Hitting the Romans

Abstract: 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.