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Algorithms"],"published-print":{"date-parts":[[2018,4,30]]},"abstract":"<jats:p>Motivated by applications in cancer genomics and following the work of Hajirasouliha and Raphael (WABI 2014), Hujdurovi\u0107 et al. (IEEE TCBB, 2018) 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. Hujdurovi\u0107 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.<\/jats:p>\n          <jats:p>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 (1) a strengthening of the heuristic by Hujdurovi\u0107 et al. via a new min-max result in digraphs generalizing Dilworth\u2019s theorem, which may be of independent interest; (2) APX-hardness results for both problems; (3) approximation algorithms; and (4) exponential-time algorithms solving the two problems to optimality faster than the na\u00efve 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.<\/jats:p>","DOI":"10.1145\/3182178","type":"journal-article","created":{"date-parts":[[2018,4,18]],"date-time":"2018-04-18T17:21:50Z","timestamp":1524072110000},"page":"1-26","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":6,"title":["Perfect Phylogenies via Branchings in Acyclic Digraphs and a Generalization of Dilworth\u2019s Theorem"],"prefix":"10.1145","volume":"14","author":[{"given":"Ademir","family":"Hujdurovi\u0107","sequence":"first","affiliation":[{"name":"University of Primorska, Koper, Slovenia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Edin","family":"Husi\u0107","sequence":"additional","affiliation":[{"name":"London School of Economics and University of Primorska, Koper, Slovenia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Martin","family":"Milani\u0107","sequence":"additional","affiliation":[{"name":"University of Primorska, Koper, Slovenia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Romeo","family":"Rizzi","sequence":"additional","affiliation":[{"name":"University of Verona, Verona, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexandru I.","family":"Tomescu","sequence":"additional","affiliation":[{"name":"Helsinki Institute for Information Technology HIIT and University of Helsinki, Finland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2018,4,16]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1137\/0201008"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(98)00158-3"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1137\/140954167"},{"key":"e_1_2_1_4_1","doi-asserted-by":"crossref","unstructured":"G. Ausiello P. Crescenzi G. Gambosi V. Kann A. Marchetti-Spaccamela and M. Protasi. 1999. Complexity and Approximation. Springer-Verlag Berlin. xx+524 pages.  G. Ausiello P. Crescenzi G. Gambosi V. Kann A. Marchetti-Spaccamela and M. Protasi. 1999. Complexity and Approximation. Springer-Verlag Berlin. xx+524 pages.","DOI":"10.1007\/978-3-642-58412-1"},{"volume":"1643","volume-title":"Proceedings of the 7th Annual European Symposium on Algorithms (ESA\u201999)","author":"Cheriyan J.","key":"e_1_2_1_5_1"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2008.06.013"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.2307\/1969503"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ipl.2005.09.013"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/0025-5564(75)90040-1"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1006\/jcss.1995.1065"},{"key":"e_1_2_1_11_1","volume-title":"DIMACS Ser. Discrete Math. Theoret. Comput. Sci.","volume":"49","author":"Frank A.","year":"1999"},{"key":"e_1_2_1_12_1","first-page":"701","article-title":"Note on Dilworth\u2019s decomposition theorem for partially ordered sets","volume":"7","author":"Fulkerson D. R.","year":"1956","journal-title":"Proc. Amer. Math. Soc."},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01585864"},{"key":"e_1_2_1_14_1","doi-asserted-by":"crossref","unstructured":"M. C. Golumbic. 2004. Algorithmic Graph Theory and Perfect Graphs (second ed.). Elsevier Science B.V. Amsterdam.   M. C. Golumbic. 2004. Algorithmic Graph Theory and Perfect Graphs (second ed.). Elsevier Science B.V. Amsterdam.","DOI":"10.1016\/S0167-5060(04)80051-7"},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-0190(97)00002-1"},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.1002\/net.3230210104"},{"key":"e_1_2_1_17_1","doi-asserted-by":"crossref","unstructured":"D. Gusfield. 1997. Algorithms on Strings Trees and Sequences: Computer Science and Computational Biology. Cambridge University Press.   D. Gusfield. 1997. Algorithms on Strings Trees and Sequences: Computer Science and Computational Biology. 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