Host Community Respecting Refugee Housing

TL;DR

This work studies a local refugee housing problem on a topology $G$ with fixed inhabitants and a set of refugees to be housed, aiming to find stable allocations under three models: Anonymous Refugees Housing (ARH), Hedonic Refugees Housing (HRH), and Diversity Refugees Housing (DRH). The authors formalize these models and analyze stability and equilibrium existence using both NP-hardness proofs and parameterized complexity, leveraging ETH-based lower bounds and structural graph parameters such as $vc(G)$ and $ ext{tw}(G)$. Key contributions include a polynomial-time algorithm for ARH on graphs of maximum degree $2$, $NP$-hardness for ARH at degree $3$, $W[2]$-hardness for ARH parameterized by $|R|$, and a comprehensive set of HRH and DRH tractability/intractability results across various parameters (e.g., $|I|$, $|R|$, $|V_U|$, $vc(G)$, $ au$). The work connects Schelling-style stability and hedonic games to local refugee housing, offering a rigorous framework with implications for computational social choice and integration policy, and outlining future directions such as swap-stability and restricted preference classes.

Abstract

We propose a novel model for refugee housing respecting the preferences of the accepting community and refugees themselves. In particular, we are given a topology representing the local community, a set of inhabitants occupying some vertices of the topology, and a set of refugees that should be housed on the empty vertices of the graph. Both the inhabitants and the refugees have preferences over the structure of their neighborhood. We are specifically interested in the problem of finding housing such that the preferences of every individual are met; using game-theoretical words, we are looking for housing that is stable with respect to some well-defined notion of stability. We investigate conditions under which the existence of equilibria is guaranteed and study the computational complexity of finding such a stable outcome. As the problem is NP-hard even in very simple settings, we employ the parameterized complexity framework to give a finer-grained view of the problem's complexity with respect to natural parameters and structural restrictions of the given topology.

Figures (3)

• Figure 1: An instance of the HRH problem from \ref{['ex:ARH']} and two possible housings. On the left, we have housing that is not respecting---inhabitant $h_2$ does not accept any refugee in its neighborhood. On the right, the housing is respecting as $h_1$ has one refugee in the neighborhood and $1\in A_{h_1}$. Note that the approval sets of both inhabitants form intervals.
• Figure 5: An instance of the HRH problem from \ref{['ex:HRH']} and two possible housings. On the left, we have housing that is not respecting, as, e.g., refugee $r_1$ does not approve inhabitant $h_2$ in its neighborhood. On the right, the housing is respecting as each agent approves its neighborhood. Underlined elements of the approval sets represent agents' neighborhoods in the respective housing.
• Figure 6: An instance of the DRH problem from \ref{['ex:DRH']} and two possible housings. On the left, we have housing that is not respecting. While inhabitant $h_2$ is satisfied as in its neighborhood, there are only agents of type $T_1$. This is not true for the remaining agents. They have only agent $h_2$ in the neighborhood, and the palette corresponding to $\{h_2\}$ is $(0,1)$, which is in the approval set of neither $h_1$ nor $r$. The housing on the right is, on the other hand, respecting.

Theorems & Definitions (36)

• definition 1: Parameterized reduction CyganFKLMPPS2015
• theorem 1: ChenCFHJKX2005
• theorem 2: ImpagliazzoP2001ImpagliazzoPZ2001
• definition 2: Tree decomposition
• definition 3
• proposition 1
• proposition 2
• proposition 3
• theorem 3
• theorem 4