The TreeRank measure is not (directly) applicable as objective function for an ILP model, since it is highly non-linear. Hence we will either apply the UpDown distance or the Weighted Triple score. Unfortunately, as we will see later on, especially an ILP model with the Up-Down distance as objective function is in general only applicable to smaller instances as it soon requires too much computational effort.
3.10.1 Triple Model
The first proposed ILP model, denoted asTriple model, usestab|cvariables representing the presence or absence of each possible triple, and further utilizes the Weighted Triple score (3.5) as objective function to be maximized (3.9); it is similar to the model for quartets presented in [234].
max X
tab|c:ab|c∈R(T)
wTab|ctab|c (3.9)
s.t. tab|c+tbc|a+tac|b= 1 ∀a < b < c∈ L (3.10) tab|c+tad|c−tbd|c≤1 ∀{a, b, c, d} ⊂ L (3.11) tab|c+tac|d−tab|d≤1 ∀{a, b, c, d} ⊂ L (3.12) tab|c=tba|c ∀a < b, c∈ L (3.13) tab|c∈ {0,1} ∀{a, b, c} ⊂ L (3.14) Constraints (3.10) ensure that exactly one possible tripleab|c,bc|aorac|bis realized in the resulting tree. The inequalities (3.11) and (3.12) express the triple transitivity and so-called telescopic conditions, which are used in order to derive new necessary triples from other ones to ensure consistency. Equality (3.13) states that triplesab|candba|care equivalent. Finally, constraints (3.14) enforce binary triple variables. This model impliesΘ(n3)variables and Θ(n4)constraints and has the advantage of being relatively efficient, since also the objective function is easy to calculate. As we will see, moderately-sized instances can be solved well in practice.
3.10.2 Combined Triple and UpDown Distance Model
Our second ILP model, denoted asTriple+UDD model, is an extension of the previous one and includes bothtab|cvariables as well asuabvariables representing the values in the Up-52
Down matrix of the final solution tree. The purpose of this is to allow the use of an objec-tive function based on the values of the UpDown matrix—which is thus finer grained and assumed to be more realistic—and at the same time obtain the tree defined by the triple variables, in order to avoid the conversion from the UpDown matrix to the triple variables (since this algorithm is not yet known, and might be a topic of future investigation). Also for ensuring consistency of the UpDown matrix thetab|cvariables are used.
min X
T∈T
X
a,b∈L
δT ab (3.15)
s.t. UT[a, b]−uab≤δT ab ∀T ∈ T;∀a, b∈ L (3.16) uab−UT[a, b]≤δT ab ∀T ∈ T;∀a, b∈ L (3.17)
uaa= 0 ∀a∈ L (3.18)
uab≥1 ∀a, b∈ L (3.19)
uab≤n−1 ∀a, b∈ L (3.20)
uab−uac< M(1−tab|c) ∀{a, b, c} ⊂ L (3.21) uba−ubc< M(1−tab|c) ∀{a, b, c} ⊂ L (3.22) uca−ucb≤M(1−tab|c) ∀{a, b, c} ⊂ L (3.23) ucb−uca≤M(1−tab|c) ∀{a, b, c} ⊂ L (3.24) uac−uab−(ubc−uba)≤M(1−tab|c) ∀{a, b, c} ⊂ L (3.25) ubc−uba−(uac−uab)≤M(1−tab|c) ∀{a, b, c} ⊂ L (3.26) min{uab |b∈ L \ {a}}= 1 ∀a∈ L (3.27)
uab∈N0 ∀a, b∈ L (3.28)
δT ab∈N0 ∀T ∈ T;∀a, b∈ L (3.29)
(3.10)–(3.14)
Here the UpDown distance (3.2) is used as an objective function: we introduce additional variablesδT ab (3.29) which hold the absolute values linearized via (3.16) and (3.17), finally allowing to minimize the sum of them (3.15). The distance from a taxa to itself is zero (3.18) and the distance between two different taxa is at least one (3.19) and at most the number of taxa minus one (3.20), since the latter is the largest depth of a binary tree havingnleaves.
Constraints (3.21)-(3.26) ensure that the UpDown matrix is consistent. In these constraints M is a sufficiently large constant (hence often also referred to asbig-Mconstraints) which can be set to the number of taxa n. This ensures that the constraints are activated only when the triple ab|cis present (i.e. when tab|c = 1). Constraints (3.25)-(3.26) are called path constraints. They ensure that the values themselves of the UpDown matrix, and not only the relative distances, are consistent. The row-min constraints (3.27) ensure that no
“artificial” inner nodes may be added to lower specific taxa, which might otherwise happen (also depending on the objective function in use). This model still impliesΘ(n3)variables andΘ(n4)constraints.
