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This article concerns the following question arising in computationalevolutionary biology. For a given subclass of phylogenetic networks, what isthe maximum value of 0 <= p <= 1 such that for every input set T of rootedtriplets, there exists some network N(T) from the subclass such that at leastp|T| of the triplets are consistent with N(T)? Here we prove that the setcontaining all triplets (the full triplet set) in some sense defines p, andmoreover that any network N achieving fraction p' for the full triplet set canbe 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 forthe field of phylogenetics by giving worst-case optimal algorithms for level-1phylogenetic 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. Wenote that all the results in this article also apply to weighted triplet sets.

Worst-case optimal approximation algorithms for maximizing triplet consistency within phylogenetic networks