  no 
15  10286.1  26.92  773521  no  
i2  40  977  10  781.85  20.00  37511  no 
k3  100  757  10  150  7.65  353  no 
13  588.38  14.29  11265  no  
m4  26  48  10  5  5.88  109  no 
p5  21  16548  10  22283.4  50.79  6125448  no 
Table3 shows that, apart from the instances f1 and m4, the FlowRM was unable to exactly solve, within the limit time, the considered instances for values of k ∈ {13,15}. Specifically, The FlowRM exactly solved in less than a minute the instance f1 when considering values of k ∈ {10,13} ; in 20 minutes the instance i2 when considering k = 10 ; in less than 3 minutes the instance k3 when considering k = 10; and the instance m4 in 5 seconds. In contrast, the FlowRM was unable to solve the instance p5, regardless of the value of k considered. In fact, already when considering k = 10, the Xpress Optimizer took more than 12 hours to exactly solve the instance p5 and explored over 10 million nodes in the search tree. A more detailed analysis of the instance showed that, despite the presence of the strengthening valid inequalities, p5 is characterized by highly fractional relaxations. This fact implies the existence of equivalent optimal solutions to the instance that, on the one hand, delay the finding of a primal bound and, on the other hand, force the Optimizer to explore many more nodes in the tree search. This situation in more pronounced in p5 but also occurs in the instances i2 and k3. To improve the tightness of the formulation we tried to include in the FlowRM also classical facets and strengthening valid inequalities for the Steiner tree problem in a graph (see[23, 36–38]). However, we did not observe any benefit from the inclusion. We suspect that the presence of highly fractional solutions to the problem could be caused both by poor lower bounds on the number of Steiner vertices considered in the FlowRM and by the existence of a number of non trivial classes of symmetries still present in the problem. Investigating such issues warrants future research efforts.
In order to measure the performance of the model on multistate character data we also considered[2] set of instances of the MPPEPSNP. Specifically, we considered the following instances: a set of 41 sequences of O.rufipogon DNA (red rice) having 1043 sites each; 80 human mtDNA sequences having 245 sites each; 50 HIV1 reverse transcriptase amino acid sequences having 176 sites each; a set of 500 sequences of mtDNA from the NCBI BLASTN best aligned taxa having 3000 sites each; a set of 500 sequences of mtDNA from the NCBI BLASTN best aligned taxa having 5000 sites each; and a set of 500 sequences of mtDNA from the NCBI BLASTN best aligned taxa having 10000 sites each. When running the same experiments described in[2] we registered a very poor performance for the FlowRM, mainly due to the large dimension of the considered instances and the presence of symmetries despite the use of constraints (13)(15). We observed that the combination of these two factors increased the runtime performance of the FlowRM of 23 orders of magnitude with respect to[2] approach. However, we also observed that when performing[2]’s “window analysis” (i.e., when decomposing into blocks of 10 SNPs the input matrix) the FlowRM performed better than[2]’s, being characterized by an average solution time of 8.27 seconds. This fact suggests that, when considering instances constituted by less than a dozen sites, an exact approach entirely based on integer programming may perform better than the implicit enumeration of the vertices of the generalized Buneman graph. Viceversa, for larger instances the implicit enumeration of the vertices of the generalized Buneman graph appears more suitable.
Conclusion
In this article we investigated the Most Parsimonious Phylogeny Estimation Problem from Single Nucleotide Polymorphism (SNP) haplotypes (MPPEPSNP), a recent version of the phylogeny estimation problem that arises when input data consist of SNPs extracted from a given population of individuals. The MPPEPSNP is$\mathcal{N}\mathcal{P}$hard and this fact has justified the development of exact and approximate solution approaches such as those described in[1, 19, 22, 26–28]. We explored the prospects for improving on the strategy of[1, 2] using a novel problem formulation and a series of additional constraints to more precisely bound the solution space and accelerate implicit enumeration of possible optimal phylogenies. We present a formulation for the problem based on an adaptation of[23]’s mixed integer formulation for the Steiner tree problem extended with a number of preprocessing techniques and reduction rules to further decrease its size. We then show that it is possible to exploit the high symmetry inherent in most instances of the problem, through a series of strengthening valid inequalities, to eliminate redundant solutions and reduce the practical search space. We demonstrate through a series of empirical tests on real and artificial data that these novel insights into the symmetry of the problem often leads to significant reductions in the gap between the optimal solution and its nonintegral linear programming bound relative to the prior art as well as often substantially faster processing of moderately hard problem instances. More generally, the work provides an indication of the conditions under which such an optimal enumeration approach is likely to be feasible, suggesting that these strategies are usable for relatively large numbers of taxa, although with stricter limits on numbers of variable sites. The work thus provides methodology suitable for provably optimal solution of some harder instances that resist all prior approaches. Our results may provide also useful guidance for strategies and prospects of similar optimization methods for other variants of phylogeny inference. In fact, if appropriately adapted, some of the results we presented here (e.g., symmetry breaking strategies) can be generalized with respect to other phylogenetic estimation criteria (e.g., the likelihood criterion) and provide important computational benefits.
