Roman domination is one of few examples where the related extension problem is polynomial-time solvable even if the original decision problem is NP-complete. This is interesting, as it allows to establish polynomial-delay enumeration algorithms for finding minimal Roman dominating functions, while it is open for more than four decades if all minimal dominating sets of a graph or if all hitting sets of a hypergraph can be enumerated with polynomial delay. To find the reason why this is the case, we combine the idea of hitting set with the idea of Roman domination. We hence obtain and study two new problems, called Roman Hitting Function and Roman Hitting Set, both generalizing Roman Domination. This allows us to delineate the borderline of polynomial-delay enumerability. Here, we assume what we call the Hitting Set Transversal Thesis, claiming that it is impossible to enumerate all minimal hitting sets of a hypergraph with polynomial delay. Our first focus is on the extension versions of these problems. While doing this, we find some conditions under which the Extension Roman Hitting Function problem is NP-complete. We then use parameterized complexity to get a better understanding of why Extension Roman Hitting Function behaves in this way. Furthermore, we analyze the parameterized and approximation complexity of the underlying optimization problems. We also discuss consequences for Roman variants of other problems like Vertex Cover.

翻译：罗马支配是少数几个即使原始决策问题为NP完全，其相关扩展问题仍可在多项式时间内求解的实例之一。这一点颇具意义，因为它允许建立多项式延迟枚举算法来寻找极小罗马支配函数，而关于一个图的所有极小支配集或超图的所有极小击中集能否在多项式延迟下枚举的问题，至今已悬而未决逾四十年。为了探明其原因，我们将击中集思想与罗马支配思想相结合，由此提出并研究了两个新问题——罗马击中函数与罗马击中集——两者均推广了罗马支配概念。这使我们得以界定多项式延迟可枚举性的分界线。在此，我们假设所谓的击中集传递论题，即断言无法以多项式延迟枚举超图的所有极小击中集。我们首先聚焦于这些问题的扩展版本，在此过程中发现了扩展罗马击中函数问题为NP完全的条件。随后利用参数化复杂性深入理解扩展罗马击中函数如此表现的原因。此外，我们分析了底层优化问题的参数化与近似复杂性。最后讨论了这些结果对诸如顶点覆盖等其他问题罗马变体的影响。