Motivated by applications in cancer genomics and following the work of Hajirasouliha and Raphael (WABI 2014), Hujdurovi\'c 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. Hujdurovi\'c 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 Hujdurovi\'c 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 na\"ive 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.

翻译：受癌症基因组学应用和Hajiirasouliha 和 Raphael (WABI,2014年) 和 Hujdurović 等人 (IEEE TCBB, 即将出现) 工作的影响, 引入了最小无冲突行分割(MCRS) 问题: 将给定的二进制矩阵的每行分割成一个略微的或一组行, 从而由此形成的矩阵符合完美的血压特征, 并在与该属性相关的所有矩阵中拥有最低可能的行数。 Hajirasouliha 和 Raphael 也提议研究一个类似的问题, 任务在于将由此产生的矩阵中不同行的直流数最小化。 Hujdurovi 和 al. 证明这两个问题都是硬化的, 给出了过渡或可调整的图表中的多时超值算算法, 给出了两个问题的新更透明的配法。 我们给出了两个问题, 显示问题相当于两个直径的直径的直径, 直系的直径, 根部的直径, 直系的直径, 直系的直系的直径, 直系的直系, 直系的直系, 直系, 直系的直系, 直系的直系的直系, 直系, 直系, 直系, 直系, 直系, 直系, 直系的直系, 直系, 直系,直系, 直系, 直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直,直,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直,直,直,直,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系,直系