Declarations
Acknowledgements
The first author acknowledges support from the Belgian National Fund for Scientific Research (FNRS) of which he is Chargé de Recherches, the Belgian American Educational Foundation (BAEF) of which he is honorary fellow, and the Carnegie Mellon University (CMU). Part of this work has been developed while Dr. Catanzaro was visiting the Tepper School of Business and the Department of Biological Sciences of the CMU during the academic years 20102011 and 20112012. This work was supported in part by U.S. National Institutes of Health awards 1R01CA140214 and 1R01AI076318.
Authors’ Affiliations
References
 Sridhar S, Lam F, Blelloch GE, Ravi R, Schwartz R: Mixed integer linear programming for maximum parsimony phylogeny inference. IEEE/ACM Trans Comput Biol Bioinformatics. 2008, 5 (3): 323331.View ArticleGoogle Scholar
 Misra N, Blelloch GE, Ravi R, Schwartz R: Generalized Buneman pruning for inferring the most parsimonious multistate phylogeny. J Comput Biol. 2011, 18 (3): 445457.PubMed CentralView ArticlePubMedGoogle Scholar
 Pachter L, Sturmfels B: The mathematics of phylogenomics. SIAM Rev. 2007, 49: 331. 10.1137/050632634.View ArticleGoogle Scholar
 Bush RM, Bender CA, Subbarao K, Cox NJ, Fitch WM: Predicting the evolution of human influenza A. Science. 1999, 286 (5446): 19211925.View ArticlePubMedGoogle Scholar
 Ross HA, Rodrigo AG: Immunemediated positive selection drives human immunodeficency virus type 1 molecular variation and predicts disease duration. J Virol. 2002, 76 (22): 1171511720.PubMed CentralView ArticlePubMedGoogle Scholar
 Ou CY, Ciesielski CA, Myers G, Bandea CI, Luo CC, Korber BTM, Mullins JI, Schochetman G, Berkelman RL, Economou AN, Witte JJ, Furman LJ, Satten GA, Maclnnes KA, Curran JW, Jaffe HW: Molecular epidemiology of HIV transmission in a dental practice. Science. 1992, 256 (5060): 11651171.View ArticlePubMedGoogle Scholar
 Marra MA, Jones SJ, Astell CR, Holt RA, BrooksWilson A, Butterfield YS, Khattra J, Asano JK, Barber SA, Chan SY, Cloutier A, Coughlin SM, Freeman D, Girn N, Griffith OL, Leach SR, Mayo M, McDonald H, Montgomery SB, Pandoh PK, Petrescu AS, Robertson AG, Schein JE, Siddiqui A, Smailus DE, Stott JM, Yang GS, Plummer F, Andonov A, Artsob H, Bastien N, Bernard K, Booth TF, Bowness D, Czub M, Drebot M, Fernando L, Flick R, Garbutt M, Gray M, Grolla A, Jones S, Feldmann H, Meyers A, Kabani A, Li Y, Normand S, Stroher U, Tipples GA, Tyler S, Vogrig R, Ward D, Watson B, Brunham RC, Krajden M, Petric M, Skowronski DM, Upton C, Roper RL: The genome sequence of the SARSassociated coronavirus. Science. 2003, 300 (5624): 13991404.View ArticlePubMedGoogle Scholar
 Chang BSW, Donoghue MJ: Recreating ancestral proteins. Trends Ecol Evol. 2000, 15 (3): 109114.View ArticlePubMedGoogle Scholar
 Bader DA, Moret BME, Vawter L: Industrial applications of highperformance computing for phylogeny reconstruction. SPIE ITCom 4528. SPIE, Denver 2001, 159168.Google Scholar
 Harvey PH, Brown AJL, Smith JM, Nee S: New Uses for New Phylogenies. Oxford University Press, Oxford 1996.Google Scholar
 Catanzaro D: Estimating phylogenies from molecular data. Mathematical Approaches to Polymer Sequence Analysis and Related Problems. Edited by: Bruni R. Springer, New York 2011.Google Scholar
 Beyer WA, Stein M, Smith T, Ulam S: A molecular sequence metric and evolutionary trees. Math Biosci. 1974, 19: 925. 10.1016/00255564(74)900285.View ArticleGoogle Scholar
 Waterman MS, Smith TF, Singh M, Beyer WA: Additive evolutionary trees. J Theor Biol. 1977, 64: 199213.View ArticlePubMedGoogle Scholar
 Albert VA: Parsimony, Phylogeny, and Genomics. Oxford University Press, Oxford 2005.Google Scholar
 Ding Z, Filkov V, Gusfield D: A linear time algorithm for Perfect Phylogeny Haplotyping (PPH) problem. J Comput Biol. 2006, 13 (2): 522553.View ArticlePubMedGoogle Scholar
 Bonizzoni P: A linear time algorithm for the Perfect Phylogeny Haplotype problem. Algorithmica. 2007, 48 (3): 267285. 10.1007/s0045300700943.View ArticleGoogle Scholar
