A linear time approximation scheme for maximum quartet consistency on sparse sampled inputs

Phylogenetic tree reconstruction is a fundamental biological problem. Quartet amalgamation-combining a set of trees over four taxa into a tree over the full set-stands at the heart of many phylogenetic reconstruction methods. This task has attracted many theoretical as well as practical works. However, even reconstruction from a consistent set of quartet trees, i.e., all quartets agree with some tree, is NP-hard, and the best approximation ratio known is 1/3. For a dense input of θ (n ⁴) quartets that are not necessarily consistent, the problem has a polynomial time approximation scheme. When the number of taxa grows, considering such dense inputs is impractical and some sampling approach is imperative. It is known that given a randomly sampled consistent set of quartets from an unknown phylogeny, one can find, in polynomial time and with high probability, a tree satisfying a 0.425 fraction of them, an improvement over the 1/3 ratio. In this paper we further show that given a randomly sampled consistent set of quartets from an unknown phylogeny, where the size of the sample is at least θ(n ² log n), there is a randomized approximation scheme that runs in linear time in the number of quartets. The previously known polynomial approximation scheme for that problem required a very dense sample of size θ (n ⁴). We note that samples of size θ (n ² log n) are sparse in the full quartet set. The result is obtained by a combinatorial technique that may be of independent interest.