Best match graphs (BMGs) are vertex-colored directed graphs that were introduced to model the relationships of genes (vertices) from different species (colors) given an underlying evolutionary tree that is assumed to be unknown. In real-life applications, BMGs are estimated from sequence similarity data. Measurement noise and approximation errors usually result in empirically determined graphs that in general violate characteristic properties of BMGs. The arc modification problems for BMGs aim at correcting such violations and thus provide a means to improve the initial estimates of best match data. We show here that the arc deletion, arc completion and arc editing problems for BMGs are NP-complete and that they can be formulated and solved as integer linear programs. To this end, we provide a novel characterization of BMGs in terms of triples (binary trees on three leaves) and a characterization of BMGs with two colors in terms of forbidden subgraphs.

中文翻译：

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最佳匹配图修改问题的复杂性

最佳匹配图 (BMG) 是顶点着色的有向图，在给定假定未知的潜在进化树的情况下，引入该图以对来自不同物种（颜色）的基因（顶点）的关系进行建模。在实际应用中，BMGs 是从序列相似性数据中估计出来的。测量噪声和近似误差通常会导致根据经验确定的图形，这些图形通常违反 BMG 的特性。BMG 的弧修改问题旨在纠正此类违规，从而提供一种改进最佳匹配数据初始估计的方法。我们在这里展示了 BMG 的弧删除、弧完成和弧编辑问题是 NP 完全的，并且它们可以被制定和解决为整数线性程序。为此，