Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2506.18578

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

Narmina Baghirova·Esther Galby·Martin Milanič
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

cs.DMcs.DSmath.COq-bio.PE

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Narmina Baghirova, Esther Galby, Martin Milanič. 2025-06-23. Perfect phylogenies via the Minimum Uncovering Branching problem: efficiently solvable cases. https://doi.org/10.1109/tcbbio.2025.3580476

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

An FPTAS for Two-Machine Open-Shop Scheduling with a Single Unavailability Interval

We consider the two-machine open-shop scheduling problem in which one machine is unavailable during a fixed interval. We study the resumable setting: an operation interrupted by the unavailability interval may resume, without penalty, when the machine becomes available. The objective is to minimize the makespan. Although the problem is NP-hard and several approximation algorithms are known, whether it admits a fully polynomial-time approximation scheme (FPTAS) has remained open for two decades. We resolve this question affirmatively by giving the first FPTAS, thereby strengthening the previously known polynomial-time approximation scheme (PTAS). As an intermediate result, we develop a new pseudo-polynomial dynamic program with seven state dimensions, improving on the ten-dimensional formulation in the literature.

cs.DM↗
arXiv2026

Generalized Graph Search Trees

Graph search algorithms and their corresponding graph search trees are commonly used in algorithmic graph theory. In recent years, the recognition problem of these graph search trees has received significant attention. So far, the research has focused on two types of search trees: first-in trees that behave like BFS-trees and last-in trees that behave like DFS-trees. The search tree paradigms differ from each other by the parent a vertex is connected to. In first-in trees, it is the first visited neighbor, while in last-in trees it is the last neighbor visited before that vertex. Here, we will generalize these concepts of graph search trees by allowing every preceding neighbor of a vertex to be the parent. We study the complexity of the recognition problem of these generalized graph search trees. We present NP-completeness proofs for most searches. We also show that the problem is trivial for Generic Search and polynomial-time solvable for several searches on bipartite graphs and chordal graphs. We also study the question how fixing the start vertex influences the complexity of the problem.

cs.DM↗
arXiv2026

The exact asymptotic constant in the metric dimension of Jaccard space

Let $X$ be a finite set with $|X|=n$ and let $\mathrm{Jac}(a,b)=|a\,\triangle\, b|/|a\cup b|$ be the Jaccard distance on the power set $2^X$. Lladser and Paradise recently proved that the metric dimension of $(2^X,\mathrm{Jac})$ is $\Theta(n/\ln n)$, with the constant left open; their bounds are $(\ln 2)\,n/\ln n\lesssim \beta(2^X,\mathrm{Jac})\lesssim 2\ln(2e)\,n/\ln n$. We determine the constant: \[ \beta(2^X,\mathrm{Jac})=\frac{2n}{\log_2 n}\,(1+o(1))=(2\ln 2)\,\frac{n}{\ln n}\,(1+o(1)). \] The proof identifies the problem, on each ``slice'' of subsets of fixed cardinality, with the Erd\H{o}s--R\'enyi coin-weighing problem for a spring scale (the problem of \emph{detecting matrices}). The lower bound is the Erd\H{o}s--R\'enyi entropy argument applied to the middle slice; the upper bound follows from the explicit detecting families of Lindstr\"om and of Cantor and Mills, augmented by a single extra landmark that reveals cardinality.

cs.DM↗