 Catanzaro D: The minimum evolution problem: Overview and classification. Networks. 2009, 53 (2): 112125. 10.1002/net.20280.View ArticleGoogle Scholar
 Felsenstein J: Inferring Phylogenies. Sinauer Associates, Sunderland 2004.Google Scholar
 Bafna V, Gusfield D, Lancia G, Yooseph S: Haplotyping as perfect phylogeny: A direct approach. J Comput Biol. 2003, 10: 323340.View ArticlePubMedGoogle Scholar
 Pennington G, Smith CA, Shackney S, Schwartz R: Reconstructing tumor phylogenies from heterogeneous singlecell data. J Bioinformatics Comput Biol. 2006, 5 (2a): 407427.View ArticleGoogle Scholar
 Riester M, Attolini CSO, Downey RJ, Singer S, Michor F: A differentiationbased phylogeny of cancer subtypes. PLoS Comput Biol. 2010, 6: e100077.View ArticleGoogle Scholar
 Sridhar S, Dhamdhere K, Blelloch GE, Halperin E, Ravi R, Schwartz R: Algorithms for efficient nearperfect phylogenetic tree reconstruction in theory and practice. IEEE/ACM Trans Comput Biol Bioinformatics. 2007, 4 (4): 561571.View ArticleGoogle Scholar
 Goemans MX, Myung YS: A catalog of Steiner tree formulations. Networks. 1993, 23: 1928. 10.1002/net.3230230104.View ArticleGoogle Scholar
 Zhang XS, Wang RS, Wu LY, Chen L: Models and algorithms for haplotyping problem. Curr Bioinformatics. 2006, 1: 105114. 10.2174/157489306775330570.View ArticleGoogle Scholar
 Catanzaro D, Labbé M: The pure parsimony haplotyping problem: Overview and computational advances. Int Trans Oper Res. 2009, 16 (5): 561584. 10.1111/j.14753995.2009.00716.x.View ArticleGoogle Scholar
 Gusfield D: Efficient algorithms for inferring evolutionary trees. Networks. 1991, 21: 1928. 10.1002/net.3230210104.View ArticleGoogle Scholar
 Argawala R, FernandezBaca D: A polynomial time algorithm for the perfect phylogeny problem when the number of character states is fixed. SIAM J Comput. 1994, 23: 12161224. 10.1137/S0097539793244587.View ArticleGoogle Scholar
 Kannan S, Warnow T: A fast algorithm for the computation and enumeration of perfect phylogenies. SIAM J Comput. 1997, 26: 17491763. 10.1137/S0097539794279067.View ArticleGoogle Scholar
 Garey MR, Johnson DS: Computers and Intractability. 2003, Freeman, New York,Google Scholar
 Du DZ, Smith JM, Rubinstein JH: Advances in Steiner trees. Kluwer Academic Publisher, Boston 2000.View ArticleGoogle Scholar
 Cheng X, Du DZ: Steiner Trees in Industry. 2001, Kluwer Academic Publishers, Boston,Google Scholar
 Fischetti M, SalazarGonzáles JJ, Toth P: A branch and cut algorithm for the symmetric generalized traveling salesman problem. Oper Res. 1997, 45 (30): 378395.View ArticleGoogle Scholar
 Halldórsson BV, Bafna V, Edwards N, Lippert R: Combinatorial problems arising in SNP and haplotype analysis. Discrete Mathematics and Theoretical Computer Science, Volume 2731 of Lecture Note in Computer Science. Edited by: Calude CS. SpringerVerlag, Berlin 2003, 2647.Google Scholar
 Robins G, Zelikovsky A: Tighter bounds for graph Steiner tree approximation. SIAM J Discrete Math. 2005, 19: 122134. 10.1137/S0895480101393155.View ArticleGoogle Scholar
 , : The international hapmap project. Nature. 2003, 426 (18): 789796.Google Scholar
 Chopra S, Rao MR: The Steiner tree problem I: Formulations, compositions and extension of facets. Math Programming. 1994, 64 (13): 209229. 10.1007/BF01582573.View ArticleGoogle Scholar
 Chopra S, Rao MR: The Steiner tree problem II: Properties and classes of facets. Math Programming. 1994, 64 (13): 231246. 10.1007/BF01582574.View ArticleGoogle Scholar
 Koch T, Martin A, Voß S: SteinLib: An Updated Library on Steiner Tree Problems in Graphs. Berlin: Tech. Rep. ZIBReport 0037, KonradZuseZentrum für Informationstechnik Berlin, Takustr. 7; 2000. [http://elib.zib.de/steinlib], Berlin: Tech. Rep. ZIBReport 0037, KonradZuseZentrum für Informationstechnik Berlin, Takustr. 7; 2000. []
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.