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Perfect phylogenies via branchings in acyclic digraphs and a generalization of Dilworth's theorem

Hujdurović, Ademir and Husić, Edin and Milanič, Martin and Rizzi, Romeo and Tomescu, Alexandru (2018) Perfect phylogenies via branchings in acyclic digraphs and a generalization of Dilworth's theorem. ACM Transactions on Algorithms, 14 (2). p. 20. ISSN 1549-6325

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Identification Number: 10.1145/3182178

Abstract

Motivated by applications in cancer genomics and following the work of Hajirasouliha and Raphael (WABI 2014), Hujdurović et al. (IEEE TCBB, to appear) introduced the minimum conflict-free row split (MCRS) problem: split each row of a given binary matrix into a bitwise OR of a set of rows so that the resulting matrix corresponds to a perfect phylogeny and has the minimum possible number of rows among all matrices with this property. Hajirasouliha and Raphael also proposed the study of a similar problem, in which the task is to minimize the number of distinct rows of the resulting matrix. Hujdurović et al. proved that both problems are NP-hard, gave a related characterization of transitively orientable graphs, and proposed a polynomial-time heuristic algorithm for the MCRS problem based on coloring cocomparability graphs. We give new, more transparent formulations of the two problems, showing that the problems are equivalent to two optimization problems on branchings in a derived directed acyclic graph. Building on these formulations, we obtain new results on the two problems, including: (i) a strengthening of the heuristic by Hujdurović et al. via a new min-max result in digraphs generalizing Dilworth's theorem, which may be of independent interest, (ii) APX-hardness results for both problems, (iii) approximation algorithms, and (iv) exponential-time algorithms solving the two problems to optimality faster than the naive brute-force approach. Our work relates to several well studied notions in combinatorial optimization: chain partitions in partially ordered sets, laminar hypergraphs, and (classical and weighted) colorings of graphs.

Item Type: Article
Official URL: https://talg.acm.org/index.cfm
Additional Information: © 2018 ACM, Inc.
Subjects: Q Science > QA Mathematics
Sets: Departments > Mathematics
Date Deposited: 12 Jul 2018 15:23
Last Modified: 12 Jul 2018 15:26
Projects: 274977, I0-0035, P1-0285, N1-0032, N1-0038, N1-0062, J1-6720, J1-7051
Funders: Academy of Finland, Slovenian Research Agency
URI: http://eprints.lse.ac.uk/id/eprint/89231

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