The problem Defensive $δ$-Covering, for some covering range $δ> 0$, is a continuous facility location problem on undirected graphs where all edges have unit length. It is a generalization of Defensive Dominating Set and $δ$-Covering. An attack and defense are sets of points, which are on vertices or on the interior of an edge. A defense counters an attack, if there is a matching of the points in the defense to the points in the attack, such that any matched points have distance at most $δ$, and every point in the attack is matched. The task is, given a graph $G$ and numbers $\ell, k \in \mathbb N$, to find a defense of size at most $\ell$ that counters every possible attack of size at most $k$. We study the complexity of this problem in various different settings. We show that if the attack is restricted to vertices, the problem is $Σ^P_2$-complete for large $δ$, but if the attack may consist of any points on the graph, it is NP-complete. Additionally, we analyze how the complexity changes if the attacks or defenses may be a multiset. If the defense is allowed to be a multiset, the complexity does not change in any case we consider, while if the attack is allowed to be a multiset, the problem often becomes easier. To show containment in the various complexity classes, we introduce a number of discretization arguments, which show that solutions with a regular structure must always exist.

翻译：摘要：对于覆盖范围 δ > 0，防御 δ-覆盖问题是一类在单位边长的无向图上的连续设施选址问题，它是防御支配集与 δ-覆盖问题的推广。攻击与防御均为位于顶点或边内部的点集。若存在从防御点到攻击点的匹配，使得任意匹配点对的距离不超过 δ，且攻击点均被匹配，则称防御成功抵御攻击。给定图 G 及自然数 ℓ, k，任务是寻找规模不超过 ℓ 的防御，使其能抵御所有规模不超过 k 的攻击。我们在多种不同设定下研究该问题的复杂性。结果表明：当攻击仅限于顶点时，对于较大的 δ 问题为 Σ^P_2-完全；而当攻击点可位于图上任意位置时，问题为 NP-完全。此外，我们分析了攻击或防御为多重集时复杂性的变化：若防御允许为多重集，我们所考虑的所有情形下复杂性均不变；若攻击允许为多重集，则问题通常变得更简单。为证明各类复杂性类别的包含关系，我们引入了一系列离散化论证，表明具有规整结构的解必然存在。