  This article concerns the following question arising in computational
evolutionary biology. For a given subclass of phylogenetic networks, what is
the maximum value of 0 <= p <= 1 such that for every input set T of rooted
triplets, there exists some network N(T) from the subclass such that at least
p|T| of the triplets are consistent with N(T)? Here we prove that the set
containing all triplets (the full triplet set) in some sense defines p, and
moreover that any network N achieving fraction p' for the full triplet set can
be converted in polynomial time into an isomorphic network N'(T) achieving >=
p' for an arbitrary triplet set T. We demonstrate the power of this result for
the field of phylogenetics by giving worst-case optimal algorithms for level-1
phylogenetic networks (a much-studied extension of phylogenetic trees),
improving considerably upon the 5/12 fraction obtained recently by Jansson,
Nguyen and Sung. For level-2 phylogenetic networks we show that p >= 0.61. We
note that all the results in this article also apply to weighted triplet sets.
