Reconciling a gene tree to a species tree under the duplication cost model

The general problem of reconciling the information from evolutionary trees representing the relationships between distinct gene families is of great importance in bioinformatics and has been popularized among the computer science researchers by Ma et al. [From gene trees to species trees, SIAM J. Comput. 30(3) (2000) 729-752] where the authors pose the intriguing question if a certain definition of minimum tree that reconciles a gene tree and a species tree is correct. We answer affirmatively to this question; moreover, we show an efficient algorithm for computing such minimum-leaf reconciliation trees and prove the uniqueness of such trees. We then tackle some different versions of the biological problem by showing that the exemplar problem, arising from the exemplar analysis of multigene genomes, is NP-hard even when the number of copies of a given label is at most two. Finally, we introduce two novel formulations for the problem of recombining evolutionary trees, extending the gene duplication problem studied in [Ma et al., From gene trees to species trees, SIAM J. Comput. 30(3) (2000) 729-752; M. Fellows et al., On the multiple gene duplication problem, in: Proc. Ninth Internat. Symp. on Algorithms and Computation (ISAAC98), 1998; R. Page, Maps between trees and cladistic analysis of historical associations among genes, Systematic Biology 43 (1994) 58-77; R.M. Page, J. Cotton, Vertebrate phylogenomics: reconciled trees and gene duplications, in: Proc. Pacific Symp. on Biocomputing 2002 (PSB2002), 2002, pp. 536-547; R. Guigò et al., Reconstruction of ancient molecular phylogeny, Mol. Phy. and Evol. 6(2) (1996) 189-213], and we give an exact algorithm (via dynamic programming) for one of these formulations.