Abstract
Measuring tree dissimilarity and studying the shape of trees are important tasks in phylogenetics. One of the most studied shape properties is the notion of tree imbalance, which can be quantified by different indicators, such as the Colless index. Here, we study the generalization of the Colless index to mobiles, i.e., full binary trees in which each leaf has been assigned a positive integer weight. In particular, we focus on the problem Balanced Mobiles, which given as input n weights and a full binary tree on n leaves, asks to find an assignment of the weights to the leaves that minimizes the Colless index, i.e., the sum of the imbalances of the internal nodes (computed as the difference between the total weight of the left and right subtrees of the node considered). We prove that this problem is strongly \(\textsf{NP}\)-hard, answering an open question given at IWOCA 2016.
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References
Bartoszek, K., Coronado, T.M., Mir, A., Rosselló, F.: Squaring within the Colless index yields a better balance index. Math. Biosci. 331, 108503 (2021)
Blum, M.G., François, O., Janson, S.: The mean, variance and limiting distribution of two statistics sensitive to phylogenetic tree balance. Ann. Appl. Probab. 16(4), 2195–2214 (2006)
Colless, D.H.: Phylogenetics: The Theory and Practice of Phylogenetic Systematics (1982)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press, Cambridge (2009)
Coronado, T.M., Fischer, M., Herbst, L., Rosselló, F., Wicke, K.: On the minimum value of the Colless index and the bifurcating trees that achieve it. J. Math. Biol. 80(7), 1993–2054 (2020)
Fischer, M., Herbst, L., Kersting, S., Khn, L., Wicke, K.: Tree balance indices: a comprehensive survey (2021)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman (1979)
Hamoudi, Y., Laplante, S., Mantaci, R.: Balanced mobiles (2016). https://www.iwoca.org. Problems Section
Hamoudi, Y., Laplante, S., Mantaci, R.: Balanced mobiles with applications to phylogenetic trees and Huffman-like problems. Technical report (2017). https://hal.science/hal-04047256
Huffman, D.A.: A method for the construction of minimum-redundancy codes. Proc. IRE 40(9), 1098–1101 (1952)
Mir, A., Rotger, L., Rosselló, F.: Sound Colless-like balance indices for multifurcating trees. PLoS ONE 13(9), e0203401 (2018)
Sackin, M.J.: “Good’’ and “bad’’ phenograms. Syst. Biol. 21(2), 225–226 (1972)
Shao, K.-T., Sokal, R.R.: Tree balance. Syst. Biol. 39(3), 266–276 (1990)
Acknowledgements
Part of this work was conducted when RR was an invited professor at Université Paris-Dauphine. This work was partially supported by the ANR project ANR-21-CE48-0022 (“S-EX-AP-PE-AL”).
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Ardévol Martínez, V., Rizzi, R., Sikora, F. (2023). Hardness of Balanced Mobiles. In: Hsieh, SY., Hung, LJ., Lee, CW. (eds) Combinatorial Algorithms. IWOCA 2023. Lecture Notes in Computer Science, vol 13889. Springer, Cham. https://doi.org/10.1007/978-3-031-34347-6_3
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DOI: https://doi.org/10.1007/978-3-031-34347-6_3
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