Towards More Realistic Models for Refugee Integration: Anonymous Refugee Housing with Upper-Bounds

TL;DR

ARH-UB introduces upper-bound based preferences for anonymous refugee housing on a city topology graph $G=(V,E)$ with inhabitants $I$ and refugees $R$, formalizing a housing $\pi\subseteq V_U$ that must satisfy $|N_G(\iota(i))\cap \pi|\le \operatorname{ub}(i)$ for every inhabitant $i$. The paper establishes a broad complexity landscape: the problem is NP-hard even for planar graphs with maximum degree $3$ via a reduction from Independent Set, yet it is polynomial-time solvable on forests and certain dense topologies such as complete bipartite graphs; it also provides fixed-parameter tractability (FPT) results with respect to parameters like the feedback-edge set number, a modulator to complete bipartite graphs, and the number of inhabitants, along with XP algorithms parameterized by tree-width $\operatorname{tw}(G)$ and corresponding W[1]-hardness results. In addition, it studies a relaxation, Relaxed ARH-UB (RARH-UB), which permits a total excess $\operatorname{exc}$ of at most $t$ and shows NP-hardness for every $t\ge 0$ but offers XP-type algorithms for bounded tree-width instances. The work thus delineates the practical tractability frontier for refugee housing under restricted upper-bound preferences and points to rich future directions, including tighter FPT results and more nuanced models that blend equilibrium considerations with real-world constraints.

Abstract

[Knop and Schierreich; AAMAS~'23] recently introduced a novel model for refugee housing, where we are given a city's topology represented as a graph, a set of inhabitants who occupy some vertices of the topology, and a set of refugees that we need to house in unoccupied vertices. Moreover, each inhabitant approves specific numbers of refugees in his neighbourhood, and our goal is to find housing such that every inhabitant approves the number of refugees housed in his neighbourhood. However, such a model suffers from several problems: (i) it is computationally hard to find desirable housing, and (ii) the inhabitants' preferences are not necessarily continuous, which hardly occurs in practice. To avoid these objections, we introduce a restricted variant of the problem that is more closely aligned with practical scenarios of refugee housing and study its complexity.