Cytoplasmic incompatibility (CI) relates to the manipulation by the parasite Wolbachia of its host reproduction. Despite its widespread occurrence, the molecular basis of CI remains unclear and theoretical models have been proposed to understand the phenomenon. We consider in this paper the quantitative Lock-Key model which currently represents a good hypothesis that is consistent with the data available. CI is in this case modelled as the problem of covering the edges of a bipartite graph with the minimum number of chain subgraphs. This problem is already known to be NP-hard, and we provide an exponential algorithm with a non trivial complexity. It is frequent that depending on the dataset, there may be many optimal solutions which can be biologically quite different among them. To rely on a single optimal solution may therefore be problematic. To this purpose, we address the problem of enumerating (listing) all minimal chain subgraph covers of a bipartite graph and show that it can be solved in quasi-polynomial time. Interestingly, in order to solve the above problems, we considered also the problem of enumerating all the maximal chain subgraphs of a bipartite graph and improved on the current results in the literature for the latter. Finally, to demonstrate the usefulness of our methods we show an application on a real dataset.

中文翻译：

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细胞质不相容性定量Lock / Key模型的算法。

细胞质不相容性（CI）与寄生虫Wolbachia对其宿主繁殖的操纵有关。尽管它的广泛发生，CI的分子基础仍然不清楚，并提出了理论模型来理解这种现象。我们在本文中考虑了定量的锁匙模型，该模型目前代表了一个很好的假设，与可用数据一致。在这种情况下，CI被建模为用最少数量的链子图覆盖二部图的边缘的问题。众所周知，这个问题是NP难题的，我们提供了一个复杂度不高的指数算法。通常，根据数据集，可能存在许多最佳解决方案，这些解决方案在生物学上可能完全不同。因此，依靠单个最佳解决方案可能会成问题。为此，我们解决了枚举（列出）二部图的所有最小链子图覆盖的问题，并表明可以在准多项式时间内解决该问题。有趣的是，为了解决上述问题，我们还考虑了枚举二部图的所有最大链子图的问题，并对后者的文献结果进行了改进。最后，为了展示我们方法的有用性，我们在真实数据集上展示了一个应用程序。我们还考虑了枚举二部图的所有最大链子图的问题，并对后者的文献中的现有结果进行了改进。最后，为了展示我们方法的有用性，我们在真实数据集上展示了一个应用程序。我们还考虑了枚举二部图的所有最大链子图的问题，并对后者的文献中的现有结果进行了改进。最后，为了展示我们方法的有用性，我们在真实数据集上展示了一个应用程序。