Knop and Schierreich [AAMAS '23] recently introduced a novel model of refugee housing and specifically asked for the computational complexity picture of the following variant. Given a topology modelled as an undirected graph, a set of inhabitants, a number of refugees $R$, an assignment of inhabitants to houses of the topology, and an upper-bound for every inhabitant, find a set $\pi$ of unoccupied houses of size $R$ intended such that the number of refugees in the neighbourhood of every inhabitant is at most its upper-bound. If such a set $\pi$ exists, we say that the instance admits an inhabitant-respecting housing. In this paper, we show that the existence of inhabitant-respecting housing is not guaranteed even under several further restrictions of the upper-bounds. Then, we focus on the computational complexity of deciding whether inhabitant-respecting housing exists. To this end, we provide tractable algorithms for several restrictions of the topology. We complement these results with appropriate hardness results and running-time lower-bounds. Furthermore, we introduce a relaxed (or approximate) version of the inhabitant-respecting housing, where we allow at most $t$ upper-bounds to be exceeded.

翻译：Knop和Schierreich [AAMAS '23] 近期提出了一种新颖的难民安置模型，并特别询问了以下变体问题的计算复杂性。给定一个由无向图建模的拓扑结构、一组居民、难民数量$R$、居民在拓扑结构中房屋的分配方案，以及每个居民的上界，需要找到一组由$R$个空置房屋构成的集合$\pi$，使得每个居民邻域内的难民数量不超过其对应的上界。若存在这样的集合$\pi$，则称该实例存在居民尊重型安置方案。本文证明，即使对上界施加多种进一步限制，居民尊重型安置方案的存在性仍无法保证。继而，我们聚焦于判定居民尊重型安置方案是否存在这一问题的计算复杂性。为此，针对拓扑结构的多种限制情形，我们提出了可解算法。这些结果通过相应的难度结论和运行时间下界加以补充。此外，我们引入了居民尊重型安置方案的松弛（或近似）版本，允许至多$t$个上界被突破。