Perfect phylogenies via the Minimum Uncovering Branching problem: efficiently solvable cases

TL;DR

This paper identifies new efficiently solvable cases of the Minimum Uncovering Branching problem, which has applications in cancer genomics, by analyzing structural properties and reducing it to known polynomial-time problems.

Contribution

The paper proves that the Minimum Uncovering Branching problem is polynomial-time solvable for certain instances, resolving an open question about bounded-width cases.

Findings

The problem is solvable in polynomial time for bounded-width instances.

Structural properties of optimal solutions enable reduction to maximum matchings and antichains.

Introduces a new polynomially computable lower bound and additional solvability conditions.

Abstract

In this paper, we present new efficiently solvable cases of the Minimum Uncovering Branching problem, an optimization problem with applications in cancer genomics introduced by Hujdurovi\'c, Husi\'c, Milani\v{c}, Rizzi, and Tomescu in 2018. The problem involves a family of finite sets, and the goal is to map each non-maximal set to exactly one set that contains it, minimizing the sum of uncovered elements across all sets in the family. Hujdurovi\'c et al. formulated the problem in terms of branchings of the digraph formed by the proper set inclusion relation on the input sets and studied the problem complexity based on properties of the corresponding partially ordered set, in particular, with respect to its height and width, defined respectively as the maximum cardinality of a chain and an antichain. They showed that the problem is APX-complete for instances of bounded height and that a constant-factor approximation algorithm exists for instances of bounded width, but left the exact complexity for bounded-width instances open. In this paper, we answer this question by proving that the problem is solvable in polynomial time. We derive this result by examining the structural properties of optimal solutions and reducing the problem to computing maximum matchings in bipartite graphs and maximum weight antichains in partially ordered sets. We also introduce a new polynomially computable lower bound and identify another condition for polynomial-time solvability.