3. CONSENSUSTREEPROBLEM
3.10.3 Reduce Computational Effort with Lazy Constraints
For some ILP models quite a lot constraints are introduced to guarantee feasibility. They can be regarded consistency constraints. In such cases it might be beneficial to initially omit some of these constraints, but of course they would have to be respected during solving in an appropriate way. One rather naïve possibility is to check the feasibility of the solution obtained after solving the reduced model to optimality. After adding all violated constraints one has to do a complete re-solving, probably in an iterated manner, causing (most likely) too much overhead. Hence in order to gain more speed-up compared to solving the initial complete model, these constraints are at best added in an incremental way during the solving process. Ilog CPLEX offers an elegant way to do this via a pool oflazy constraints. For this one adds the desired consistency constraints in a pool (set), which in our case are the tran-sitivity (3.11) and telescopic constraints (3.12), and they are henceforth treated differently.
Whenever an integer solution is found it is checked whether one of these constraints is vio-lated and adds them to the model if required; consult the user manual for more information.
This scheme is in some sense analogue to the cutting plane approach when solving a linear program. Depending on the problem, this can dramatically improve the performance of the solver.
3.10.4 Heuristic Generation of Variables
Instead of reducing the number of (initial) constraints the optimization algorithm has to deal with, the number of variables considered for solving the problem can also be restricted at the beginning. Triple variablestab|ccorresponding to triples ab|cfor which it is assumed that they have value zero in the final solution are eliminated from the problem. This step is referred to as pruning the variable set. The important question is: How is the variable set to be pruned, so as not to discard the optimal solution? Since there is no theoretical background to answer this question, the variable set is pruned in a heuristic way: triples that are conjectured to most likely not appear in the optimal solution are pruned. These are exactly the triples not appearing in any input tree. Since the tree which best summarizes the information contained in the input trees is sought, it is expected that triples not appearing in any input tree will not be present in the final solution.
The idea of beginning with a reduced set of variables is closely related to column generation approaches, see Section 2.3.1 for a general outline and Section 4.7 for a concrete application of it. However, since for the CTP the pricing problem is most likely only solvable via com-plete enumeration of all variables (i.e. showing no special structure for which an effective algorithm is known), we settled here to propose a heuristic method in which variables are added in case they appear in an incumbent solution. For this purpose, an iterative rounding and repair procedure is applied to each non-integer candidate solution in order to find a fea-sible tree. In a first step this procedure investigates the triples in lexicographic order and sets those corresponding triple variable having the highest (fractional) value in the LP solution to one (randomly selecting a variable in case of equal values) and the other two variables to zero. Then for each subset of four taxa we check all according transitivity and telescopic constraints and in case of a violation repair them individually by setting the corresponding 54
triple variable to one and the other two to zero. This repair step is consecutively applied until no more violations occur.
We denote this whole method as heuristic column generation. In this approach, the initial set of variables is chosen by the previously described pruning method.
3.10.5 Hybridization of Heuristic and Exact Methods
Initially we planned to combine the strengths of the heuristic and exact approaches in a suitable way. We especially had in mind to select a subtree of a meaningful size (i.e. small enough to be solved in reasonable time yet large enough to allow for an improvement) and solve it to (near) optimality or to derive some sort of exact recombination operator (i.e.
basically a solution merging). Unfortunately, for several reasons this did not work out. The major problem was the non-linearity of the favored TreeRank similarity measure and the fact, that optimizing using alternative linear measures does not imply maximizing the TreeRank measure. This makes it virtually impossible that the proposed exact methods are of a benefit for the metaheuristics. Considering the contrary hybridization, i.e. in some way boosting the performance of the exact approaches via utilizing the heuristics, also yielded no meaningful possibilities here. Examples of the latter would be to offer better initial solutions than the best input trees as well as to provide the exact methods with improved feasible solutions during its execution. On the one hand these approaches would not really represent something new from a methodical point of view and on the other hand the applicability of the exact approaches strongly depends on the problem size (also memory consumption becomes an issue due to the large number of constraints) and it is unlikely that good bounds from feasible solutions will allow to tackle larger instances.