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

## TL;DR

This paper introduces new formulations and algorithms for the minimum conflict-free row split problem in perfect phylogeny, connecting it to branchings in acyclic digraphs and generalizing Dilworth's theorem, with implications for computational complexity and optimization.

## Contribution

It provides transparent formulations linking the problem to acyclic digraph branchings, extends Dilworth's theorem, and offers improved algorithms and complexity results.

## Key findings

- Strengthened heuristic via a new min-max theorem in digraphs
- Proved APX-hardness of the problems
- Developed approximation and exponential-time algorithms

## Abstract

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.