Bioinformatics Algorithms: Techniques and Applications

RECONCILIATION OF GENE TREES AND SPECIES TREES 171
In [6], it is shown that Definition 7.6 computes a tree that satisfies the properties
for the reconciled tree and such a tree has also minimum size. Furthermore, given a
gene tree TG and a species tree TS, there exists a unique minimum reconciled tree for
TG and TS [6], which is also the tree inducing the minimum number of duplications
and losses [21]. Next, we introduce the main problem for the lateral gene transfer
model.
PROBLEM 7.12 α-Active Lateral Gene Transfer Problem
Input: gene tree TG and species tree TS.
Output: find a minimum cost α-active lateral transfer scenario for TS and TG.
The restriction of α-active Lateral Gene Transfer Problem where α = 1 is
APX-hard [11,23], while it has a O(2 4T |S| 2 ) fixed-parameter algorithm [23],
where the parameter T is the cost of the scenario. For arbitrary α there is an
O(4 α (4T (α + T )) T |L| 2 ) time algorithm [23].
The extension of the problem that considers both duplications and lateral gene
transfers is known to be NP-hard [24] and admits a fixed-parameter tractable
algorithm [24] that computes the minimum number of duplications and lateral
transfers.
7.5.3 Open Problems
A deep understanding of the approximation complexity of the agreement problems
is still needed. More precisely, the only known result is the 2-factor approximation
algorithm for the variant of Optimal Species Tree Reconstruction with Duplication
Cost (see Problem 7.8) in which the duplication cost is slightly modified to
obtain a metric d [31]. In this variant, all gene trees are uniquely labeled. Moreover,
given a gene tree TG and a species tree TS, the symmetric duplication cost
between TG and TS is defined as d(TG,TS) = 1 2 (dup(TS,TG) + dup(TG,TS)). The
new version of the problem remains NP-hard while admitting a 2-approximation
algorithm [31].
An interesting open problem on species trees and gene trees is the computational
complexity of reconstructing a species tree from a set of gene trees over instances
consisting of a constant number of gene trees or even of two gene trees only.
An extension of the reconciliation approach (see Definition 7.5 and
Problem 7.11) is proposed in [20] by introducing a notion of extended reconciled
tree allowing the identification of lateral gene transfers, in addition to duplication
and losses. A notion of scenario is also introduced to identify lateral
transfers. A dynamic programming algorithm to compute a scenario inducing
a minimum reconciliation cost is given [20]. Also notice that no approximation
algorithms are known for the event inference problems presented in this
section.