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

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 number of rows among all matrices with this property. Hajirasouliha and Raphael also proposed the study of a similar problem, referred to as the minimum distinct conflict-free row split (MDCRS) 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 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) two approximation algorithms for the MCRS problem, and (iv) a 2-approximation algorithm for the MDCRS problem. The branching formulations also lead to exact exponential-time algorithms for solving the two problems to optimality faster than the na\"ive brute-force approach.