Proceedings of the seventh annual ACM-SIAM symposium on Discrete algorithms
Approximation algorithms
Reconstructing approximate phylogenetic trees from quartet samples
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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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. 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(n4) quartets (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. In this paper we show that if the number of quartets sampled is at least Θ(n2) 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(n4). We note that samples of size Θ(n2) log n) are sparse in the full quartet set.