Abstract
BackgroundGiven a gene family, the relations between genes (orthology/paralogy), are represented by a relation graph, where edges connect pairs of orthologous genes and “missing” edges represent paralogs. While a gene tree directly induces a relation graph, the converse is not always true. Indeed, a relation graph is not necessarily “satisfiable”, i.e. does not necessarily correspond to a gene tree. And even if that holds, it may not be “consistent”, i.e. the tree may not represent a true history in agreement with a species tree. Previous studies have addressed the problem of correcting a relation graph for satisfiability and consistency. Here we consider the weighted version of the problem, where a degree of confidence is assigned to each orthology or paralogy relation. We also consider a maximization variant of the unweighted version of the problem.ResultsWe provide complexity and algorithmic results for the approximation of the considered problems. We show that minimizing the correction of a weighted graph does not admit a constant factor approximation algorithm assuming the unique game conjecture, and we give an n-approximation algorithm, n being the number of vertices in the graph. We also provide polynomial time approximation schemes for the maximization variant for unweighted graphs.ConclusionsWe provided complexity and algorithmic results for variants of the problem of correcting a relation graph for satisfiability and consistency. For the maximization variants we were able to design polynomial time approximation schemes, while for the weighted minimization variants we were able to provide the first inapproximability results.
Highlights
Given a gene family, the relations between genes, are represented by a relation graph, where edges connect pairs of orthologous genes and “missing” edges represent paralogs
We show that a result in [16] implies a polynomial time approximation scheme (PTAS) for satisfiability
A bounded approximation algorithm for minimum weighted editing for satisfiability and consistency While Minimum weighted editing for satisfiability (MinWES) and Minimum weighted editing for consistency (MinWEC) are not approximable within a constant factor, we show here that they can be approximated within factor n = |V (R)|, and we give the corresponding algorithms
Summary
We provide complexity and algorithmic results for the approximation of the considered problems. We provide polynomial time approximation schemes for the maximization variant for unweighted graphs
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