  In 1996, Guigo et al. [Mol. Phylogenet. Evol., 6 (1996), 189-203] posed the
following problem: for a given species tree and a number of gene trees, what is
the minimum number of duplication episodes, where several genes could have
undergone duplication together to generate the observed situation. (Gene order
is neglected, but duplication of genes could have happened only on certain
segments that duplicated). We study two versions of this problem, one of which
was algorithmically solved not long ago by Bansal and Eulenstein
[Bioinformatics, 24(13), (2008), 132-138]. We provide min-max theorems for both
versions that generalize Gallai's archetypal min-max theorem on intervals,
allowing simplified proofs to the correctness of the algorithms (as it always
happens with duality) and deeper understanding. An interesting feature of our
approach is that its recursive nature requires a generality that
bioinformaticians attempting to solve a particular problem usually avoid.
