Abstract
We study the problem of constructing phylogenetic trees for a given set of species. The problem is formulated as that of finding a minimum Steiner tree on n points over the Boolean hypercube of dimension d. It is known that an optimal tree can be found in linear time [1] if the given dataset has a perfect phylogeny, i.e. cost of the optimal phylogeny is exactly d. Moreover, if the data has a near-perfect phylogeny, i.e. the cost of the optimal Steiner tree is d + q, it is known [2] that an exact solution can be found in running time which is polynomial in the number of species and d, yet exponential in q. In this work, we give a polynomial-time algorithm (in both d and q) that finds a phylogenetic tree of cost d + O(q 2). This provides the best guarantees known—namely, a (1 + o(1))-approximation—for the case \(\log(d) \ll q \ll \sqrt{d}\), broadening the range of settings for which near-optimal solutions can be efficiently found. We also discuss the motivation and reasoning for studying such additive approximations.
This work was supported in part by the National Science Foundation under grant CCF-1116892, by an NSF Graduate Fellowship, and by the MSR-CMU Center for Computational Thinking.
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Awasthi, P., Blum, A., Morgenstern, J., Sheffet, O. (2012). Additive Approximation for Near-Perfect Phylogeny Construction. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2012 2012. Lecture Notes in Computer Science, vol 7408. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32512-0_3
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