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

2025-06-23

AI Summary

This paper resolves a long-standing open complexity question for the Minimum Uncovering Branching (MUB) problem: the exact computational complexity when the width—the size of the largest antichain—of the input set family is bounded. While the height-bounded case is known to be APX-complete, only a constant-factor approximation algorithm was previously available for the width-bounded case, leaving its precise complexity unresolved. We establish that any MUB instance of width $w$ admits an exact solution in $O(n^{2w+1})$ time. Our approach models MUB as an optimization problem over a partially ordered set and leverages Dilworth’s decomposition, maximum bipartite matching, and maximum-weight antichain computation to reduce it to a polynomial-time solvable graph-theoretic problem. This result closes a fundamental theoretical gap in the complexity landscape of MUB and yields the first polynomial-time exact algorithm framework for reconstructing perfect phylogenies in cancer genome evolution inference.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 Hujdurović, Husić, Milanič, 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. Hujdurović 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.

Problem

• Efficiently solving Minimum Uncovering Branching problem cases
• Mapping non-maximal sets to minimize uncovered elements
• Proving polynomial-time solvability for bounded-width instances

Innovation

• Polynomial-time solution for bounded-width instances
• Reduction to maximum bipartite graph matchings
• New polynomially computable lower bound